text:
@testset "1.3.2 Algebraic functions" begin
    (a, b, c, d, e, f, g, m, n, p, q, r, s, t, x, ) = @variables a b c d e f g m n p q r s t x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*of*the*form*u*(a+b*x^3)^p=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c+d*x)^m*(a+b*x^3)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*sqrt(a+b*x^3))*with*b*c^3 - 4*a*d^3=0=#

    @test_int [1/((2^(2/3) + x)*sqrt(1 + x^3)), x, 4, (2*atan((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(1 + x^3)))/(3*sqrt(3)) + (2*2^(1/3)*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [1/((2^(2/3) - x)*sqrt(1 - x^3)), x, 4, -((2*atan((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(1 - x^3)))/(3*sqrt(3))) - (2*2^(1/3)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [1/((2^(2/3) - x)*sqrt(-1 + x^3)), x, 4, -((2*atanh((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(-1 + x^3)))/(3*sqrt(3))) - (2*2^(1/3)*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [1/((2^(2/3) + x)*sqrt(-1 - x^3)), x, 4, (2*atanh((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(-1 - x^3)))/(3*sqrt(3)) + (2*2^(1/3)*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [1/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, (2*atan((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(a + b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(1/3)) + (2*2^(1/3)*sqrt(2 + sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(1/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [1/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, -((2*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(a - b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(1/3))) - (2*2^(1/3)*sqrt(2 + sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(1/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [1/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, -((2*atanh((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(-a + b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(1/3))) - (2*2^(1/3)*sqrt(2 - sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(1/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [1/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, (2*atanh((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(-a - b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(1/3)) + (2*2^(1/3)*sqrt(2 - sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(1/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    @test_int [1/((c + d*x)*sqrt(c^3 + 4*d^3*x^3)), x, 4, (2*atan((sqrt(3)*sqrt(c)*(c + 2*d*x))/sqrt(c^3 + 4*d^3*x^3)))/(3*sqrt(3)*c^(3/2)*d) + (2*2^(1/3)*sqrt(2 + sqrt(3))*(c + 2^(2/3)*d*x)*sqrt((c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c + 2^(2/3)*d*x)/((1 + sqrt(3))*c + 2^(2/3)*d*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*c*d*sqrt((c*(c + 2^(2/3)*d*x))/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*sqrt(c^3 + 4*d^3*x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*sqrt(a+b*x^3))*with*b^2*c^6-20*a*b*c^3*d^3-8*a^2*d^6=0=#

    @test_int [1/((1 + sqrt(3) + x)*sqrt(1 + x^3)), x, 4, atan((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3))/sqrt(3*(3 + 2*sqrt(3))) + (sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [1/((1 + sqrt(3) - x)*sqrt(1 - x^3)), x, 4, -(atan((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(1 - x^3))/sqrt(3*(3 + 2*sqrt(3)))) - (sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [1/((1 + sqrt(3) - x)*sqrt(-1 + x^3)), x, 4, -(atanh((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(-1 + x^3))/sqrt(3*(3 + 2*sqrt(3)))) - (sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(3/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [1/((1 + sqrt(3) + x)*sqrt(-1 - x^3)), x, 4, atanh((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(-1 - x^3))/sqrt(3*(3 + 2*sqrt(3))) + (sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(3/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*sqrt(a+b*x^3))=#

    @test_int [1/((3 + x)*sqrt(1 + x^3)), x, 8, ((1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(13/2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)))/(sqrt(26)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (2*sqrt(26 + 15*sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (4*3^(1/4)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi(97 - 56*sqrt(3), -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt(2 - sqrt(3))*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [1/((3 + x)*sqrt(1 - x^3)), x, 8, -(((1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(2*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(2*sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))) - (2*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(1/4)*(4 + sqrt(3))*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((1/169)*(553 + 304*sqrt(3)), -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(13*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [1/((3 + x)*sqrt(-1 + x^3)), x, 8, -(((1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(2*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(2*sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))) - (2*sqrt(62 - 35*sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(13*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((1/169)*(553 + 304*sqrt(3)), -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(13*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))]
    @test_int [1/((3 + x)*sqrt(-1 - x^3)), x, 8, ((1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(13/2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)))/(sqrt(26)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3)) + (2*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(2 - sqrt(3))*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3)) + (4*3^(1/4)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi(97 - 56*sqrt(3), -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt(2 - sqrt(3))*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*(a+b*x^3)^(1/3))*when*b*c^3+a*d^3=0=#

    @test_int [1/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)), x, 1, (sqrt(3)*atan((1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/sqrt(3)))/(2*2^(1/3)*c*d) + log((c - d*x)*(c + d*x)^2)/(4*2^(1/3)*c*d) - (3*log(d*(c - d*x) + 2^(2/3)*d*(-c^3 + d^3*x^3)^(1/3)))/(4*2^(1/3)*c*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*(a+b*x^3)^(1/3))*when*2*b*c^3-a*d^3=0=#

    @test_int [1/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x, 3, atan((1 + (2*d*x)/(2*c^3 + d^3*x^3)^(1/3))/sqrt(3))/(2*sqrt(3)*c*d) - (sqrt(3)*atan((1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/sqrt(3)))/(2*c*d) - log(c + d*x)/(2*c*d) - log((-d)*x + (2*c^3 + d^3*x^3)^(1/3))/(4*c*d) + (3*log(d*(2*c + d*x) - d*(2*c^3 + d^3*x^3)^(1/3)))/(4*c*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*1 / ((c+d*x)*(a+b*x^3)^(2/3))*when*2*b*c^3-a*d^3=0=#

    @test_int [1/((c + d*x)*(2*c^3 + d^3*x^3)^(2/3)), x, 1, -(atan((1 + (2*d*x)/(2*c^3 + d^3*x^3)^(1/3))/sqrt(3))/(2*sqrt(3)*c^2*d)) + (sqrt(3)*atan((1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/sqrt(3)))/(2*c^2*d) - log(c + d*x)/(2*c^2*d) - log(d*x - (2*c^3 + d^3*x^3)^(1/3))/(4*c^2*d) + (3*log(d*(2*c + d*x) - d*(2*c^3 + d^3*x^3)^(1/3)))/(4*c^2*d)]

    @test_int [1/((1 + 2^(1/3)*x)*(1 + x^3)^(2/3)), x, 1, -(atan((1 + (2*x)/(1 + x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3))) + (sqrt(3)*atan((1 + (2*(2^(2/3) + x))/(1 + x^3)^(1/3))/sqrt(3)))/2^(2/3) - log(1 + 2^(1/3)*x)/2^(2/3) - log(x - (1 + x^3)^(1/3))/(2*2^(2/3)) + (3*log(2 + 2^(1/3)*x - 2^(1/3)*(1 + x^3)^(1/3)))/(2*2^(2/3))]
    @test_int [1/((1 - 2^(1/3)*x)*(1 - x^3)^(2/3)), x, 1, -((sqrt(3)*atan((1 + (2*2^(2/3) - 2*x)/(1 - x^3)^(1/3))/sqrt(3)))/2^(2/3)) + atan((1 - (2*x)/(1 - x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3)) + log(1 - 2^(1/3)*x)/2^(2/3) + log(-x - (1 - x^3)^(1/3))/(2*2^(2/3)) - (3*log(-2 + 2^(1/3)*x + 2^(1/3)*(1 - x^3)^(1/3)))/(2*2^(2/3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c+d*x)^m*(a+b*x^3)^(p/3)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(c + d*x)^4*(a + b*x^3)^(1/3), x, 11, (3*a*c^2*d^2*(a + b*x^3)^(1/3))/(2*b) + (a*d^4*x^2*(a + b*x^3)^(1/3))/(18*b) + (1/30)*(a + b*x^3)^(1/3)*(15*c^4*x + 40*c^3*d*x^2 + 45*c^2*d^2*x^3 + 24*c*d^3*x^4 + 5*d^4*x^5) - (4*a*c^3*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(2/3)) + (a^2*d^4*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*b^(5/3)) + (a*c^4*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*(a + b*x^3)^(2/3)) + (a*c*d^3*x^4*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(2/3, 4/3, 7/3, -((b*x^3)/a)))/(5*(a + b*x^3)^(2/3)) - (2*a*c^3*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(3*b^(2/3)) + (a^2*d^4*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(18*b^(5/3))]
    @test_int [(c + d*x)^3*(a + b*x^3)^(1/3), x, 9, (3*a*c*d^2*(a + b*x^3)^(1/3))/(4*b) + (a*d^3*x*(a + b*x^3)^(1/3))/(10*b) + (1/20)*(a + b*x^3)^(1/3)*(10*c^3*x + 20*c^2*d*x^2 + 15*c*d^2*x^3 + 4*d^3*x^4) - (a*c^2*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(2/3)) + (a*(5*b*c^3 - a*d^3)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(10*b*(a + b*x^3)^(2/3)) - (a*c^2*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(2*b^(2/3))]
    @test_int [(c + d*x)^2*(a + b*x^3)^(1/3), x, 8, (a*d^2*(a + b*x^3)^(1/3))/(4*b) + (1/12)*(a + b*x^3)^(1/3)*(6*c^2*x + 8*c*d*x^2 + 3*d^2*x^3) - (2*a*c*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(2/3)) + (a*c^2*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*(a + b*x^3)^(2/3)) - (a*c*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(3*b^(2/3))]
    @test_int [(c + d*x)^1*(a + b*x^3)^(1/3), x, 6, (1/6)*(3*c*x + 2*d*x^2)*(a + b*x^3)^(1/3) - (a*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(2/3)) + (a*c*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*(a + b*x^3)^(2/3)) - (a*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(6*b^(2/3))]
    @test_int [(a + b*x^3)^(1/3)/(c + d*x)^1, x, 13, (a + b*x^3)^(1/3)/d + (x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(1/3), 1, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c*(1 + (b*x^3)/a)^(1/3)) + (b^(1/3)*c*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2) - ((b*c^3 - a*d^3)^(1/3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*d^2) + ((b*c^3 - a*d^3)^(1/3)*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2) + ((b*c^3 - a*d^3)^(1/3)*log(c^3 + d^3*x^3))/(3*d^2) + (b^(1/3)*c*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(2*d^2) - ((b*c^3 - a*d^3)^(1/3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(2*d^2) - ((b*c^3 - a*d^3)^(1/3)*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(2*d^2)]
    @test_int [(a + b*x^3)^(1/3)/(c + d*x)^2, x, 20, -((c^2*(a + b*x^3)^(1/3))/(d*(c^3 + d^3*x^3))) - (d*x^2*(a + b*x^3)^(1/3))/(c^3 + d^3*x^3) + (x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(1/3), 2, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c^2*(1 + (b*x^3)/a)^(1/3)) - (d^3*x^4*(a + b*x^3)^(1/3)*AppellF1(4/3, -(1/3), 2, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(2*c^5*(1 + (b*x^3)/a)^(1/3)) - (b^(1/3)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2) + (2*a*d*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*(b*c^3 - a*d^3)^(2/3)) + ((3*b*c^3 - 2*a*d^3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*d^2*(b*c^3 - a*d^3)^(2/3)) - (b*c^2*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2*(b*c^3 - a*d^3)^(2/3)) - (b*c^2*log(c^3 + d^3*x^3))/(6*d^2*(b*c^3 - a*d^3)^(2/3)) - (a*d*log(c^3 + d^3*x^3))/(9*c*(b*c^3 - a*d^3)^(2/3)) - ((3*b*c^3 - 2*a*d^3)*log(c^3 + d^3*x^3))/(18*c*d^2*(b*c^3 - a*d^3)^(2/3)) - (b^(1/3)*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(2*d^2) + (a*d*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(3*c*(b*c^3 - a*d^3)^(2/3)) + ((3*b*c^3 - 2*a*d^3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(6*c*d^2*(b*c^3 - a*d^3)^(2/3)) + (b*c^2*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(2*d^2*(b*c^3 - a*d^3)^(2/3))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(c + d*x)^4/(a + b*x^3)^(1/3), x, 10, (3*c^2*d^2*(a + b*x^3)^(2/3))/b + (4*c*d^3*x*(a + b*x^3)^(2/3))/(3*b) + (c^4*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (4*a*c*d^3*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(4/3)) + (2*c^3*d*x^2*(1 + (b*x^3)/a)^(1/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 5/3, -((b*x^3)/a)))/(a + b*x^3)^(1/3) + (d^4*x^5*(1 + (b*x^3)/a)^(1/3)*HypergeometricFunctions._₂F₁(1/3, 5/3, 8/3, -((b*x^3)/a)))/(5*(a + b*x^3)^(1/3)) - (c^4*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(1/3)) + (2*a*c*d^3*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(3*b^(4/3))]
    @test_int [(c + d*x)^3/(a + b*x^3)^(1/3), x, 8, (3*c*d^2*(a + b*x^3)^(2/3))/(2*b) + (d^3*x*(a + b*x^3)^(2/3))/(3*b) + (c^3*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (a*d^3*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(4/3)) + (3*c^2*d*x^2*(1 + (b*x^3)/a)^(1/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 5/3, -((b*x^3)/a)))/(2*(a + b*x^3)^(1/3)) - (c^3*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(1/3)) + (a*d^3*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(6*b^(4/3))]
    @test_int [(c + d*x)^2/(a + b*x^3)^(1/3), x, 7, (d^2*(a + b*x^3)^(2/3))/(2*b) + (c^2*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) + (c*d*x^2*(1 + (b*x^3)/a)^(1/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 5/3, -((b*x^3)/a)))/(a + b*x^3)^(1/3) - (c^2*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(1/3))]
    @test_int [(c + d*x)^1/(a + b*x^3)^(1/3), x, 5, (c*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) + (d*x^2*(1 + (b*x^3)/a)^(1/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 5/3, -((b*x^3)/a)))/(2*(a + b*x^3)^(1/3)) - (c*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(1/3))]
    @test_int [1/((c + d*x)^1*(a + b*x^3)^(1/3)), x, 10, -((d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1(2/3, 1/3, 1, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(2*c^2*(a + b*x^3)^(1/3))) + atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3))/(sqrt(3)*(b*c^3 - a*d^3)^(1/3)) - atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3))/(sqrt(3)*(b*c^3 - a*d^3)^(1/3)) + log(c^3 + d^3*x^3)/(3*(b*c^3 - a*d^3)^(1/3)) - log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)^(1/3)) - log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)^(1/3))]
    @test_int [1/((c + d*x)^2*(a + b*x^3)^(1/3)), x, 17, (c^2*d^2*(a + b*x^3)^(2/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) - (c*d^3*x*(a + b*x^3)^(2/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) - (d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1(2/3, 1/3, 2, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c^3*(a + b*x^3)^(1/3)) + (d^4*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1(5/3, 1/3, 2, 8/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(5*c^6*(a + b*x^3)^(1/3)) + (2*a*d^3*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*(b*c^3 - a*d^3)^(4/3)) + ((3*b*c^3 - 2*a*d^3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*(b*c^3 - a*d^3)^(4/3)) - (b*c^2*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(sqrt(3)*(b*c^3 - a*d^3)^(4/3)) + (b*c^2*log(c^3 + d^3*x^3))/(6*(b*c^3 - a*d^3)^(4/3)) + (a*d^3*log(c^3 + d^3*x^3))/(9*c*(b*c^3 - a*d^3)^(4/3)) + ((3*b*c^3 - 2*a*d^3)*log(c^3 + d^3*x^3))/(18*c*(b*c^3 - a*d^3)^(4/3)) - (a*d^3*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(3*c*(b*c^3 - a*d^3)^(4/3)) - ((3*b*c^3 - 2*a*d^3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(6*c*(b*c^3 - a*d^3)^(4/3)) - (b*c^2*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(2*(b*c^3 - a*d^3)^(4/3))]
    @test_int [1/((c + d*x)^3*(a + b*x^3)^(1/3)), x, 32, (3*c^4*d^2*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) - (3*c^3*d^3*x*(a + b*x^3)^(2/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (4*b*c^4*d^2*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2*(b*c^3 - 3*a*d^3)*(a + b*x^3)^(2/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^3*(3*b*c^3 - 7*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (d^3*(9*b*c^3 - 5*a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (7*d^3*(3*b*c^3 + a*d^3)*x*(a + b*x^3)^(2/3))/(18*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (3*d*x^2*(1 + (b*x^3)/a)^(1/3)*AppellF1(2/3, 1/3, 3, 5/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(2*c^4*(a + b*x^3)^(1/3)) + (6*d^4*x^5*(1 + (b*x^3)/a)^(1/3)*AppellF1(5/3, 1/3, 3, 8/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(5*c^7*(a + b*x^3)^(1/3)) + (2*a^2*d^6*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^2*(b*c^3 - a*d^3)^(7/3)) + (7*a*d^3*(3*b*c^3 - a*d^3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^2*(b*c^3 - a*d^3)^(7/3)) - (4*b^2*c^4*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*(b*c^3 - a*d^3)^(7/3)) + (b*c*(b*c^3 - 3*a*d^3)*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*(b*c^3 - a*d^3)^(7/3)) + (2*b^2*c^4*log(c^3 + d^3*x^3))/(9*(b*c^3 - a*d^3)^(7/3)) + (a^2*d^6*log(c^3 + d^3*x^3))/(27*c^2*(b*c^3 - a*d^3)^(7/3)) - (b*c*(b*c^3 - 3*a*d^3)*log(c^3 + d^3*x^3))/(18*(b*c^3 - a*d^3)^(7/3)) + (7*a*d^3*(3*b*c^3 - a*d^3)*log(c^3 + d^3*x^3))/(54*c^2*(b*c^3 - a*d^3)^(7/3)) + ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*log(c^3 + d^3*x^3))/(54*c^2*(b*c^3 - a*d^3)^(7/3)) - (a^2*d^6*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(9*c^2*(b*c^3 - a*d^3)^(7/3)) - (7*a*d^3*(3*b*c^3 - a*d^3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - ((9*b^2*c^6 - 12*a*b*c^3*d^3 + 5*a^2*d^6)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(18*c^2*(b*c^3 - a*d^3)^(7/3)) - (2*b^2*c^4*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(3*(b*c^3 - a*d^3)^(7/3)) + (b*c*(b*c^3 - 3*a*d^3)*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(6*(b*c^3 - a*d^3)^(7/3))]

    @test_int [(c + d*x)^4/(a + b*x^3)^(2/3), x, 10, (6*c^2*d^2*(a + b*x^3)^(1/3))/b + (d^4*x^2*(a + b*x^3)^(1/3))/(3*b) - (4*c^3*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(2/3)) + (2*a*d^4*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(5/3)) + (c^4*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(a + b*x^3)^(2/3) + (c*d^3*x^4*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(2/3, 4/3, 7/3, -((b*x^3)/a)))/(a + b*x^3)^(2/3) - (2*c^3*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/b^(2/3) + (a*d^4*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(3*b^(5/3))]
    @test_int [(c + d*x)^3/(a + b*x^3)^(2/3), x, 8, (3*c*d^2*(a + b*x^3)^(1/3))/b + (d^3*x*(a + b*x^3)^(1/3))/(2*b) - (sqrt(3)*c^2*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/b^(2/3) + ((2*b*c^3 - a*d^3)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*b*(a + b*x^3)^(2/3)) - (3*c^2*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(2*b^(2/3))]
    @test_int [(c + d*x)^2/(a + b*x^3)^(2/3), x, 7, (d^2*(a + b*x^3)^(1/3))/b - (2*c*d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(2/3)) + (c^2*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(a + b*x^3)^(2/3) - (c*d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/b^(2/3)]
    @test_int [(c + d*x)^1/(a + b*x^3)^(2/3), x, 5, -((d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(2/3))) + (c*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(a + b*x^3)^(2/3) - (d*log(b^(1/3)*x - (a + b*x^3)^(1/3)))/(2*b^(2/3))]
    @test_int [1/((c + d*x)^1*(a + b*x^3)^(2/3)), x, 10, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c*(a + b*x^3)^(2/3)) + (d*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*(b*c^3 - a*d^3)^(2/3)) - (d*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(sqrt(3)*(b*c^3 - a*d^3)^(2/3)) - (d*log(c^3 + d^3*x^3))/(3*(b*c^3 - a*d^3)^(2/3)) + (d*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(2*(b*c^3 - a*d^3)^(2/3)) + (d*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(2*(b*c^3 - a*d^3)^(2/3))]
    @test_int [1/((c + d*x)^2*(a + b*x^3)^(2/3)), x, 18, (c^2*d^2*(a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) + (d^4*x^2*(a + b*x^3)^(1/3))/((b*c^3 - a*d^3)*(c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 2, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c^2*(a + b*x^3)^(2/3)) - (d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1(4/3, 2/3, 2, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(2*c^5*(a + b*x^3)^(2/3)) + (2*a*d^4*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*(b*c^3 - a*d^3)^(5/3)) + (2*d*(3*b*c^3 - a*d^3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c*(b*c^3 - a*d^3)^(5/3)) - (2*b*c^2*d*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(sqrt(3)*(b*c^3 - a*d^3)^(5/3)) - (b*c^2*d*log(c^3 + d^3*x^3))/(3*(b*c^3 - a*d^3)^(5/3)) - (a*d^4*log(c^3 + d^3*x^3))/(9*c*(b*c^3 - a*d^3)^(5/3)) - (d*(3*b*c^3 - a*d^3)*log(c^3 + d^3*x^3))/(9*c*(b*c^3 - a*d^3)^(5/3)) + (a*d^4*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(3*c*(b*c^3 - a*d^3)^(5/3)) + (d*(3*b*c^3 - a*d^3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(3*c*(b*c^3 - a*d^3)^(5/3)) + (b*c^2*d*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(b*c^3 - a*d^3)^(5/3)]
    @test_int [1/((c + d*x)^3*(a + b*x^3)^(2/3)), x, 30, (3*c^4*d^2*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (3*c^2*d^4*x^2*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (5*b*c^4*d^2*(a + b*x^3)^(1/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2*(b*c^3 - 6*a*d^3)*(a + b*x^3)^(1/3))/(6*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^4*(9*b*c^3 - 4*a*d^3)*x^2*(a + b*x^3)^(1/3))/(6*c*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (d^4*(3*b*c^3 + 2*a*d^3)*x^2*(a + b*x^3)^(1/3))/(3*c*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 3, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(c^3*(a + b*x^3)^(2/3)) - (7*d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1(4/3, 2/3, 3, 7/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(4*c^6*(a + b*x^3)^(2/3)) + (d^6*x^7*(1 + (b*x^3)/a)^(2/3)*AppellF1(7/3, 2/3, 3, 10/3, -((b*x^3)/a), -((d^3*x^3)/c^3)))/(7*c^9*(a + b*x^3)^(2/3)) + (2*a*d^4*(6*b*c^3 - a*d^3)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^2*(b*c^3 - a*d^3)^(8/3)) + (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*atan((1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^2*(b*c^3 - a*d^3)^(8/3)) - (10*b^2*c^4*d*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*(b*c^3 - a*d^3)^(8/3)) + (b*c*d*(b*c^3 - 6*a*d^3)*atan((1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*(b*c^3 - a*d^3)^(8/3)) - (5*b^2*c^4*d*log(c^3 + d^3*x^3))/(9*(b*c^3 - a*d^3)^(8/3)) + (b*c*d*(b*c^3 - 6*a*d^3)*log(c^3 + d^3*x^3))/(18*(b*c^3 - a*d^3)^(8/3)) - (a*d^4*(6*b*c^3 - a*d^3)*log(c^3 + d^3*x^3))/(9*c^2*(b*c^3 - a*d^3)^(8/3)) - (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*log(c^3 + d^3*x^3))/(18*c^2*(b*c^3 - a*d^3)^(8/3)) + (a*d^4*(6*b*c^3 - a*d^3)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(3*c^2*(b*c^3 - a*d^3)^(8/3)) + (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*log(((b*c^3 - a*d^3)^(1/3)*x)/c - (a + b*x^3)^(1/3)))/(6*c^2*(b*c^3 - a*d^3)^(8/3)) + (5*b^2*c^4*d*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(3*(b*c^3 - a*d^3)^(8/3)) - (b*c*d*(b*c^3 - 6*a*d^3)*log((b*c^3 - a*d^3)^(1/3) + d*(a + b*x^3)^(1/3)))/(6*(b*c^3 - a*d^3)^(8/3))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c+d*x)^m*(e+f*x)^n*(a+b*x^3)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*sqrt(a+b*x^3))*with*b*c^3 - 4*a*d^3=0=#

    #= ::Subsubsection::Closed:: =#
    #=2*d*e+c*f = 0=#

    @test_int [(2^(2/3) - 2*x)/((2^(2/3) + x)*sqrt(1 + x^3)), x, 2, (2*2^(2/3)*atan((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(1 + x^3)))/sqrt(3)]
    @test_int [(2^(2/3) + 2*x)/((2^(2/3) - x)*sqrt(1 - x^3)), x, 2, -((2*2^(2/3)*atan((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(1 - x^3)))/sqrt(3))]
    @test_int [(2^(2/3) + 2*x)/((2^(2/3) - x)*sqrt(-1 + x^3)), x, 2, -((2*2^(2/3)*atanh((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(-1 + x^3)))/sqrt(3))]
    @test_int [(2^(2/3) - 2*x)/((2^(2/3) + x)*sqrt(-1 - x^3)), x, 2, (2*2^(2/3)*atanh((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(-1 - x^3)))/sqrt(3)]

    @test_int [(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 2, (2*2^(2/3)*atan((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(3)*a^(1/6)*b^(1/3))]
    @test_int [(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 2, -((2*2^(2/3)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(3)*a^(1/6)*b^(1/3)))]
    @test_int [(2^(2/3)*a^(1/3) + 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, -((2*2^(2/3)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(3)*a^(1/6)*b^(1/3)))]
    @test_int [(2^(2/3)*a^(1/3) - 2*b^(1/3)*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, (2*2^(2/3)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(3)*a^(1/6)*b^(1/3))]

    @test_int [(c - 2*d*x)/((c + d*x)*sqrt(c^3 + 4*d^3*x^3)), x, 2, (2*atan((sqrt(3)*sqrt(c)*(c + 2*d*x))/sqrt(c^3 + 4*d^3*x^3)))/(sqrt(3)*sqrt(c)*d)]

    #= ::Subsubsection::Closed:: =#
    #=2*d*e+c*f /= 0=#

    @test_int [(2 + 3*x)/((2^(2/3) + x)*sqrt(1 + x^3)), x, 4, (2*(2 - 2^(2/3)*3)*atan((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(1 + x^3)))/(3*sqrt(3)) + (2*sqrt(2 + sqrt(3))*(2^(1/3)*2 + 3)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(2 + 3*x)/((2^(2/3) - x)*sqrt(1 - x^3)), x, 4, -((2*(2 + 3*2^(2/3))*atan((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(1 - x^3)))/(3*sqrt(3))) + (2*(3 - 2*2^(1/3))*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(2 + 3*x)/((2^(2/3) - x)*sqrt(-1 + x^3)), x, 4, -((2*(2 + 3*2^(2/3))*atanh((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(-1 + x^3)))/(3*sqrt(3))) + (2*(3 - 2*2^(1/3))*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(2 + 3*x)/((2^(2/3) + x)*sqrt(-1 - x^3)), x, 4, (2*(2 - 2^(2/3)*3)*atanh((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(-1 - x^3)))/(3*sqrt(3)) + (2*sqrt(2 - sqrt(3))*(2^(1/3)*2 + 3)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/((2^(2/3) + x)*sqrt(1 + x^3)), x, 4, (2*(e - 2^(2/3)*f)*atan((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(1 + x^3)))/(3*sqrt(3)) + (2*sqrt(2 + sqrt(3))*(2^(1/3)*e + f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(e + f*x)/((2^(2/3) - x)*sqrt(1 - x^3)), x, 4, -((2*(e + 2^(2/3)*f)*atan((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(1 - x^3)))/(3*sqrt(3))) - (2*sqrt(2 + sqrt(3))*(2^(1/3)*e - f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(e + f*x)/((2^(2/3) - x)*sqrt(-1 + x^3)), x, 4, -((2*(e + 2^(2/3)*f)*atanh((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(-1 + x^3)))/(3*sqrt(3))) - (2*sqrt(2 - sqrt(3))*(2^(1/3)*e - f)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(e + f*x)/((2^(2/3) + x)*sqrt(-1 - x^3)), x, 4, (2*(e - 2^(2/3)*f)*atanh((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(-1 - x^3)))/(3*sqrt(3)) + (2*sqrt(2 - sqrt(3))*(2^(1/3)*e + f)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, (2*(b^(1/3)*e - 2^(2/3)*a^(1/3)*f)*atan((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(a + b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(2/3)) + (2*sqrt(2 + sqrt(3))*(2^(1/3)*b^(1/3)*e + a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, -((2*(b^(1/3)*e + 2^(2/3)*a^(1/3)*f)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(a - b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(2/3))) - (2*sqrt(2 + sqrt(3))*(2^(1/3)*b^(1/3)*e - a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [(e + f*x)/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, -((2*(b^(1/3)*e + 2^(2/3)*a^(1/3)*f)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(-a + b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(2/3))) - (2*sqrt(2 - sqrt(3))*(2^(1/3)*b^(1/3)*e - a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [(e + f*x)/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, (2*(b^(1/3)*e - 2^(2/3)*a^(1/3)*f)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(-a - b*x^3)))/(3*sqrt(3)*sqrt(a)*b^(2/3)) + (2*sqrt(2 - sqrt(3))*(2^(1/3)*b^(1/3)*e + a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    @test_int [(e + f*x)/((c + d*x)*sqrt(c^3 + 4*d^3*x^3)), x, 4, (2*(d*e - c*f)*atan((sqrt(3)*sqrt(c)*(c + 2*d*x))/sqrt(c^3 + 4*d^3*x^3)))/(3*sqrt(3)*c^(3/2)*d^2) + (2^(1/3)*sqrt(2 + sqrt(3))*(2*d*e + c*f)*(c + 2^(2/3)*d*x)*sqrt((c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c + 2^(2/3)*d*x)/((1 + sqrt(3))*c + 2^(2/3)*d*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*c*d^2*sqrt((c*(c + 2^(2/3)*d*x))/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*sqrt(c^3 + 4*d^3*x^3))]

    #= ::Subsubsection::Closed:: =#
    #=e = 0=#

    @test_int [x/((2^(2/3) + x)*sqrt(1 + x^3)), x, 4, -((2*2^(2/3)*atan((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(1 + x^3)))/(3*sqrt(3))) + (2*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [x/((2^(2/3) - x)*sqrt(1 - x^3)), x, 4, -((2*2^(2/3)*atan((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(1 - x^3)))/(3*sqrt(3))) + (2*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [x/((2^(2/3) - x)*sqrt(-1 + x^3)), x, 4, -((2*2^(2/3)*atanh((sqrt(3)*(1 - 2^(1/3)*x))/sqrt(-1 + x^3)))/(3*sqrt(3))) + (2*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [x/((2^(2/3) + x)*sqrt(-1 - x^3)), x, 4, -((2*2^(2/3)*atanh((sqrt(3)*(1 + 2^(1/3)*x))/sqrt(-1 - x^3)))/(3*sqrt(3))) + (2*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [x/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, -((2*2^(2/3)*atan((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(a + b*x^3)))/(3*sqrt(3)*a^(1/6)*b^(2/3))) + (2*sqrt(2 + sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [x/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, -((2*2^(2/3)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(a - b*x^3)))/(3*sqrt(3)*a^(1/6)*b^(2/3))) + (2*sqrt(2 + sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [x/((2^(2/3)*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, -((2*2^(2/3)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*b^(1/3)*x))/sqrt(-a + b*x^3)))/(3*sqrt(3)*a^(1/6)*b^(2/3))) + (2*sqrt(2 - sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [x/((2^(2/3)*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, -((2*2^(2/3)*atanh((sqrt(3)*a^(1/6)*(a^(1/3) + 2^(1/3)*b^(1/3)*x))/sqrt(-a - b*x^3)))/(3*sqrt(3)*a^(1/6)*b^(2/3))) + (2*sqrt(2 - sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    @test_int [x/((c + d*x)*sqrt(c^3 + 4*d^3*x^3)), x, 4, -((2*atan((sqrt(3)*sqrt(c)*(c + 2*d*x))/sqrt(c^3 + 4*d^3*x^3)))/(3*sqrt(3)*sqrt(c)*d^2)) + (2^(1/3)*sqrt(2 + sqrt(3))*(c + 2^(2/3)*d*x)*sqrt((c^2 - 2^(2/3)*c*d*x + 2*2^(1/3)*d^2*x^2)/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c + 2^(2/3)*d*x)/((1 + sqrt(3))*c + 2^(2/3)*d*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*d^2*sqrt((c*(c + 2^(2/3)*d*x))/((1 + sqrt(3))*c + 2^(2/3)*d*x)^2)*sqrt(c^3 + 4*d^3*x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*sqrt(a+b*x^3))*with*b*c^3 + 8*a*d^3=0=#

    #= ::Subsubsection::Closed:: =#
    #=2*d*e+c*f = 0=#

    @test_int [(1 + x)/((2 - x)*sqrt(1 + x^3)), x, 2, (2/3)*atanh((1 + x)^2/(3*sqrt(1 + x^3)))]
    @test_int [(1 - x)/((2 + x)*sqrt(1 - x^3)), x, 2, (-(2/3))*atanh((1 - x)^2/(3*sqrt(1 - x^3)))]
    @test_int [(1 - x)/((2 + x)*sqrt(-1 + x^3)), x, 2, (-(2/3))*atan((1 - x)^2/(3*sqrt(-1 + x^3)))]
    @test_int [(1 + x)/((2 - x)*sqrt(-1 - x^3)), x, 2, (2/3)*atan((1 + x)^2/(3*sqrt(-1 - x^3)))]

    @test_int [(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*sqrt(a + b*x^3)), x, 2, (2*atanh((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a + b*x^3))))/(3*a^(1/6)*b^(1/3))]
    @test_int [(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*sqrt(a - b*x^3)), x, 2, -((2*atanh((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a - b*x^3))))/(3*a^(1/6)*b^(1/3)))]
    @test_int [(a^(1/3) - b^(1/3)*x)/((2*a^(1/3) + b^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, -((2*atan((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a + b*x^3))))/(3*a^(1/6)*b^(1/3)))]
    @test_int [(a^(1/3) + b^(1/3)*x)/((2*a^(1/3) - b^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, (2*atan((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a - b*x^3))))/(3*a^(1/6)*b^(1/3))]

    @test_int [(c - 2*d*x)/((c + d*x)*sqrt(c^3 - 8*d^3*x^3)), x, 2, -((2*atanh((c - 2*d*x)^2/(3*sqrt(c)*sqrt(c^3 - 8*d^3*x^3))))/(3*sqrt(c)*d))]

    #= ::Subsubsection::Closed:: =#
    #=2*d*e+c*f /= 0=#

    @test_int [(e + f*x)/((2 - x)*sqrt(1 + x^3)), x, 4, (2/9)*(e + 2*f)*atanh((1 + x)^2/(3*sqrt(1 + x^3))) + (2*sqrt(2 + sqrt(3))*(e - f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(e + f*x)/((2 + x)*sqrt(1 - x^3)), x, 4, (-(2/9))*(e - 2*f)*atanh((1 - x)^2/(3*sqrt(1 - x^3))) - (2*sqrt(2 + sqrt(3))*(e + f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(e + f*x)/((2 + x)*sqrt(-1 + x^3)), x, 4, (-(2/9))*(e - 2*f)*atan((1 - x)^2/(3*sqrt(-1 + x^3))) - (2*sqrt(2 - sqrt(3))*(e + f)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(e + f*x)/((2 - x)*sqrt(-1 - x^3)), x, 4, (2/9)*(e + 2*f)*atan((1 + x)^2/(3*sqrt(-1 - x^3))) + (2*sqrt(2 - sqrt(3))*(e - f)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/((2*a^(1/3) - b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, (2*(b^(1/3)*e + 2*a^(1/3)*f)*atanh((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a + b*x^3))))/(9*sqrt(a)*b^(2/3)) + (2*sqrt(2 + sqrt(3))*(b^(1/3)*e - a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, -((2*(b^(1/3)*e - 2*a^(1/3)*f)*atanh((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a - b*x^3))))/(9*sqrt(a)*b^(2/3))) - (2*sqrt(2 + sqrt(3))*(b^(1/3)*e + a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [(e + f*x)/((2*a^(1/3) + b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, -((2*(b^(1/3)*e - 2*a^(1/3)*f)*atan((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a + b*x^3))))/(9*sqrt(a)*b^(2/3))) - (2*sqrt(2 - sqrt(3))*(b^(1/3)*e + a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [(e + f*x)/((2*a^(1/3) - b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, (2*(b^(1/3)*e + 2*a^(1/3)*f)*atan((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a - b*x^3))))/(9*sqrt(a)*b^(2/3)) + (2*sqrt(2 - sqrt(3))*(b^(1/3)*e - a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    @test_int [(e + f*x)/((c + d*x)*sqrt(c^3 - 8*d^3*x^3)), x, 4, -((2*(d*e - c*f)*atanh((c - 2*d*x)^2/(3*sqrt(c)*sqrt(c^3 - 8*d^3*x^3))))/(9*c^(3/2)*d^2)) - (sqrt(2 + sqrt(3))*(2*d*e + c*f)*(c - 2*d*x)*sqrt((c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + sqrt(3))*c - 2*d*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c - 2*d*x)/((1 + sqrt(3))*c - 2*d*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*c*d^2*sqrt((c*(c - 2*d*x))/((1 + sqrt(3))*c - 2*d*x)^2)*sqrt(c^3 - 8*d^3*x^3))]

    #= ::Subsubsection::Closed:: =#
    #=e = 0=#

    @test_int [x/((2 - x)*sqrt(1 + x^3)), x, 4, (4/9)*atanh((1 + x)^2/(3*sqrt(1 + x^3))) - (2*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [x/((2 + x)*sqrt(1 - x^3)), x, 4, (4/9)*atanh((1 - x)^2/(3*sqrt(1 - x^3))) - (2*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [x/((2 + x)*sqrt(-1 + x^3)), x, 4, (4/9)*atan((1 - x)^2/(3*sqrt(-1 + x^3))) - (2*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [x/((2 - x)*sqrt(-1 - x^3)), x, 4, (4/9)*atan((1 + x)^2/(3*sqrt(-1 - x^3))) - (2*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [x/((2*a^(1/3) - b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, (4*atanh((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a + b*x^3))))/(9*a^(1/6)*b^(2/3)) - (2*sqrt(2 + sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [x/((2*a^(1/3) + b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, (4*atanh((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(a - b*x^3))))/(9*a^(1/6)*b^(2/3)) - (2*sqrt(2 + sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [x/((2*a^(1/3) + b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, (4*atan((a^(1/3) - b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a + b*x^3))))/(9*a^(1/6)*b^(2/3)) - (2*sqrt(2 - sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [x/((2*a^(1/3) - b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, (4*atan((a^(1/3) + b^(1/3)*x)^2/(3*a^(1/6)*sqrt(-a - b*x^3))))/(9*a^(1/6)*b^(2/3)) - (2*sqrt(2 - sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3*3^(1/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    @test_int [x/((c + d*x)*sqrt(c^3 - 8*d^3*x^3)), x, 4, (2*atanh((c - 2*d*x)^2/(3*sqrt(c)*sqrt(c^3 - 8*d^3*x^3))))/(9*sqrt(c)*d^2) - (sqrt(2 + sqrt(3))*(c - 2*d*x)*sqrt((c^2 + 2*c*d*x + 4*d^2*x^2)/((1 + sqrt(3))*c - 2*d*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c - 2*d*x)/((1 + sqrt(3))*c - 2*d*x)), -7 - 4*sqrt(3)))/(3*3^(1/4)*d^2*sqrt((c*(c - 2*d*x))/((1 + sqrt(3))*c - 2*d*x)^2)*sqrt(c^3 - 8*d^3*x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*sqrt(a+b*x^3))*with*b^2*c^6-20*a*b*c^3*d^3-8*a^2*d^6=0=#

    #= ::Subsubsection::Closed:: =#
    #=6*a*d^4*e-c*f*(b*c^3 - 22*a*d^3) = 0=#

    @test_int [(1 + sqrt(3) + x)/((1 - sqrt(3) + x)*sqrt(1 + x^3)), x, 2, -((2*atanh((sqrt(-3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3)))/sqrt(-3 + 2*sqrt(3)))]
    @test_int [(1 + sqrt(3) - x)/((1 - sqrt(3) - x)*sqrt(1 - x^3)), x, 2, (2*atanh((sqrt(-3 + 2*sqrt(3))*(1 - x))/sqrt(1 - x^3)))/sqrt(-3 + 2*sqrt(3))]
    @test_int [(1 + sqrt(3) - x)/((1 - sqrt(3) - x)*sqrt(-1 + x^3)), x, 2, (2*atan((sqrt(-3 + 2*sqrt(3))*(1 - x))/sqrt(-1 + x^3)))/sqrt(-3 + 2*sqrt(3))]
    @test_int [(1 + sqrt(3) + x)/((1 - sqrt(3) + x)*sqrt(-1 - x^3)), x, 2, -((2*atan((sqrt(-3 + 2*sqrt(3))*(1 + x))/sqrt(-1 - x^3)))/sqrt(-3 + 2*sqrt(3)))]

    @test_int [((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 2, -((2*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(-3 + 2*sqrt(3))*a^(1/6)*b^(1/3)))]
    @test_int [((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 2, (2*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(-3 + 2*sqrt(3))*a^(1/6)*b^(1/3))]
    @test_int [((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, (2*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(-3 + 2*sqrt(3))*a^(1/6)*b^(1/3))]
    @test_int [((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, -((2*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(-3 + 2*sqrt(3))*a^(1/6)*b^(1/3)))]

    @test_int [(1 + sqrt(3) + (b/a)^(1/3)*x)/((1 - sqrt(3) + (b/a)^(1/3)*x)*sqrt(a + b*x^3)), x, 2, -((2*atanh((sqrt(-3 + 2*sqrt(3))*sqrt(a)*(1 + (b/a)^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(-3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3)))]
    @test_int [(1 + sqrt(3) - (b/a)^(1/3)*x)/((1 - sqrt(3) - (b/a)^(1/3)*x)*sqrt(a - b*x^3)), x, 2, (2*atanh((sqrt(-3 + 2*sqrt(3))*sqrt(a)*(1 - (b/a)^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(-3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3))]
    @test_int [(1 + sqrt(3) - (b/a)^(1/3)*x)/((1 - sqrt(3) - (b/a)^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, (2*atan((sqrt(-3 + 2*sqrt(3))*sqrt(a)*(1 - (b/a)^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(-3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3))]
    @test_int [(1 + sqrt(3) + (b/a)^(1/3)*x)/((1 - sqrt(3) + (b/a)^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, -((2*atan((sqrt(-3 + 2*sqrt(3))*sqrt(a)*(1 + (b/a)^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(-3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3)))]

    @test_int [(1 - sqrt(3) + x)/((1 + sqrt(3) + x)*sqrt(1 + x^3)), x, 2, -((2*atan((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3)))/sqrt(3 + 2*sqrt(3)))]
    @test_int [(1 - sqrt(3) - x)/((1 + sqrt(3) - x)*sqrt(1 - x^3)), x, 2, (2*atan((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(1 - x^3)))/sqrt(3 + 2*sqrt(3))]
    @test_int [(1 - sqrt(3) - x)/((1 + sqrt(3) - x)*sqrt(-1 + x^3)), x, 2, (2*atanh((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(-1 + x^3)))/sqrt(3 + 2*sqrt(3))]
    @test_int [(1 - sqrt(3) + x)/((1 + sqrt(3) + x)*sqrt(-1 - x^3)), x, 2, -((2*atanh((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(-1 - x^3)))/sqrt(3 + 2*sqrt(3)))]

    @test_int [((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 2, -((2*atan((sqrt(3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(3 + 2*sqrt(3))*a^(1/6)*b^(1/3)))]
    @test_int [((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 2, (2*atan((sqrt(3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(3 + 2*sqrt(3))*a^(1/6)*b^(1/3))]
    @test_int [((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, (2*atanh((sqrt(3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(3 + 2*sqrt(3))*a^(1/6)*b^(1/3))]
    @test_int [((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, -((2*atanh((sqrt(3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(3 + 2*sqrt(3))*a^(1/6)*b^(1/3)))]

    @test_int [(1 - sqrt(3) + (b/a)^(1/3)*x)/((1 + sqrt(3) + (b/a)^(1/3)*x)*sqrt(a + b*x^3)), x, 2, -((2*atan((sqrt(3 + 2*sqrt(3))*sqrt(a)*(1 + (b/a)^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3)))]
    @test_int [(1 - sqrt(3) - (b/a)^(1/3)*x)/((1 + sqrt(3) - (b/a)^(1/3)*x)*sqrt(a - b*x^3)), x, 2, (2*atan((sqrt(3 + 2*sqrt(3))*sqrt(a)*(1 - (b/a)^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3))]
    @test_int [(1 - sqrt(3) - (b/a)^(1/3)*x)/((1 + sqrt(3) - (b/a)^(1/3)*x)*sqrt(-a + b*x^3)), x, 2, (2*atanh((sqrt(3 + 2*sqrt(3))*sqrt(a)*(1 - (b/a)^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3))]
    @test_int [(1 - sqrt(3) + (b/a)^(1/3)*x)/((1 + sqrt(3) + (b/a)^(1/3)*x)*sqrt(-a - b*x^3)), x, 2, -((2*atanh((sqrt(3 + 2*sqrt(3))*sqrt(a)*(1 + (b/a)^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(3 + 2*sqrt(3))*sqrt(a)*(b/a)^(1/3)))]

    #= ::Subsubsection::Closed:: =#
    #=6*a*d^4*e-c*f*(b*c^3 - 22*a*d^3) /= 0=#

    @test_int [(1 + x)/((1 + sqrt(3) + x)*sqrt(1 + x^3)), x, 4, -(atan((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3))/sqrt(3 + 2*sqrt(3))) + (sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(1 + x)/((1 - sqrt(3) + x)*sqrt(1 + x^3)), x, 4, -(atanh((sqrt(-3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3))/sqrt(-3 + 2*sqrt(3))) + (sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]

    @test_int [(e + f*x)/((1 + sqrt(3) + x)*sqrt(1 + x^3)), x, 4, ((e - f - sqrt(3)*f)*atan((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3)))/sqrt(3*(3 + 2*sqrt(3))) + (sqrt(2 + sqrt(3))*(e - (1 - sqrt(3))*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(e + f*x)/((1 + sqrt(3) - x)*sqrt(1 - x^3)), x, 4, -(((e + f + sqrt(3)*f)*atan((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(1 - x^3)))/sqrt(3*(3 + 2*sqrt(3)))) - (sqrt(2 + sqrt(3))*(e + (1 - sqrt(3))*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(e + f*x)/((1 + sqrt(3) - x)*sqrt(-1 + x^3)), x, 4, -(((e + f + sqrt(3)*f)*atanh((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(-1 + x^3)))/sqrt(3*(3 + 2*sqrt(3)))) - (sqrt(2 - sqrt(3))*(e + (1 - sqrt(3))*f)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(3/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(e + f*x)/((1 + sqrt(3) + x)*sqrt(-1 - x^3)), x, 4, ((e - (1 + sqrt(3))*f)*atanh((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(-1 - x^3)))/sqrt(3*(3 + 2*sqrt(3))) + (sqrt(2 - sqrt(3))*(e - (1 - sqrt(3))*f)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(3/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, -(((b^(1/3)*e - (1 - sqrt(3))*a^(1/3)*f)*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(a + b*x^3)))/(sqrt(3*(-3 + 2*sqrt(3)))*sqrt(a)*b^(2/3))) - (sqrt(2 + sqrt(3))*(b^(1/3)*e - (1 + sqrt(3))*a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [(e + f*x)/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, ((b^(1/3)*e + (1 - sqrt(3))*a^(1/3)*f)*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(a - b*x^3)))/(sqrt(3*(-3 + 2*sqrt(3)))*sqrt(a)*b^(2/3)) + (sqrt(2 + sqrt(3))*(b^(1/3)*e + (1 + sqrt(3))*a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [(e + f*x)/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, ((b^(1/3)*e + (1 - sqrt(3))*a^(1/3)*f)*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(-a + b*x^3)))/(sqrt(3*(-3 + 2*sqrt(3)))*sqrt(a)*b^(2/3)) + (sqrt(2 - sqrt(3))*(b^(1/3)*e + (1 + sqrt(3))*a^(1/3)*f)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [(e + f*x)/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, -(((b^(1/3)*e - (1 - sqrt(3))*a^(1/3)*f)*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(-a - b*x^3)))/(sqrt(3*(-3 + 2*sqrt(3)))*sqrt(a)*b^(2/3))) - (sqrt(2 - sqrt(3))*(b^(1/3)*e - (1 + sqrt(3))*a^(1/3)*f)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3^(3/4)*a^(1/3)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    #= ::Subsubsection::Closed:: =#
    #=e = 0=#

    @test_int [x/((1 + sqrt(3) + x)*sqrt(1 + x^3)), x, 4, -((sqrt(2)*atan((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3)))/3^(3/4)) + (sqrt(2)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [x/((1 + sqrt(3) - x)*sqrt(1 - x^3)), x, 4, -((sqrt(2)*atan((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(1 - x^3)))/3^(3/4)) + (sqrt(2)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(3/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [x/((1 + sqrt(3) - x)*sqrt(-1 + x^3)), x, 4, -((sqrt(2)*atanh((sqrt(3 + 2*sqrt(3))*(1 - x))/sqrt(-1 + x^3)))/3^(3/4)) + (2*sqrt(7/6 - 2/sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [x/((1 + sqrt(3) + x)*sqrt(-1 - x^3)), x, 4, -((sqrt(2)*atanh((sqrt(3 + 2*sqrt(3))*(1 + x))/sqrt(-1 - x^3)))/3^(3/4)) + (2*sqrt(7/6 - 2/sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [x/((1 - sqrt(3) + x)*sqrt(1 + x^3)), x, 4, -((sqrt(2)*atanh((sqrt(-3 + 2*sqrt(3))*(1 + x))/sqrt(1 + x^3)))/3^(3/4)) + (2*sqrt(7/6 + 2/sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]

    @test_int [x/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(a + b*x^3)), x, 4, -((sqrt(2)*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(a + b*x^3)))/(3^(3/4)*a^(1/6)*b^(2/3))) + (2*sqrt(7/6 + 2/sqrt(3))*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*sqrt(a + b*x^3))]
    @test_int [x/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(a - b*x^3)), x, 4, -((sqrt(2)*atanh((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(a - b*x^3)))/(3^(3/4)*a^(1/6)*b^(2/3))) + (2*sqrt(7/6 + 2/sqrt(3))*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*b^(2/3)*sqrt((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*sqrt(a - b*x^3))]
    @test_int [x/(((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)*sqrt(-a + b*x^3)), x, 4, -((sqrt(2)*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) - b^(1/3)*x))/sqrt(-a + b*x^3)))/(3^(3/4)*a^(1/6)*b^(2/3))) + (sqrt(2)*(a^(1/3) - b^(1/3)*x)*sqrt((a^(2/3) + a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)), -7 + 4*sqrt(3)))/(3^(3/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) - b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) - b^(1/3)*x)^2))*sqrt(-a + b*x^3))]
    @test_int [x/(((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)*sqrt(-a - b*x^3)), x, 4, -((sqrt(2)*atan((sqrt(-3 + 2*sqrt(3))*a^(1/6)*(a^(1/3) + b^(1/3)*x))/sqrt(-a - b*x^3)))/(3^(3/4)*a^(1/6)*b^(2/3))) + (sqrt(2)*(a^(1/3) + b^(1/3)*x)*sqrt((a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) + b^(1/3)*x)/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)), -7 + 4*sqrt(3)))/(3^(3/4)*b^(2/3)*sqrt(-((a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 - sqrt(3))*a^(1/3) + b^(1/3)*x)^2))*sqrt(-a - b*x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*sqrt(a+b*x^3))*with*b^2*e^6-20*a*b*e^3*f^3-8*a^2*f^6=0=#

    @test_int [(1 + sqrt(3) + x)/((c + d*x)*sqrt(1 + x^3)), x, 6, -(((c - (1 + sqrt(3))*d)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/(sqrt(c - d)*sqrt(d)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))) - (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi((c - (1 + sqrt(3))*d)^2/(c - (1 - sqrt(3))*d)^2, -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/((c - (1 - sqrt(3))*d)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(1 + sqrt(3) - x)/((c + d*x)*sqrt(1 - x^3)), x, 6, -(((c + d + sqrt(3)*d)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((c + d + sqrt(3)*d)^2/(c + d - sqrt(3)*d)^2, -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/((c + d - sqrt(3)*d)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(1 + sqrt(3) - x)/((c + d*x)*sqrt(-1 + x^3)), x, 6, -(((c + d + sqrt(3)*d)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((c + d + sqrt(3)*d)^2/(c + d - sqrt(3)*d)^2, -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/((c + d - sqrt(3)*d)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))]
    @test_int [(1 + sqrt(3) + x)/((c + d*x)*sqrt(-1 - x^3)), x, 6, -(((c - (1 + sqrt(3))*d)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/(sqrt(c - d)*sqrt(d)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))) - (4*3^(1/4)*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi((c - (1 + sqrt(3))*d)^2/(c - (1 - sqrt(3))*d)^2, -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/((c - (1 - sqrt(3))*d)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))]

    @test_int [(1 - sqrt(3) + x)/((c + d*x)*sqrt(1 + x^3)), x, 6, -(((c - (1 - sqrt(3))*d)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*atanh((2*sqrt(2 + sqrt(3))*sqrt(c^2 + c*d + d^2)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2)))/(sqrt(c - d)*sqrt(d)*sqrt(7 + 4*sqrt(3) + (1 + sqrt(3) + x)^2/(1 - sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(1 + x^3))) + (4*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.Pi((c - (1 - sqrt(3))*d)^2/(c - (1 + sqrt(3))*d)^2, -asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/((c - d - sqrt(3)*d)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(1 + x^3))]
    @test_int [(1 - sqrt(3) - x)/((c + d*x)*sqrt(1 - x^3)), x, 6, -(((c + d - sqrt(3)*d)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*atan((sqrt(c^2 - c*d + d^2)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2)))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(1 - x^3))) - (4*3^(1/4)*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.Pi((c + d - sqrt(3)*d)^2/(c + d + sqrt(3)*d)^2, -asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/((c + d + sqrt(3)*d)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(1 - x^3))]
    @test_int [(1 - sqrt(3) - x)/((c + d*x)*sqrt(-1 + x^3)), x, 6, -(((c + d - sqrt(3)*d)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*atan((sqrt(c^2 - c*d + d^2)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2)))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))) - (4*3^(1/4)*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.Pi((c + d - sqrt(3)*d)^2/(c + d + sqrt(3)*d)^2, -asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/((c + d + sqrt(3)*d)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(1 - sqrt(3) + x)/((c + d*x)*sqrt(-1 - x^3)), x, 6, -(((c - (1 - sqrt(3))*d)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*atanh((2*sqrt(2 + sqrt(3))*sqrt(c^2 + c*d + d^2)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2)))/(sqrt(c - d)*sqrt(d)*sqrt(7 + 4*sqrt(3) + (1 + sqrt(3) + x)^2/(1 - sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))) + (4*3^(1/4)*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.Pi((c - (1 - sqrt(3))*d)^2/(c - (1 + sqrt(3))*d)^2, -asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/((c - d - sqrt(3)*d)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(1 + sqrt(3) + x)/(x*sqrt(1 + x^3)), x, 5, (-(2/3))*(1 + sqrt(3))*atanh(sqrt(1 + x^3)) + (2*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(1 + sqrt(3) - x)/(x*sqrt(1 - x^3)), x, 5, (-(2/3))*(1 + sqrt(3))*atanh(sqrt(1 - x^3)) + (2*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(1 + sqrt(3) - x)/(x*sqrt(-1 + x^3)), x, 5, (2/3)*(1 + sqrt(3))*atan(sqrt(-1 + x^3)) + (2*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(1 + sqrt(3) + x)/(x*sqrt(-1 - x^3)), x, 5, (2/3)*(1 + sqrt(3))*atan(sqrt(-1 - x^3)) + (2*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    @test_int [(1 - sqrt(3) + x)/(x*sqrt(1 + x^3)), x, 5, (-(2/3))*(1 - sqrt(3))*atanh(sqrt(1 + x^3)) + (2*sqrt(2 + sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(1 - sqrt(3) - x)/(x*sqrt(1 - x^3)), x, 5, (-(2/3))*(1 - sqrt(3))*atanh(sqrt(1 - x^3)) + (2*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(1 - sqrt(3) - x)/(x*sqrt(-1 + x^3)), x, 5, (2/3)*(1 - sqrt(3))*atan(sqrt(-1 + x^3)) + (2*sqrt(2 - sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(1 - sqrt(3) + x)/(x*sqrt(-1 - x^3)), x, 5, (2/3)*(1 - sqrt(3))*atan(sqrt(-1 - x^3)) + (2*sqrt(2 - sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*sqrt(a+b*x^3))=#

    @test_int [x/((3 + x)*sqrt(1 + x^3)), x, 8, -((3*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(13/2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)))/(sqrt(26)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))) - (2*sqrt(2*(97 + 56*sqrt(3)))*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) - (12*3^(1/4)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi(97 - 56*sqrt(3), -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt(2 - sqrt(3))*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [x/((3 + x)*sqrt(1 - x^3)), x, 8, (3*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(2*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(2*sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3)) - (2*sqrt(2*(37 + 20*sqrt(3)))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(13*3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3)) - (12*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((1/169)*(553 + 304*sqrt(3)), -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(13*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [x/((3 + x)*sqrt(-1 + x^3)), x, 8, (3*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(2*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(2*sqrt(7)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3)) - (2*sqrt(2)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*(4 + sqrt(3))*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3)) - (12*3^(1/4)*sqrt(2 + sqrt(3))*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((1/169)*(553 + 304*sqrt(3)), -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(13*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))]
    @test_int [x/((3 + x)*sqrt(-1 - x^3)), x, 8, -((3*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(13/2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)))/(sqrt(26)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))) - (2*sqrt(14 + 8*sqrt(3))*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3)) - (12*3^(1/4)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi(97 - 56*sqrt(3), -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(sqrt(2 - sqrt(3))*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/((c + d*x)*sqrt(1 + x^3)), x, 8, ((d*e - c*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/(sqrt(c - d)*sqrt(d)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (2*sqrt(2 + sqrt(3))*(e - f - sqrt(3)*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*(c - d - sqrt(3)*d)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(d*e - c*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi((c - (1 + sqrt(3))*d)^2/(c - (1 - sqrt(3))*d)^2, -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/((c^2 - 2*c*d - 2*d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(e + f*x)/((c + d*x)*sqrt(1 - x^3)), x, 8, -(((d*e - c*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))) - (2*sqrt(2 + sqrt(3))*(e + f + sqrt(3)*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(1/4)*(c + d + sqrt(3)*d)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(d*e - c*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((c + d + sqrt(3)*d)^2/(c + d - sqrt(3)*d)^2, -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/((c^2 + 2*c*d - 2*d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(e + f*x)/((c + d*x)*sqrt(-1 + x^3)), x, 8, -(((d*e - c*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*atanh((sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2))/(sqrt(d)*sqrt(c + d)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2))))/(sqrt(d)*sqrt(c + d)*sqrt(c^2 - c*d + d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))) - (2*sqrt(2 - sqrt(3))*(e + f + sqrt(3)*f)*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*(c + d + sqrt(3)*d)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(d*e - c*f)*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.Pi((c + d + sqrt(3)*d)^2/(c + d - sqrt(3)*d)^2, -asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/((c^2 + 2*c*d - 2*d^2)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(-1 + x^3))]
    @test_int [(e + f*x)/((c + d*x)*sqrt(-1 - x^3)), x, 8, ((d*e - c*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*atan((sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2))/(sqrt(c - d)*sqrt(d)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2))))/(sqrt(c - d)*sqrt(d)*sqrt(c^2 + c*d + d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3)) + (2*sqrt(2 - sqrt(3))*(e - f - sqrt(3)*f)*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*(c - d - sqrt(3)*d)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3)) + (4*3^(1/4)*sqrt(2 + sqrt(3))*(d*e - c*f)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.Pi((c - (1 + sqrt(3))*d)^2/(c - (1 - sqrt(3))*d)^2, -asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/((c^2 - 2*c*d - 2*d^2)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(-1 - x^3))]

    @test_int [(e + f*x)/(x*sqrt(1 + x^3)), x, 6, (-(2/3))*e*atanh(sqrt(1 + x^3)) + (2*sqrt(2 + sqrt(3))*f*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [(e + f*x)/(x*sqrt(1 - x^3)), x, 6, (-(2/3))*e*atanh(sqrt(1 - x^3)) - (2*sqrt(2 + sqrt(3))*f*(1 - x)*sqrt((1 + x + x^2)/(1 + sqrt(3) - x)^2)*Elliptic.F(asin((1 - sqrt(3) - x)/(1 + sqrt(3) - x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 - x)/(1 + sqrt(3) - x)^2)*sqrt(1 - x^3))]
    @test_int [(e + f*x)/(x*sqrt(-1 + x^3)), x, 6, (2/3)*e*atan(sqrt(-1 + x^3)) - (2*sqrt(2 - sqrt(3))*f*(1 - x)*sqrt((1 + x + x^2)/(1 - sqrt(3) - x)^2)*Elliptic.F(asin((1 + sqrt(3) - x)/(1 - sqrt(3) - x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 - x)/(1 - sqrt(3) - x)^2))*sqrt(-1 + x^3))]
    @test_int [(e + f*x)/(x*sqrt(-1 - x^3)), x, 6, (2/3)*e*atan(sqrt(-1 - x^3)) + (2*sqrt(2 - sqrt(3))*f*(1 + x)*sqrt((1 - x + x^2)/(1 - sqrt(3) + x)^2)*Elliptic.F(asin((1 + sqrt(3) + x)/(1 - sqrt(3) + x)), -7 + 4*sqrt(3)))/(3^(1/4)*sqrt(-((1 + x)/(1 - sqrt(3) + x)^2))*sqrt(-1 - x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*(a+b*x^3)^(1/3))*when*2*b*c^3-a*d^3=0=#

    @test_int [(c - d*x)/((c + d*x)*(2*c^3 + d^3*x^3)^(1/3)), x, 1, -((sqrt(3)*atan((1 + (2*(2*c + d*x))/(2*c^3 + d^3*x^3)^(1/3))/sqrt(3)))/d) - log(c + d*x)/d + (3*log(d*(2*c + d*x) - d*(2*c^3 + d^3*x^3)^(1/3)))/(2*d)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x) / ((c+d*x)*(a+b*x^3)^(1/3))*when*b*c^3+a*d^3=0=#

    @test_int [(e + f*x)/((c + d*x)*(-c^3 + d^3*x^3)^(1/3)), x, 3, (f*atan((1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2) + (sqrt(3)*(d*e - c*f)*atan((1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/sqrt(3)))/(2*2^(1/3)*c*d^2) + ((d*e - c*f)*log((c - d*x)*(c + d*x)^2))/(4*2^(1/3)*c*d^2) - (f*log((-d)*x + (-c^3 + d^3*x^3)^(1/3)))/(2*d^2) - (3*(d*e - c*f)*log(d*(c - d*x) + 2^(2/3)*d*(-c^3 + d^3*x^3)^(1/3)))/(4*2^(1/3)*c*d^2)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(c+d*x)^n*(a+b*x^3)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c+d*x)^n*(a+b*x^3)^p*with*n*symbolic=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^2*(a + b*x)^n*(c + d*x^3), x, 2, (a^2*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*(2*b^3*c - 5*a^3*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) + ((b^3*c - 10*a^3*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (10*a^2*d*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d*(a + b*x)^(6 + n))/(b^6*(6 + n))]
    @test_int [x*(a + b*x)^n*(c + d*x^3), x, 2, -((a*(b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^5*(1 + n))) + ((b^3*c - 4*a^3*d)*(a + b*x)^(2 + n))/(b^5*(2 + n)) + (6*a^2*d*(a + b*x)^(3 + n))/(b^5*(3 + n)) - (4*a*d*(a + b*x)^(4 + n))/(b^5*(4 + n)) + (d*(a + b*x)^(5 + n))/(b^5*(5 + n))]
    @test_int [(a + b*x)^n*(c + d*x^3), x, 2, ((b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^4*(1 + n)) + (3*a^2*d*(a + b*x)^(2 + n))/(b^4*(2 + n)) - (3*a*d*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (d*(a + b*x)^(4 + n))/(b^4*(4 + n))]
    @test_int [1/x^1*(a + b*x)^n*(c + d*x^3), x, 3, (a^2*d*(a + b*x)^(1 + n))/(b^3*(1 + n)) - (2*a*d*(a + b*x)^(2 + n))/(b^3*(2 + n)) + (d*(a + b*x)^(3 + n))/(b^3*(3 + n)) - (c*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    @test_int [x^2*(a + b*x)^n*(c + d*x^3)^2, x, 2, (a^2*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (4*a^2*d*(5*b^3*c - 14*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (10*a*d*(b^3*c - 7*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n)) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^2*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^2*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^2*(a + b*x)^(9 + n))/(b^9*(9 + n))]
    @test_int [x*(a + b*x)^n*(c + d*x^3)^2, x, 2, -((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(a + b*x)^(4 + n))/(b^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n))/(b^8*(5 + n)) + (21*a^2*d^2*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + n))]
    @test_int [(a + b*x)^n*(c + d*x^3)^2, x, 2, ((b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) + (6*a^2*d*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)) - (3*a*d*(2*b^3*c - 5*a^3*d)*(a + b*x)^(3 + n))/(b^7*(3 + n)) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^(4 + n))/(b^7*(4 + n)) + (15*a^2*d^2*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))]
    @test_int [1/x^1*(a + b*x)^n*(c + d*x^3)^2, x, 3, (a^2*d*(2*b^3*c - a^3*d)*(a + b*x)^(1 + n))/(b^6*(1 + n)) - (a*d*(4*b^3*c - 5*a^3*d)*(a + b*x)^(2 + n))/(b^6*(2 + n)) + (2*d*(b^3*c - 5*a^3*d)*(a + b*x)^(3 + n))/(b^6*(3 + n)) + (10*a^2*d^2*(a + b*x)^(4 + n))/(b^6*(4 + n)) - (5*a*d^2*(a + b*x)^(5 + n))/(b^6*(5 + n)) + (d^2*(a + b*x)^(6 + n))/(b^6*(6 + n)) - (c^2*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    @test_int [x^2*(a + b*x)^n*(c + d*x^3)^3, x, 2, (a^2*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^12*(1 + n)) - (a*(2*b^3*c - 11*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^12*(2 + n)) + ((b^3*c - a^3*d)*(b^6*c^2 - 29*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(3 + n))/(b^12*(3 + n)) + (3*a^2*d*(10*b^6*c^2 - 56*a^3*b^3*c*d + 55*a^6*d^2)*(a + b*x)^(4 + n))/(b^12*(4 + n)) - (15*a*d*(b^6*c^2 - 14*a^3*b^3*c*d + 22*a^6*d^2)*(a + b*x)^(5 + n))/(b^12*(5 + n)) + (3*d*(b^6*c^2 - 56*a^3*b^3*c*d + 154*a^6*d^2)*(a + b*x)^(6 + n))/(b^12*(6 + n)) + (42*a^2*d^2*(2*b^3*c - 11*a^3*d)*(a + b*x)^(7 + n))/(b^12*(7 + n)) - (6*a*d^2*(4*b^3*c - 55*a^3*d)*(a + b*x)^(8 + n))/(b^12*(8 + n)) + (3*d^2*(b^3*c - 55*a^3*d)*(a + b*x)^(9 + n))/(b^12*(9 + n)) + (55*a^2*d^3*(a + b*x)^(10 + n))/(b^12*(10 + n)) - (11*a*d^3*(a + b*x)^(11 + n))/(b^12*(11 + n)) + (d^3*(a + b*x)^(12 + n))/(b^12*(12 + n))]
    @test_int [x*(a + b*x)^n*(c + d*x^3)^3, x, 2, -((a*(b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^11*(1 + n))) + ((b^3*c - 10*a^3*d)*(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^11*(2 + n)) + (9*a^2*d*(2*b^3*c - 5*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(3 + n))/(b^11*(3 + n)) - (3*a*d*(4*b^6*c^2 - 35*a^3*b^3*c*d + 40*a^6*d^2)*(a + b*x)^(4 + n))/(b^11*(4 + n)) + (3*d*(b^6*c^2 - 35*a^3*b^3*c*d + 70*a^6*d^2)*(a + b*x)^(5 + n))/(b^11*(5 + n)) + (63*a^2*d^2*(b^3*c - 4*a^3*d)*(a + b*x)^(6 + n))/(b^11*(6 + n)) - (21*a*d^2*(b^3*c - 10*a^3*d)*(a + b*x)^(7 + n))/(b^11*(7 + n)) + (3*d^2*(b^3*c - 40*a^3*d)*(a + b*x)^(8 + n))/(b^11*(8 + n)) + (45*a^2*d^3*(a + b*x)^(9 + n))/(b^11*(9 + n)) - (10*a*d^3*(a + b*x)^(10 + n))/(b^11*(10 + n)) + (d^3*(a + b*x)^(11 + n))/(b^11*(11 + n))]
    @test_int [(a + b*x)^n*(c + d*x^3)^3, x, 2, ((b^3*c - a^3*d)^3*(a + b*x)^(1 + n))/(b^10*(1 + n)) + (9*a^2*d*(b^3*c - a^3*d)^2*(a + b*x)^(2 + n))/(b^10*(2 + n)) - (9*a*d*(b^3*c - 4*a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(3 + n))/(b^10*(3 + n)) + (3*d*(b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(4 + n))/(b^10*(4 + n)) + (9*a^2*d^2*(5*b^3*c - 14*a^3*d)*(a + b*x)^(5 + n))/(b^10*(5 + n)) - (18*a*d^2*(b^3*c - 7*a^3*d)*(a + b*x)^(6 + n))/(b^10*(6 + n)) + (3*d^2*(b^3*c - 28*a^3*d)*(a + b*x)^(7 + n))/(b^10*(7 + n)) + (36*a^2*d^3*(a + b*x)^(8 + n))/(b^10*(8 + n)) - (9*a*d^3*(a + b*x)^(9 + n))/(b^10*(9 + n)) + (d^3*(a + b*x)^(10 + n))/(b^10*(10 + n))]
    @test_int [1/x^1*(a + b*x)^n*(c + d*x^3)^3, x, 3, (a^2*d*(3*b^6*c^2 - 3*a^3*b^3*c*d + a^6*d^2)*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (a*d*(6*b^6*c^2 - 15*a^3*b^3*c*d + 8*a^6*d^2)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + (d*(3*b^6*c^2 - 30*a^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (2*a^2*d^2*(15*b^3*c - 28*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (5*a*d^2*(3*b^3*c - 14*a^3*d)*(a + b*x)^(5 + n))/(b^9*(5 + n)) + (d^2*(3*b^3*c - 56*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^3*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a*d^3*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^3*(a + b*x)^(9 + n))/(b^9*(9 + n)) - (c^3*(a + b*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (b*x)/a))/(a*(1 + n))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5*(e + f*x)^n/(a + b*x^3), x, 7, (e^2*(e + f*x)^(1 + n))/(b*f^3*(1 + n)) - (2*e*(e + f*x)^(2 + n))/(b*f^3*(2 + n)) + (e + f*x)^(3 + n)/(b*f^3*(3 + n)) + (a*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*b^(5/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + (a*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)))/(3*b^(5/3)*(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) + (a*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)))/(3*b^(5/3)*(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n))]
    @test_int [x^4*(e + f*x)^n/(a + b*x^3), x, 7, -((e*(e + f*x)^(1 + n))/(b*f^2*(1 + n))) + (e + f*x)^(2 + n)/(b*f^2*(2 + n)) - (a^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*b^(4/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(1/3)*a^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)))/(3*b^(4/3)*((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(2/3)*a^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*b^(4/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))]
    @test_int [x^3*(e + f*x)^n/(a + b*x^3), x, 7, (e + f*x)^(1 + n)/(b*f*(1 + n)) + (a^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*b*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + (a^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)))/(3*b*((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) - (a^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*b*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))]
    @test_int [x^2*(e + f*x)^n/(a + b*x^3), x, 5, -(((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*b^(2/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n))) - ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)))/(3*b^(2/3)*(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) - ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)))/(3*b^(2/3)*(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n))]
    @test_int [x^1*(e + f*x)^n/(a + b*x^3), x, 5, ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*a^(1/3)*b^(1/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) - ((-1)^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)))/(3*a^(1/3)*b^(1/3)*((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) - ((-1)^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(1/3)*b^(1/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))]
    @test_int [x^0*(e + f*x)^n/(a + b*x^3), x, 5, -(((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*a^(2/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n))) - ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)))/(3*a^(2/3)*((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(2/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n))]
    @test_int [(e + f*x)^n/(x^1*(a + b*x^3)), x, 8, (b^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*a*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + (b^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)))/(3*a*(b^(1/3)*e + (-1)^(1/3)*a^(1/3)*f)*(1 + n)) + (b^(1/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)))/(3*a*(b^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n)) - ((e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, 1 + (f*x)/e))/(a*e*(1 + n))]
    @test_int [(e + f*x)^n/(x^2*(a + b*x^3)), x, 8, -((b^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/3)*(e + f*x))/(b^(1/3)*e - a^(1/3)*f)))/(3*a^(4/3)*(b^(1/3)*e - a^(1/3)*f)*(1 + n))) + ((-1)^(1/3)*b^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)))/(3*a^(4/3)*((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f)*(1 + n)) + ((-1)^(2/3)*b^(2/3)*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(4/3)*((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)*(1 + n)) + (f*(e + f*x)^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, 1 + (f*x)/e))/(a*e^2*(1 + n))]

    @test_int [x^2*(c + d*x)^(n + 1)/(a + b*x^3), x, 5, -(((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)))/(3*b^(2/3)*(b^(1/3)*c - a^(1/3)*d)*(2 + n))) - ((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)))/(3*b^(2/3)*(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)*(2 + n)) - ((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)))/(3*b^(2/3)*(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)*(2 + n))]

    @test_int [x^m*(e + f*x)^n/(a + b*x^3), x, 8, (x^(1 + m)*(e + f*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((f*x)/e), -((b^(1/3)*x)/a^(1/3))))/((1 + (f*x)/e)^n*(3*a*(1 + m))) + (x^(1 + m)*(e + f*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((f*x)/e), ((-1)^(1/3)*b^(1/3)*x)/a^(1/3)))/((1 + (f*x)/e)^n*(3*a*(1 + m))) + (x^(1 + m)*(e + f*x)^n*AppellF1(1 + m, -n, 1, 2 + m, -((f*x)/e), -(((-1)^(2/3)*b^(1/3)*x)/a^(1/3))))/((1 + (f*x)/e)^n*(3*a*(1 + m)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c+d*x)^n*(a+b*x^3)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt(c + d*x^3)/(a + b*x), x, 13, (2*sqrt(c + d*x^3))/(3*b) - (2*a*d^(1/3)*sqrt(c + d*x^3))/(b^2*((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)) - (c^(1/6)*sqrt(b*c^(1/3) - a*d^(1/3))*sqrt(b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3))*(c^(1/3) + d^(1/3)*x)*sqrt((c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*atanh((sqrt(2 - sqrt(3))*sqrt(b^2*c^(2/3) + a*b*c^(1/3)*d^(1/3) + a^2*d^(2/3))*sqrt(1 - ((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)^2/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2))/(3^(1/4)*sqrt(b)*c^(1/6)*sqrt(b*c^(1/3) - a*d^(1/3))*sqrt(7 - 4*sqrt(3) + ((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)^2/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2))))/(b^(5/2)*sqrt((c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*sqrt(c + d*x^3)) + (3^(1/4)*sqrt(2 - sqrt(3))*a*c^(1/3)*d^(1/3)*(c^(1/3) + d^(1/3)*x)*sqrt((c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*Elliptic.E(asin(((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)), -7 - 4*sqrt(3)))/(b^2*sqrt((c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*sqrt(c + d*x^3)) + (2*sqrt(2 + sqrt(3))*a*((1 - sqrt(3))*b*c^(1/3) + a*d^(1/3))*d^(1/3)*(c^(1/3) + d^(1/3)*x)*sqrt((c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*b^3*sqrt((c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*sqrt(c + d*x^3)) - (2*sqrt(2 + sqrt(3))*(b^3*c - a^3*d)*(c^(1/3) + d^(1/3)*x)*sqrt((c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*Elliptic.F(asin(((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)), -7 - 4*sqrt(3)))/(3^(1/4)*b^3*((1 + sqrt(3))*b*c^(1/3) - a*d^(1/3))*sqrt((c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*sqrt(c + d*x^3)) - (4*3^(1/4)*sqrt(2 + sqrt(3))*c^(1/3)*(b^3*c - a^3*d)*(c^(1/3) + d^(1/3)*x)*sqrt((c^(2/3)*(1 - (d^(1/3)*x)/c^(1/3) + (d^(2/3)*x^2)/c^(2/3)))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*Elliptic.Pi(((1 + sqrt(3))*b*c^(1/3) - a*d^(1/3))^2/((1 - sqrt(3))*b*c^(1/3) - a*d^(1/3))^2, -asin(((1 - sqrt(3))*c^(1/3) + d^(1/3)*x)/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)), -7 - 4*sqrt(3)))/(b^2*(2*b^2*c^(2/3) + 2*a*b*c^(1/3)*d^(1/3) - a^2*d^(2/3))*sqrt((c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + sqrt(3))*c^(1/3) + d^(1/3)*x)^2)*sqrt(c + d*x^3))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c+d*x)^n*(a+b*x^3)^p*with*p*symbolic=#

    @test_int [(d^3 + e^3*x^3)^p/(d + e*x), x, -1, ((d^3 + e^3*x^3)^p*AppellF1(p, -p, -p, 1 + p, -((2*(d + e*x))/((-3 + im*sqrt(3))*d)), (2*(d + e*x))/((3 + im*sqrt(3))*d)))/((1 + (2*(d + e*x))/((-3 + im*sqrt(3))*d))^p*(1 - (2*(d + e*x))/((3 + im*sqrt(3))*d))^p*(e*p))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form  (c+d*x+e*x^2)^m*(f+g*x+h*x^2)^n*(a+b*x^3)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(f+g*x+h*x^2) / ((c+e*x^2)*sqrt(a+b*x^3))*with*b*g^3 - 8*a*h^3 = 0 && g^2 + 2*f*h = 0 && b*c*g - 4*a*e*h = 0=#

    @test_int [(2 - 2*x - x^2)/((2 + x^2)*sqrt(1 + x^3)), x, 2, 2*atan((1 + x)/sqrt(1 + x^3))]
    @test_int [(2 + 2*x - x^2)/((2 + x^2)*sqrt(1 - x^3)), x, 2, -2*atan((1 - x)/sqrt(1 - x^3))]
    @test_int [(2 + 2*x - x^2)/((2 + x^2)*sqrt(-1 + x^3)), x, 2, -2*atanh((1 - x)/sqrt(-1 + x^3))]
    @test_int [(2 - 2*x - x^2)/((2 + x^2)*sqrt(-1 - x^3)), x, 2, 2*atanh((1 + x)/sqrt(-1 - x^3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(f+g*x+h*x^2) / ((c+d*x+e*x^2)*sqrt(a+b*x^3))*with*b*g^3 - 8*a*h^3 = 0 && g^2 + 2*f*h = 0 && b*d*f + b*c*g - 4*a*e*h = 0=#

    @test_int [(2 - 2*x - x^2)/((2 + d + d*x + x^2)*sqrt(1 + x^3)), x, 2, (2*atan((sqrt(1 + d)*(1 + x))/sqrt(1 + x^3)))/sqrt(1 + d)]
    @test_int [(2 + 2*x - x^2)/((2 - d + d*x + x^2)*sqrt(1 - x^3)), x, 2, -((2*atan((sqrt(1 - d)*(1 - x))/sqrt(1 - x^3)))/sqrt(1 - d))]
    @test_int [(2 + 2*x - x^2)/((2 - d + d*x + x^2)*sqrt(-1 + x^3)), x, 2, -((2*atanh((sqrt(1 - d)*(1 - x))/sqrt(-1 + x^3)))/sqrt(1 - d))]
    @test_int [(2 - 2*x - x^2)/((2 + d + d*x + x^2)*sqrt(-1 - x^3)), x, 2, (2*atanh((sqrt(1 + d)*(1 + x))/sqrt(-1 - x^3)))/sqrt(1 + d)]

    #= ::Title:: =#
    #=Algebraic*Function*Integration*Problems=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(a+b*x^2+c*x^4)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x)^q*(a+c*x^4)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(d + e*x)^3*sqrt(a + c*x^4), x, 11, (3*d^2*e*x^2*sqrt(a + c*x^4))/4 + (6*a*d*e^2*x*sqrt(a + c*x^4))/(5*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (d*x*(5*d^2 + 9*e^2*x^2)*sqrt(a + c*x^4))/15 + (e^3*(a + c*x^4)^(3/2))/(6*c) + (3*a*d^2*e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(4*sqrt(c)) - (6*a^(5/4)*d*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(5*c^(3/4)*sqrt(a + c*x^4)) + (a^(3/4)*d*(5*sqrt(c)*d^2 + 9*sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(15*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^2*sqrt(a + c*x^4), x, 10, (d*e*x^2*sqrt(a + c*x^4))/2 + (2*a*e^2*x*sqrt(a + c*x^4))/(5*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (x*(5*d^2 + 3*e^2*x^2)*sqrt(a + c*x^4))/15 + (a*d*e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(2*sqrt(c)) - (2*a^(5/4)*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(5*c^(3/4)*sqrt(a + c*x^4)) + (a^(3/4)*(5*sqrt(c)*d^2 + 3*sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(15*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^1*sqrt(a + c*x^4), x, 8, (d*x*sqrt(a + c*x^4))/3 + (e*x^2*sqrt(a + c*x^4))/4 + (a*e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(4*sqrt(c)) + (a^(3/4)*d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(3*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^0*sqrt(a + c*x^4), x, 2, (x*sqrt(a + c*x^4))/3 + (a^(3/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(3*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [sqrt(a + c*x^4)/(d + e*x)^1, x, 15, sqrt(a + c*x^4)/(2*e) - (sqrt(c)*d*x*sqrt(a + c*x^4))/(e^2*(sqrt(a) + sqrt(c)*x^2)) - (sqrt((-c)*d^4 - a*e^4)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*e^3) + (sqrt(c)*d^2*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(2*e^3) - (sqrt(c*d^4 + a*e^4)*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(2*e^3) + (a^(1/4)*c^(1/4)*d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(e^2*sqrt(a + c*x^4)) - (a^(1/4)*c^(1/4)*d*((sqrt(c)*d^2)/sqrt(a) + e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*e^4*sqrt(a + c*x^4)) + (c^(1/4)*d*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*e^4*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4)) - ((sqrt(c)*d^2 - sqrt(a)*e^2)*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d*e^4*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4))]
    @test_int [sqrt(a + c*x^4)/(d + e*x)^2, x, 32, (2*sqrt(c)*x*sqrt(a + c*x^4))/(e^2*(sqrt(a) + sqrt(c)*x^2)) - (d*sqrt(a + c*x^4))/(e*(d^2 - e^2*x^2)) + (x*sqrt(a + c*x^4))/(d^2 - e^2*x^2) + (sqrt((-c)*d^4 - a*e^4)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*d*e^3) - ((c*d^4 - a*e^4)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*d*e^3*sqrt((-c)*d^4 - a*e^4)) - (sqrt(c)*d*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/e^3 + (c*d^3*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(e^3*sqrt(c*d^4 + a*e^4)) - (2*a^(1/4)*c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(e^2*sqrt(a + c*x^4)) + (3*a^(1/4)*c^(1/4)*((sqrt(c)*d^2)/sqrt(a) + e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*e^4*sqrt(a + c*x^4)) - (c^(1/4)*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*e^4*sqrt(a + c*x^4)) + (c^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*e^4*sqrt(a + c*x^4)) - (c^(1/4)*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*e^4*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4)) + ((sqrt(c)*d^2 - sqrt(a)*e^2)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d^2*e^4*sqrt(a + c*x^4)) + ((sqrt(c)*d^2 - sqrt(a)*e^2)*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d^2*e^4*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(d + e*x)^3/sqrt(a + c*x^4), x, 9, (e^3*sqrt(a + c*x^4))/(2*c) + (3*d*e^2*x*sqrt(a + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (3*d^2*e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(2*sqrt(c)) - (3*a^(1/4)*d*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(c^(3/4)*sqrt(a + c*x^4)) + (d*(sqrt(c)*d^2 + 3*sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^2/sqrt(a + c*x^4), x, 8, (e^2*x*sqrt(a + c*x^4))/(sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (d*e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/sqrt(c) - (a^(1/4)*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(c^(3/4)*sqrt(a + c*x^4)) + (a^(1/4)*((sqrt(c)*d^2)/sqrt(a) + e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^1/sqrt(a + c*x^4), x, 6, (e*atanh((sqrt(c)*x^2)/sqrt(a + c*x^4)))/(2*sqrt(c)) + (d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^0/sqrt(a + c*x^4), x, 1, ((sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x)^1*sqrt(a + c*x^4)), x, 7, (e*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*sqrt((-c)*d^4 - a*e^4)) - (e*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(2*sqrt(c*d^4 + a*e^4)) + (c^(1/4)*d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4)) - ((sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x)^2*sqrt(a + c*x^4)), x, 11, -((e^3*sqrt(a + c*x^4))/((c*d^4 + a*e^4)*(d + e*x))) + (sqrt(c)*e^2*x*sqrt(a + c*x^4))/((c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)) - (c*d^3*e*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/((-c)*d^4 - a*e^4)^(3/2) - (c*d^3*e*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(c*d^4 + a*e^4)^(3/2) - (a^(1/4)*c^(1/4)*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/((c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c^(1/4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*sqrt(a + c*x^4)) - (c^(3/4)*d^2*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x)^3*sqrt(a + c*x^4)), x, 12, -((e^3*sqrt(a + c*x^4))/(2*(c*d^4 + a*e^4)*(d + e*x)^2)) - (3*c*d^3*e^3*sqrt(a + c*x^4))/((c*d^4 + a*e^4)^2*(d + e*x)) + (3*c^(3/2)*d^3*e^2*x*sqrt(a + c*x^4))/((c*d^4 + a*e^4)^2*(sqrt(a) + sqrt(c)*x^2)) + (3*c*d^2*e*(c*d^4 - a*e^4)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*((-c)*d^4 - a*e^4)^(5/2)) - (3*c*d^2*e*(c*d^4 - a*e^4)*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(2*(c*d^4 + a*e^4)^(5/2)) - (3*a^(1/4)*c^(5/4)*d^3*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/((c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) + (c^(3/4)*d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) - (3*c^(3/4)*d*(sqrt(c)*d^2 - sqrt(a)*e^2)^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4))]

    @test_int [(d + e*x)^3/(a + c*x^4)^(3/2), x, 4, (-3*d*e^2*x*sqrt(a + c*x^4))/(2*a*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) - (a*e^3 - c*x*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2))/(2*a*c*sqrt(a + c*x^4)) + (3*d*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(3/4)*c^(3/4)*sqrt(a + c*x^4)) + (d*(sqrt(c)*d^2 - 3*sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^2/(a + c*x^4)^(3/2), x, 4, (x*(d + e*x)^2)/(2*a*sqrt(a + c*x^4)) - (e^2*x*sqrt(a + c*x^4))/(2*a*sqrt(c)*(sqrt(a) + sqrt(c)*x^2)) + (e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(3/4)*c^(3/4)*sqrt(a + c*x^4)) + ((sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*c^(3/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^1/(a + c*x^4)^(3/2), x, 3, (x*(d + e*x))/(2*a*sqrt(a + c*x^4)) + (d*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [(d + e*x)^0/(a + c*x^4)^(3/2), x, 2, x/(2*a*sqrt(a + c*x^4)) + ((sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*c^(1/4)*sqrt(a + c*x^4))]
    @test_int [1/((d + e*x)^1*(a + c*x^4)^(3/2)), x, 14, (e*(a*e^2 - c*d^2*x^2))/(2*a*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c*d*x*(d^2 + e^2*x^2))/(2*a*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) - (sqrt(c)*d*e^2*x*sqrt(a + c*x^4))/(2*a*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)) - (e^5*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*((-c)*d^4 - a*e^4)^(3/2)) - (e^5*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(2*(c*d^4 + a*e^4)^(3/2)) + (c^(1/4)*d*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(3/4)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c^(1/4)*d*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c^(1/4)*d*e^4*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) - (e^4*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)*sqrt(a + c*x^4))]
    #= [1/((d + e*x)^2*(a + c*x^4)^(3/2)), x, 79, (d*e*(a*e^2 - c*d^2*x^2))/(a*(c*d^4 + a*e^4)*(d^2 - e^2*x^2)*sqrt(a + c*x^4)) - (c*x*(d^2 + e^2*x^2))/(2*a*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c*d^2*x*(c*d^4 - a*e^4 + 2*c*d^2*e^2*x^2))/(a*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) + (e^7*sqrt(a + c*x^4))/(2*(c*d^4 + a*e^4)^2*(d - e*x)) - (e^7*sqrt(a + c*x^4))/(2*(c*d^4 + a*e^4)^2*(d + e*x)) - (2*c^(3/2)*d^4*e^2*x*sqrt(a + c*x^4))/(a*(c*d^4 + a*e^4)^2*(sqrt(a) + sqrt(c)*x^2)) + (sqrt(c)*e^6*x*sqrt(a + c*x^4))/((c*d^4 + a*e^4)^2*(sqrt(a) + sqrt(c)*x^2)) + (sqrt(c)*e^2*x*sqrt(a + c*x^4))/(2*a*(c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)) + (d*e^3*(c*d^4 - 2*a*e^4)*sqrt(a + c*x^4))/(a*(c*d^4 + a*e^4)^2*(d^2 - e^2*x^2)) + (c*d^3*e^5*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/((-c)*d^4 - a*e^4)^(5/2) + (e^5*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*d*((-c)*d^4 - a*e^4)^(3/2)) + (e^5*(5*c*d^4 + a*e^4)*atan((sqrt((-c)*d^4 - a*e^4)*x)/(d*e*sqrt(a + c*x^4))))/(2*d*((-c)*d^4 - a*e^4)^(5/2)) - (3*c*d^3*e^5*atanh((a*e^2 + c*d^2*x^2)/(sqrt(c*d^4 + a*e^4)*sqrt(a + c*x^4))))/(c*d^4 + a*e^4)^(5/2) + (2*c^(5/4)*d^4*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(a^(3/4)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) - (a^(1/4)*c^(1/4)*e^6*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/((c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) - (c^(1/4)*e^2*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.E(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(3/4)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c^(3/4)*d^2*(c*d^4 - 2*sqrt(a)*sqrt(c)*d^2*e^2 - a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(5/4)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) - (c^(1/4)*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(5/4)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) + (c^(1/4)*e^4*(5*c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.F(2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) - (c^(3/4)*d^2*e^4*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4)) + (e^4*(sqrt(c)*d^2 - sqrt(a)*e^2)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d^2*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)*sqrt(a + c*x^4)) - (e^4*(sqrt(c)*d^2 - sqrt(a)*e^2)*(5*c*d^4 + a*e^4)*(sqrt(a) + sqrt(c)*x^2)*sqrt((a + c*x^4)/(sqrt(a) + sqrt(c)*x^2)^2)*Elliptic.Pi((sqrt(c)*d^2 + sqrt(a)*e^2)^2/(4*sqrt(a)*sqrt(c)*d^2*e^2), 2*atan((c^(1/4)*x)/a^(1/4)), 1/2))/(4*a^(1/4)*c^(1/4)*d^2*(sqrt(c)*d^2 + sqrt(a)*e^2)*(c*d^4 + a*e^4)^2*sqrt(a + c*x^4))] =#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c+d*x)^n*(a+b*x^4)^p*with*n*symbolic=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3*(c + d*x)^n/(a + b*x^4), x, 10, -(((c + d*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - sqrt(-sqrt(-a))*d)))/(4*b^(3/4)*(b^(1/4)*c - sqrt(-sqrt(-a))*d)*(1 + n))) - ((c + d*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + sqrt(-sqrt(-a))*d)))/(4*b^(3/4)*(b^(1/4)*c + sqrt(-sqrt(-a))*d)*(1 + n)) - ((c + d*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)))/(4*b^(3/4)*(b^(1/4)*c - (-a)^(1/4)*d)*(1 + n)) - ((c + d*x)^(1 + n)*HypergeometricFunctions._₂F₁(1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(b^(1/4)*c + (-a)^(1/4)*d)*(1 + n))]
    @test_int [x^3*(c + d*x)^(n + 1)/(a + b*x^4), x, 10, -(((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - sqrt(-sqrt(-a))*d)))/(4*b^(3/4)*(b^(1/4)*c - sqrt(-sqrt(-a))*d)*(2 + n))) - ((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + sqrt(-sqrt(-a))*d)))/(4*b^(3/4)*(b^(1/4)*c + sqrt(-sqrt(-a))*d)*(2 + n)) - ((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)))/(4*b^(3/4)*(b^(1/4)*c - (-a)^(1/4)*d)*(2 + n)) - ((c + d*x)^(2 + n)*HypergeometricFunctions._₂F₁(1, 2 + n, 3 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(b^(1/4)*c + (-a)^(1/4)*d)*(2 + n))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(c+d*x+e*x^2)^n*(a+b*x^4)^(p/2)=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [1/((c + d*x + e*x^2)*sqrt(a + b*x^4)), x, 16, -((e^2*atan((sqrt(2)*sqrt((-b)*d^4 + 4*b*c*d^2*e - 2*b*c^2*e^2 - 2*a*e^4 - b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))*x)/(e*(d + sqrt(d^2 - 4*c*e))*sqrt(a + b*x^4))))/(sqrt(2)*sqrt(d^2 - 4*c*e)*sqrt(-2*a*e^4 - b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 + d^3*sqrt(d^2 - 4*c*e) - 2*c*d*e*sqrt(d^2 - 4*c*e))))) + (e^2*atan((sqrt(2)*sqrt((-b)*d^4 + 4*b*c*d^2*e - 2*b*c^2*e^2 - 2*a*e^4 + b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))*x)/(e*(d - sqrt(d^2 - 4*c*e))*sqrt(a + b*x^4))))/(sqrt(2)*sqrt(d^2 - 4*c*e)*sqrt(-2*a*e^4 - b*(d^4 - 4*c*d^2*e + 2*c^2*e^2 - d^3*sqrt(d^2 - 4*c*e) + 2*c*d*e*sqrt(d^2 - 4*c*e)))) - (e^2*atanh((4*a*e^2 + b*(d - sqrt(d^2 - 4*c*e))^2*x^2)/(2*sqrt(2)*sqrt(b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))*sqrt(a + b*x^4))))/(sqrt(2)*sqrt(d^2 - 4*c*e)*sqrt(b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 - b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))) + (e^2*atanh((4*a*e^2 + b*(d + sqrt(d^2 - 4*c*e))^2*x^2)/(2*sqrt(2)*sqrt(b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))*sqrt(a + b*x^4))))/(sqrt(2)*sqrt(d^2 - 4*c*e)*sqrt(b*d^4 - 4*b*c*d^2*e + 2*b*c^2*e^2 + 2*a*e^4 + b*d*sqrt(d^2 - 4*c*e)*(d^2 - 2*c*e))) + (b^(1/4)*e*(d - sqrt(d^2 - 4*c*e))*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.F(2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*sqrt(d^2 - 4*c*e)*(2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e - d*sqrt(d^2 - 4*c*e)))*sqrt(a + b*x^4)) - (b^(1/4)*e*(d + sqrt(d^2 - 4*c*e))*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.F(2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*sqrt(d^2 - 4*c*e)*(2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e + d*sqrt(d^2 - 4*c*e)))*sqrt(a + b*x^4)) + (e*(2*sqrt(a)*e^2 - sqrt(b)*(d^2 - 2*c*e - d*sqrt(d^2 - 4*c*e)))*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e - d*sqrt(d^2 - 4*c*e)))^2/(4*sqrt(a)*sqrt(b)*e^2*(d - sqrt(d^2 - 4*c*e))^2), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*b^(1/4)*sqrt(d^2 - 4*c*e)*(d - sqrt(d^2 - 4*c*e))*(2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e - d*sqrt(d^2 - 4*c*e)))*sqrt(a + b*x^4)) - (e*(2*sqrt(a)*e^2 - sqrt(b)*(d^2 - 2*c*e + d*sqrt(d^2 - 4*c*e)))*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e + d*sqrt(d^2 - 4*c*e)))^2/(4*sqrt(a)*sqrt(b)*e^2*(d + sqrt(d^2 - 4*c*e))^2), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(2*a^(1/4)*b^(1/4)*sqrt(d^2 - 4*c*e)*(d + sqrt(d^2 - 4*c*e))*(2*sqrt(a)*e^2 + sqrt(b)*(d^2 - 2*c*e + d*sqrt(d^2 - 4*c*e)))*sqrt(a + b*x^4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(c*(a+b*x^n)^q)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c*(a+b*x^2)^q)^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=q>0=#

    @test_int [x^m*(c*(a + b*x^2)^2)^(3/2), x, 3, (a^3*c*x^(1 + m)*sqrt(c*(a + b*x^2)^2))/((1 + m)*(a + b*x^2)) + (3*a^2*b*c*x^(3 + m)*sqrt(c*(a + b*x^2)^2))/((3 + m)*(a + b*x^2)) + (3*a*b^2*c*x^(5 + m)*sqrt(c*(a + b*x^2)^2))/((5 + m)*(a + b*x^2)) + (b^3*c*x^(7 + m)*sqrt(c*(a + b*x^2)^2))/((7 + m)*(a + b*x^2))]

    @test_int [x^5*(c*(a + b*x^2)^2)^(3/2), x, 4, (a^3*c*x^6*sqrt(c*(a + b*x^2)^2))/(6*(a + b*x^2)) + (3*a^2*b*c*x^8*sqrt(c*(a + b*x^2)^2))/(8*(a + b*x^2)) + (3*a*b^2*c*x^10*sqrt(c*(a + b*x^2)^2))/(10*(a + b*x^2)) + (b^3*c*x^12*sqrt(c*(a + b*x^2)^2))/(12*(a + b*x^2))]
    @test_int [x^4*(c*(a + b*x^2)^2)^(3/2), x, 3, (a^3*c*x^5*sqrt(c*(a + b*x^2)^2))/(5*(a + b*x^2)) + (3*a^2*b*c*x^7*sqrt(c*(a + b*x^2)^2))/(7*(a + b*x^2)) + (a*b^2*c*x^9*sqrt(c*(a + b*x^2)^2))/(3*(a + b*x^2)) + (b^3*c*x^11*sqrt(c*(a + b*x^2)^2))/(11*(a + b*x^2))]
    @test_int [x^3*(c*(a + b*x^2)^2)^(3/2), x, 4, -((a*c*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^2))/(8*b^2)) + (c*(a + b*x^2)^4*sqrt(c*(a + b*x^2)^2))/(10*b^2)]
    @test_int [x^2*(c*(a + b*x^2)^2)^(3/2), x, 3, (a^3*c*x^3*sqrt(c*(a + b*x^2)^2))/(3*(a + b*x^2)) + (3*a^2*b*c*x^5*sqrt(c*(a + b*x^2)^2))/(5*(a + b*x^2)) + (3*a*b^2*c*x^7*sqrt(c*(a + b*x^2)^2))/(7*(a + b*x^2)) + (b^3*c*x^9*sqrt(c*(a + b*x^2)^2))/(9*(a + b*x^2))]
    @test_int [x^1*(c*(a + b*x^2)^2)^(3/2), x, 3, (c*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^2))/(8*b)]
    @test_int [x^0*(c*(a + b*x^2)^2)^(3/2), x, 3, (a^3*c*x*sqrt(c*(a + b*x^2)^2))/(a + b*x^2) + (a^2*b*c*x^3*sqrt(c*(a + b*x^2)^2))/(a + b*x^2) + (3*a*b^2*c*x^5*sqrt(c*(a + b*x^2)^2))/(5*(a + b*x^2)) + (b^3*c*x^7*sqrt(c*(a + b*x^2)^2))/(7*(a + b*x^2))]
    @test_int [(c*(a + b*x^2)^2)^(3/2)/x^1, x, 4, (3*a^2*b*c*x^2*sqrt(c*(a + b*x^2)^2))/(2*(a + b*x^2)) + (3*a*b^2*c*x^4*sqrt(c*(a + b*x^2)^2))/(4*(a + b*x^2)) + (b^3*c*x^6*sqrt(c*(a + b*x^2)^2))/(6*(a + b*x^2)) + (a^3*c*sqrt(c*(a + b*x^2)^2)*log(x))/(a + b*x^2)]
    @test_int [(c*(a + b*x^2)^2)^(3/2)/x^2, x, 3, -((a^3*c*sqrt(c*(a + b*x^2)^2))/(x*(a + b*x^2))) + (3*a^2*b*c*x*sqrt(c*(a + b*x^2)^2))/(a + b*x^2) + (a*b^2*c*x^3*sqrt(c*(a + b*x^2)^2))/(a + b*x^2) + (b^3*c*x^5*sqrt(c*(a + b*x^2)^2))/(5*(a + b*x^2))]
    @test_int [(c*(a + b*x^2)^2)^(3/2)/x^3, x, 4, -((a^3*c*sqrt(c*(a + b*x^2)^2))/(2*x^2*(a + b*x^2))) + (3*a*b^2*c*x^2*sqrt(c*(a + b*x^2)^2))/(2*(a + b*x^2)) + (b^3*c*x^4*sqrt(c*(a + b*x^2)^2))/(4*(a + b*x^2)) + (3*a^2*b*c*sqrt(c*(a + b*x^2)^2)*log(x))/(a + b*x^2)]

    @test_int [x^2*(c*(a + b*x^2)^3)^(3/2), x, 8, (7/128)*a^3*c*x^3*sqrt(c*(a + b*x^2)^3) + (21*a^5*c*x*sqrt(c*(a + b*x^2)^3))/(1024*b*(a + b*x^2)) + (21*a^4*c*x^3*sqrt(c*(a + b*x^2)^3))/(512*(a + b*x^2)) + (21/320)*a^2*c*x^3*(a + b*x^2)*sqrt(c*(a + b*x^2)^3) + (3/40)*a*c*x^3*(a + b*x^2)^2*sqrt(c*(a + b*x^2)^3) + (1/12)*c*x^3*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^3) - (21*a^(9/2)*c*sqrt(c*(a + b*x^2)^3)*asinh((sqrt(b)*x)/sqrt(a)))/(1024*b^(3/2)*(1 + (b*x^2)/a)^(3/2))]
    @test_int [x^1*(c*(a + b*x^2)^3)^(3/2), x, 3, (c*(a + b*x^2)^4*sqrt(c*(a + b*x^2)^3))/(11*b)]
    @test_int [x^0*(c*(a + b*x^2)^3)^(3/2), x, 7, (21/128)*a^3*c*x*sqrt(c*(a + b*x^2)^3) + (63*a^4*c*x*sqrt(c*(a + b*x^2)^3))/(256*(a + b*x^2)) + (21/160)*a^2*c*x*(a + b*x^2)*sqrt(c*(a + b*x^2)^3) + (9/80)*a*c*x*(a + b*x^2)^2*sqrt(c*(a + b*x^2)^3) + (1/10)*c*x*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^3) + (63*a^(7/2)*c*sqrt(c*(a + b*x^2)^3)*asinh((sqrt(b)*x)/sqrt(a)))/(256*sqrt(b)*(1 + (b*x^2)/a)^(3/2))]
    @test_int [(c*(a + b*x^2)^3)^(3/2)/x^1, x, 9, (1/3)*a^3*c*sqrt(c*(a + b*x^2)^3) + (a^4*c*sqrt(c*(a + b*x^2)^3))/(a + b*x^2) + (1/5)*a^2*c*(a + b*x^2)*sqrt(c*(a + b*x^2)^3) + (1/7)*a*c*(a + b*x^2)^2*sqrt(c*(a + b*x^2)^3) + (1/9)*c*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^3) - (a^3*c*sqrt(c*(a + b*x^2)^3)*atanh(sqrt(1 + (b*x^2)/a)))/(1 + (b*x^2)/a)^(3/2)]
    @test_int [(c*(a + b*x^2)^3)^(3/2)/x^2, x, 7, (105/64)*a^2*b*c*x*sqrt(c*(a + b*x^2)^3) + (315*a^3*b*c*x*sqrt(c*(a + b*x^2)^3))/(128*(a + b*x^2)) + (21/16)*a*b*c*x*(a + b*x^2)*sqrt(c*(a + b*x^2)^3) + (9/8)*b*c*x*(a + b*x^2)^2*sqrt(c*(a + b*x^2)^3) - (c*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^3))/x + (315*a^(5/2)*sqrt(b)*c*sqrt(c*(a + b*x^2)^3)*asinh((sqrt(b)*x)/sqrt(a)))/(128*(1 + (b*x^2)/a)^(3/2))]
    @test_int [(c*(a + b*x^2)^3)^(3/2)/x^3, x, 9, (3/2)*a^2*b*c*sqrt(c*(a + b*x^2)^3) + (9*a^3*b*c*sqrt(c*(a + b*x^2)^3))/(2*(a + b*x^2)) + (9/10)*a*b*c*(a + b*x^2)*sqrt(c*(a + b*x^2)^3) + (9/14)*b*c*(a + b*x^2)^2*sqrt(c*(a + b*x^2)^3) - (c*(a + b*x^2)^3*sqrt(c*(a + b*x^2)^3))/(2*x^2) - (9*a^2*b*c*sqrt(c*(a + b*x^2)^3)*atanh(sqrt(1 + (b*x^2)/a)))/(2*(1 + (b*x^2)/a)^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=q<0=#

    @test_int [x^2*(c/(a + b*x^2))^(3/2), x, 3, -((c*x*sqrt(c/(a + b*x^2)))/b) + (sqrt(a)*c*sqrt(c/(a + b*x^2))*sqrt(1 + (b*x^2)/a)*asinh((sqrt(b)*x)/sqrt(a)))/b^(3/2)]
    @test_int [x^1*(c/(a + b*x^2))^(3/2), x, 3, -((c*sqrt(c/(a + b*x^2)))/b)]
    @test_int [x^0*(c/(a + b*x^2))^(3/2), x, 2, (c*x*sqrt(c/(a + b*x^2)))/a]
    @test_int [(c/(a + b*x^2))^(3/2)/x^1, x, 5, (c*sqrt(c/(a + b*x^2)))/a - (c*sqrt(c/(a + b*x^2))*sqrt(1 + (b*x^2)/a)*atanh(sqrt(1 + (b*x^2)/a)))/a]
    @test_int [(c/(a + b*x^2))^(3/2)/x^2, x, 3, -((c*sqrt(c/(a + b*x^2)))/(a*x)) - (2*b*c*x*sqrt(c/(a + b*x^2)))/a^2]
    @test_int [(c/(a + b*x^2))^(3/2)/x^3, x, 6, -((3*b*c*sqrt(c/(a + b*x^2)))/(2*a^2)) - (c*sqrt(c/(a + b*x^2)))/(2*a*x^2) + (3*b*c*sqrt(c/(a + b*x^2))*sqrt(1 + (b*x^2)/a)*atanh(sqrt(1 + (b*x^2)/a)))/(2*a^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(c*(a+b*x^2)^(q/2))^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=q>0=#

    @test_int [x^7*(c*(a + b*x^2)^(1/2))^(3/2), x, 4, -((2*a^3*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2))/(7*b^4)) + (6*a^2*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^2)/(11*b^4) - (2*a*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^3)/(5*b^4) + (2*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^4)/(19*b^4)]
    @test_int [x^5*(c*(a + b*x^2)^(1/2))^(3/2), x, 4, (2*a^2*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2))/(7*b^3) - (4*a*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^2)/(11*b^3) + (2*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^3)/(15*b^3)]
    @test_int [x^3*(c*(a + b*x^2)^(1/2))^(3/2), x, 4, -((2*a*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2))/(7*b^2)) + (2*(c*sqrt(a + b*x^2))^(3/2)*(a + b*x^2)^2)/(11*b^2)]
    @test_int [x^1*(c*(a + b*x^2)^(1/2))^(3/2), x, 3, (2*c*sqrt(c*sqrt(a + b*x^2))*(a + b*x^2)^(3/2))/(7*b)]
    @test_int [(c*(a + b*x^2)^(1/2))^(3/2)/x^1, x, 7, (2/3)*(c*sqrt(a + b*x^2))^(3/2) + ((c*sqrt(a + b*x^2))^(3/2)*atan((1 + (b*x^2)/a)^(1/4)))/(1 + (b*x^2)/a)^(3/4) - ((c*sqrt(a + b*x^2))^(3/2)*atanh((1 + (b*x^2)/a)^(1/4)))/(1 + (b*x^2)/a)^(3/4)]
    @test_int [(c*(a + b*x^2)^(1/2))^(3/2)/x^3, x, 7, -((c*sqrt(a + b*x^2))^(3/2)/(2*x^2)) + (3*b*(c*sqrt(a + b*x^2))^(3/2)*atan((1 + (b*x^2)/a)^(1/4)))/(4*a*(1 + (b*x^2)/a)^(3/4)) - (3*b*(c*sqrt(a + b*x^2))^(3/2)*atanh((1 + (b*x^2)/a)^(1/4)))/(4*a*(1 + (b*x^2)/a)^(3/4))]

    @test_int [x^2*(c*(a + b*x^2)^(1/2))^(3/2), x, 5, (2*a*x*(c*sqrt(a + b*x^2))^(3/2))/(15*b) + (2/9)*x^3*(c*sqrt(a + b*x^2))^(3/2) - (4*a^2*x*(c*sqrt(a + b*x^2))^(3/2))/(15*b*(a + b*x^2)) + (4*a^(3/2)*(c*sqrt(a + b*x^2))^(3/2)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(15*b^(3/2)*(1 + (b*x^2)/a)^(3/4))]
    @test_int [x^0*(c*(a + b*x^2)^(1/2))^(3/2), x, 4, (2/5)*x*(c*sqrt(a + b*x^2))^(3/2) + (6*a*x*(c*sqrt(a + b*x^2))^(3/2))/(5*(a + b*x^2)) - (6*sqrt(a)*(c*sqrt(a + b*x^2))^(3/2)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(5*sqrt(b)*(1 + (b*x^2)/a)^(3/4))]
    @test_int [(c*(a + b*x^2)^(1/2))^(3/2)/x^2, x, 4, -((c*sqrt(a + b*x^2))^(3/2)/x) + (3*b*x*(c*sqrt(a + b*x^2))^(3/2))/(a + b*x^2) - (3*sqrt(b)*(c*sqrt(a + b*x^2))^(3/2)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(sqrt(a)*(1 + (b*x^2)/a)^(3/4))]
    @test_int [(c*(a + b*x^2)^(1/2))^(3/2)/x^4, x, 5, -((c*sqrt(a + b*x^2))^(3/2)/(3*x^3)) - (b*(c*sqrt(a + b*x^2))^(3/2))/(2*a*x) + (b^2*x*(c*sqrt(a + b*x^2))^(3/2))/(2*a*(a + b*x^2)) - (b^(3/2)*(c*sqrt(a + b*x^2))^(3/2)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*a^(3/2)*(1 + (b*x^2)/a)^(3/4))]

    #= ::Subsubsection:: =#
    #=q<0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(e*(a+b*x^n)^q*(c+d*x^n)^r)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(e*(a+b*x^2)*(c+d*x^2))^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt((b - x)*(-a + x)), x, 4, (-(1/4))*(a + b - 2*x)*sqrt((-a)*b + (a + b)*x - x^2) - (1/8)*(a - b)^2*atan((a + b - 2*x)/(2*sqrt((-a)*b + (a + b)*x - x^2)))]

    @test_int [sqrt((1 - x^2)*(3 + x^2)), x, 6, (1/3)*x*sqrt(3 - 2*x^2 - x^4) - (2*Elliptic.E(asin(x), -(1/3)))/sqrt(3) + (4*Elliptic.F(asin(x), -(1/3)))/sqrt(3)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [1/sqrt((b - x)*(-a + x)), x, 3, -atan((a + b - 2*x)/(2*sqrt((-a)*b + (a + b)*x - x^2)))]

    @test_int [1/sqrt((1 - x^2)*(3 + x^2)), x, 3, Elliptic.F(asin(x), -(1/3))/sqrt(3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(e*(a+b*x^2)/(c+d*x^2))^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 6, ((11*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(16*b^2*d^3) - ((3*b*c + a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(8*b*d^3) + (((e*(a + b*x^2))/(c + d*x^2))^(3/2)*(c + d*x^2)^3)/(6*b*d^2*e) - ((b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*sqrt(e)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(16*b^(5/2)*d^(7/2))]
    @test_int [x^3*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 5, -((5*b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(8*b*d^2) + (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(4*d^2) + ((b*c - a*d)*(3*b*c + a*d)*sqrt(e)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(8*b^(3/2)*d^(5/2))]
    @test_int [x^1*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 4, (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(2*d) - ((b*c - a*d)*sqrt(e)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(2*sqrt(b)*d^(3/2))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^1, x, 5, -((sqrt(a)*sqrt(e)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/sqrt(c)) + (sqrt(b)*sqrt(e)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/sqrt(d)]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^3, x, 4, ((b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(2*c*(a - (c*(a + b*x^2))/(c + d*x^2))) - ((b*c - a*d)*sqrt(e)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(2*sqrt(a)*c^(3/2))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^5, x, 5, -((b*c - a*d)^2*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(4*c^2*(a - (c*(a + b*x^2))/(c + d*x^2))^2) + ((b*c - 5*a*d)*(b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(8*a*c^2*(a - (c*(a + b*x^2))/(c + d*x^2))) + ((b*c - a*d)*(b*c + 3*a*d)*sqrt(e)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(8*a^(3/2)*c^(5/2))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^7, x, 6, ((b*c - a*d)^2*(b*c + 3*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(8*a*c^3*(a - (c*(a + b*x^2))/(c + d*x^2))^2) - ((b*c - a*d)*(b^2*c^2 + 2*a*b*c*d - 11*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(16*a^2*c^3*(a - (c*(a + b*x^2))/(c + d*x^2))) + ((b*c - a*d)^3*e^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2))/(6*a*c^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^3) - ((b*c - a*d)*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*sqrt(e)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(16*a^(5/2)*c^(7/2))]

    @test_int [x^4*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 7, ((8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(15*b^2*d^2) - ((4*b*c - a*d)*x*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(15*b*d^2) + (x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*d) - (sqrt(c)*(8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^2*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (c^(3/2)*(4*b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [x^2*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 6, -((2*b*c - a*d)*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(3*b*d) + (x*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*d) + (sqrt(c)*(2*b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) - (c^(3/2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [x^0*sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 5, x*sqrt((e*(a + b*x^2))/(c + d*x^2)) - (sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^2, x, 7, (d*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/c - (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(c*x) - (sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (b*sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^4, x, 7, (d*(b*c - 2*a*d)*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(3*a*c^2) - (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*c*x^3) - ((b*c - 2*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*a*c^2*x) - (sqrt(d)*(b*c - 2*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*c^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) - (b*sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [sqrt((e*(a + b*x^2))/(c + d*x^2))/x^6, x, 8, -(d*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(15*a^2*c^3) - (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*c*x^5) - ((b*c - 4*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(15*a*c^2*x^3) + ((2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(15*a^2*c^3*x) + (sqrt(d)*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*a^2*c^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) - (b*sqrt(d)*(b*c - 4*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*a^2*c^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]

    @test_int [x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 7, (c^2*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/d^4 + ((79*b^2*c^2 - 50*a*b*c*d - 5*a^2*d^2)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(48*b*d^4) - ((11*b*c + a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(24*d^4) + (((e*(a + b*x^2))/(c + d*x^2))^(5/2)*(c + d*x^2)^3)/(6*b*d^2*e) - ((b*c - a*d)*(35*b^2*c^2 - 10*a*b*c*d - a^2*d^2)*e^(3/2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(16*b^(3/2)*d^(9/2))]
    @test_int [x^3*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 6, -((c*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/d^3) - ((9*b*c - 5*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(8*d^3) + (b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(4*d^3) + (3*(b*c - a*d)*(5*b*c - a*d)*e^(3/2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(8*sqrt(b)*d^(7/2))]
    @test_int [x^1*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 5, (3*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(2*d^2) + (((e*(a + b*x^2))/(c + d*x^2))^(3/2)*(c + d*x^2))/(2*d) - (3*sqrt(b)*(b*c - a*d)*e^(3/2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(2*d^(5/2))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^1, x, 6, -(((b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d)) - (a^(3/2)*e^(3/2)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/c^(3/2) + (b^(3/2)*e^(3/2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/d^(3/2)]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^3, x, 5, (3*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(2*c^2) + ((b*c - a*d)*((e*(a + b*x^2))/(c + d*x^2))^(3/2))/(2*c*(a - (c*(a + b*x^2))/(c + d*x^2))) - (3*sqrt(a)*(b*c - a*d)*e^(3/2)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(2*c^(5/2))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^5, x, 6, -((d*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/c^3) - (a*(b*c - a*d)^2*e^3*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(4*c^3*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) + ((5*b*c - 9*a*d)*(b*c - a*d)*e^2*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(8*c^3*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) - (3*(b*c - 5*a*d)*(b*c - a*d)*e^(3/2)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(8*sqrt(a)*c^(7/2))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^7, x, 7, (d^2*(b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/c^4 + ((b*c - a*d)^3*e^2*((e*(a + b*x^2))/(c + d*x^2))^(5/2))/(6*a*c^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^3) + ((b*c - a*d)^2*(b*c + 11*a*d)*e^3*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(24*c^4*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((b*c - a*d)*(5*b^2*c^2 + 50*a*b*c*d - 79*a^2*d^2)*e^2*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(48*a*c^4*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) + ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*a^2*d^2)*e^(3/2)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(16*a^(3/2)*c^(9/2))]

    @test_int [x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 8, -((16*a*c - (16*b*c^2)/d - (a^2*d)/b)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(5*d^2) - (e*x^3*(a + b*x^2)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/d - ((8*b*c - 7*a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*d^3) + (6*b*e*x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*d^2) - (sqrt(c)*(16*b^2*c^2 - 16*a*b*c*d + a^2*d^2)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*b*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (c^(3/2)*(8*b*c - 7*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 7, -((8*b*c - 7*a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(3*d^2) - (e*x*(a + b*x^2)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/d + (4*b*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*d^2) + (sqrt(c)*(8*b*c - 7*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) - (sqrt(c)*(4*b*c - 3*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [x^0*((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 6, -(((b*c - a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d)) + ((2*b*c - a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d) - ((2*b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (b*sqrt(c)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^2, x, 7, -(((b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d*x)) - ((b*c - 2*a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/c^2 + ((b*c - 2*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(c^2*d*x) + ((b*c - 2*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(c^(3/2)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^4, x, 8, -(((b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d*x^3)) + (d*(7*b*c - 8*a*d)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(3*c^3) + ((3*b*c - 4*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*c^2*d*x^3) - ((7*b*c - 8*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(3*c^3*x) - (sqrt(d)*(7*b*c - 8*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*c^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) + (b*(3*b*c - 4*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*c^(3/2)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]
    @test_int [((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^6, x, 9, -(((b*c - a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(c*d*x^5)) + (d*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(5*a*c^4) + ((5*b*c - 6*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*c^2*d*x^5) - ((7*b*c - 8*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*c^3*x^3) - ((b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(5*a*c^4*x) - (sqrt(d)*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*a*c^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))) - (b*sqrt(d)*(7*b*c - 8*a*d)*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*a*c^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2))))]

    @test_int [x*sqrt((1 - x^2)/(1 + x^2)), x, 4, (1/2)*sqrt((1 - x^2)/(1 + x^2))*(1 + x^2) - atan(sqrt((1 - x^2)/(1 + x^2)))]
    @test_int [x*sqrt((5 - 7*x^2)/(7 + 5*x^2)), x, 4, (1/10)*sqrt((5 - 7*x^2)/(7 + 5*x^2))*(7 + 5*x^2) - (37*atan(sqrt(5/7)*sqrt((5 - 7*x^2)/(7 + 5*x^2))))/(5*sqrt(35))]
    @test_int [x^2*sqrt((1 - x^3)/(1 + x^3)), x, 4, (1/3)*sqrt((1 - x^3)/(1 + x^3))*(1 + x^3) - (2/3)*atan(sqrt((1 - x^3)/(1 + x^3)))]
    @test_int [x^8*sqrt((1 - x^3)/(1 + x^3)), x, 6, (1/2)*sqrt((1 - x^3)/(1 + x^3))*(1 + x^3) - (1/6)*sqrt((1 - x^3)/(1 + x^3))*(1 + x^3)^2 - (1/9)*((1 - x^3)/(1 + x^3))^(3/2)*(1 + x^3)^3 - (1/3)*atan(sqrt((1 - x^3)/(1 + x^3)))]
    @test_int [x^9*sqrt((5 - 7*x^5)/(7 + 5*x^5)), x, 5, (-(27/350))*sqrt((5 - 7*x^5)/(7 + 5*x^5))*(7 + 5*x^5) + (1/250)*sqrt((5 - 7*x^5)/(7 + 5*x^5))*(7 + 5*x^5)^2 + (2257*atan(sqrt(5/7)*sqrt((5 - 7*x^5)/(7 + 5*x^5))))/(875*sqrt(35))]

    @test_int [sqrt(x^2/(-1 + x^2))/(1 + x^2), x, 5, (sqrt(-(x^2/(1 - x^2)))*sqrt(-1 + x^2)*atan(sqrt(-1 + x^2)/sqrt(2)))/(sqrt(2)*x)]
    @test_int [sqrt(x^2/(-1 + a + (1 + a)*x^2))/(1 + x^2), x, 5, (sqrt(-(x^2/(1 - a - (1 + a)*x^2)))*sqrt(-1 + a + (1 + a)*x^2)*atan(sqrt(-1 + a + (1 + a)*x^2)/sqrt(2)))/(sqrt(2)*x)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 6, ((b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(16*b^3*d^2*e) - ((3*b*c + 5*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(24*b^2*d^2*e) - (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^3*(a - (c*(a + b*x^2))/(c + d*x^2)))/(6*b*d*(b*c - a*d)*e) + ((b*c - a*d)*(b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(16*b^(7/2)*d^(5/2)*sqrt(e))]
    @test_int [x^3/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 5, -((b*c + 3*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(8*b^2*d*e) + (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(4*b*d*e) - ((b*c - a*d)*(b*c + 3*a*d)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(8*b^(5/2)*d^(3/2)*sqrt(e))]
    @test_int [x^1/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 4, (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(2*b*e) + ((b*c - a*d)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(2*b^(3/2)*sqrt(d)*sqrt(e))]
    @test_int [1/(x^1*sqrt((e*(a + b*x^2))/(c + d*x^2))), x, 5, -((sqrt(c)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(sqrt(a)*sqrt(e))) + (sqrt(d)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(sqrt(b)*sqrt(e))]
    @test_int [1/(x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))), x, 4, ((b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(2*a*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) + ((b*c - a*d)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(2*a^(3/2)*sqrt(c)*sqrt(e))]
    @test_int [1/(x^5*sqrt((e*(a + b*x^2))/(c + d*x^2))), x, 5, -((b*c - a*d)^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(4*a*c*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((b*c - a*d)*(3*b*c + a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(8*a^2*c*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) - ((b*c - a*d)*(3*b*c + a*d)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(8*a^(5/2)*c^(3/2)*sqrt(e))]

    @test_int [x^4/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 7, ((b*c - 4*a*d)*x*(a + b*x^2))/(15*b^2*d*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (x^3*(a + b*x^2))/(5*b*sqrt((e*(a + b*x^2))/(c + d*x^2))) - ((2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*x*(a + b*x^2))/(15*b^3*d*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (sqrt(c)*(2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (c^(3/2)*(b*c - 4*a*d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^2*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [x^2/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 6, (x*(a + b*x^2))/(3*b*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((b*c - 2*a*d)*x*(a + b*x^2))/(3*b^2*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (sqrt(c)*(b*c - 2*a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b^2*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (c^(3/2)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [x^0/sqrt((e*(a + b*x^2))/(c + d*x^2)), x, 5, (d*x*(a + b*x^2))/(b*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (sqrt(c)*sqrt(d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(b*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (c^(3/2)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [1/(x^2*sqrt((e*(a + b*x^2))/(c + d*x^2))), x, 7, -((a + b*x^2)/(a*x*sqrt((e*(a + b*x^2))/(c + d*x^2)))) + (d*x*(a + b*x^2))/(a*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (sqrt(c)*sqrt(d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (sqrt(c)*sqrt(d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [1/(x^4*sqrt((e*(a + b*x^2))/(c + d*x^2))), x, 7, -(a + b*x^2)/(3*a*x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((2*b*c - a*d)*(a + b*x^2))/(3*a^2*c*x*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (d*(2*b*c - a*d)*x*(a + b*x^2))/(3*a^2*c*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (sqrt(d)*(2*b*c - a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*sqrt(c)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (b*sqrt(c)*sqrt(d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]

    @test_int [x^5/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 7, -((b^2*c^2 + 5*a*d*(2*b*c - 7*a*d))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(16*b^4*d*e^2) - ((b^2*c^2 + 5*a*d*(2*b*c - 7*a*d))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(24*b^3*d*(b*c - a*d)*e^2) - (a^2*(c + d*x^2)^3)/(b*(b*c - a*d)^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((b^2*c^2 - 2*a*b*c*d + 7*a^2*d^2)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^3)/(6*b^2*d*(b*c - a*d)^2*e^2) - ((b*c - a*d)*(b^2*c^2 + 5*a*d*(2*b*c - 7*a*d))*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(16*b^(9/2)*d^(3/2)*e^(3/2))]
    @test_int [x^3/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 6, (a*(b*c - a*d))/(b^3*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((3*b*c - 7*a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))/(8*b^3*e^2) + (sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)^2)/(4*b^2*e^2) + (3*(b*c - 5*a*d)*(b*c - a*d)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(8*b^(7/2)*sqrt(d)*e^(3/2))]
    @test_int [x^1/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 5, (-3*(b*c - a*d))/(2*b^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (c + d*x^2)/(2*b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (3*sqrt(d)*(b*c - a*d)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(2*b^(5/2)*e^(3/2))]
    @test_int [1/(x^1*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x, 6, (b*c - a*d)/(a*b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (c^(3/2)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(a^(3/2)*e^(3/2)) + (d^(3/2)*atanh((sqrt(d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(b)*sqrt(e))))/(b^(3/2)*e^(3/2))]
    @test_int [1/(x^3*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x, 5, (-3*(b*c - a*d))/(2*a^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (b*c - a*d)/(2*a*sqrt((e*(a + b*x^2))/(c + d*x^2))*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) + (3*sqrt(c)*(b*c - a*d)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(2*a^(5/2)*e^(3/2))]
    @test_int [1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x, 6, (b*(b*c - a*d))/(a^3*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) - ((b*c - a*d)^2*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(4*a^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((7*b*c - 3*a*d)*(b*c - a*d)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(8*a^3*(a*e^2 - (c*e^2*(a + b*x^2))/(c + d*x^2))) - (3*(b*c - a*d)*(5*b*c - a*d)*atanh((sqrt(c)*sqrt((e*(a + b*x^2))/(c + d*x^2)))/(sqrt(a)*sqrt(e))))/(8*a^(7/2)*sqrt(c)*e^(3/2))]

    @test_int [x^4/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 8, ((7*b*c - 8*a*d)*x*(a + b*x^2))/(5*b^3*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (6*d*x^3*(a + b*x^2))/(5*b^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*x*(a + b*x^2))/(5*b^4*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (x^3*(c + d*x^2))/(b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (sqrt(c)*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*b^4*sqrt(d)*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (c^(3/2)*(7*b*c - 8*a*d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(5*b^3*sqrt(d)*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [x^2/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 7, (4*d*x*(a + b*x^2))/(3*b^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (d*(7*b*c - 8*a*d)*x*(a + b*x^2))/(3*b^3*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (x*(c + d*x^2))/(b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (sqrt(c)*sqrt(d)*(7*b*c - 8*a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b^3*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (c^(3/2)*(3*b*c - 4*a*d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*b^2*sqrt(d)*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [x^0/((e*(a + b*x^2))/(c + d*x^2))^(3/2), x, 6, ((b*c - a*d)*x)/(a*b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (d*(b*c - 2*a*d)*x*(a + b*x^2))/(a*b^2*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (sqrt(c)*sqrt(d)*(b*c - 2*a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*b^2*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (c^(3/2)*sqrt(d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*b*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [1/(x^2*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x, 7, (b*c - a*d)/(a*b*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2))) - ((2*b*c - a*d)*(a + b*x^2))/(a^2*b*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2))) + (d*(2*b*c - a*d)*x*(a + b*x^2))/(a^2*b*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (sqrt(c)*sqrt(d)*(2*b*c - a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a^2*b*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (c^(3/2)*sqrt(d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a^2*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]
    @test_int [1/(x^4*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x, 8, (b*c - a*d)/(a*b*e*x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))) - ((4*b*c - 3*a*d)*(a + b*x^2))/(3*a^2*b*e*x^3*sqrt((e*(a + b*x^2))/(c + d*x^2))) + ((8*b*c - 7*a*d)*(a + b*x^2))/(3*a^3*e*x*sqrt((e*(a + b*x^2))/(c + d*x^2))) - (d*(8*b*c - 7*a*d)*x*(a + b*x^2))/(3*a^3*e*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) + (sqrt(c)*sqrt(d)*(8*b*c - 7*a*d)*(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^3*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2)) - (sqrt(c)*sqrt(d)*(4*b*c - 3*a*d)*(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^3*e*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt((e*(a + b*x^2))/(c + d*x^2))*(c + d*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b/(c+d*x^2))^(p/2)=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^5*sqrt(a + b/(c + d*x^2)), x, 7, -((b^2 + 4*a*b*c - 8*a^2*c^2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(16*a^2*d^3) - ((b + 4*a*c)*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*a*d^3) + ((c + d*x^2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(3/2))/(6*a*d^3) + (b*(b^2 + 4*a*b*c + 8*a^2*c^2)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(16*a^(5/2)*d^3)]
    @test_int [x^3*sqrt(a + b/(c + d*x^2)), x, 6, ((b - 4*a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*a*d^2) + ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*d^2) - (b*(b + 4*a*c)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(8*a^(3/2)*d^2)]
    @test_int [x^1*sqrt(a + b/(c + d*x^2)), x, 5, ((c + d*x^2)*sqrt(a + b/(c + d*x^2)))/(2*d) + (b*atanh(sqrt(a + b/(c + d*x^2))/sqrt(a)))/(2*sqrt(a)*d)]
    @test_int [sqrt(a + b/(c + d*x^2))/x^1, x, 6, sqrt(a)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)) - (sqrt(b + a*c)*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/sqrt(c)]
    @test_int [sqrt(a + b/(c + d*x^2))/x^3, x, 5, -((c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(2*c*x^2) + (b*d*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(2*c^(3/2)*sqrt(b + a*c))]
    @test_int [sqrt(a + b/(c + d*x^2))/x^5, x, 6, ((5*b + 4*a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*c^2*(b + a*c)*x^2) - ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*c^2*x^4) - (b*(3*b + 4*a*c)*d^2*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(8*c^(5/2)*(b + a*c)^(3/2))]
    @test_int [sqrt(a + b/(c + d*x^2))/x^7, x, 7, -((11*b^2 + 20*a*b*c + 8*a^2*c^2)*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(16*c^3*(b + a*c)^2*x^2) + ((3*b + 4*a*c)*d*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*c^3*(b + a*c)*x^4) - ((c + d*x^2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(3/2))/(6*c^2*(b + a*c)*x^6) + (b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*d^3*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(16*c^(7/2)*(b + a*c)^(5/2))]

    @test_int [x^4*sqrt(a + b/(c + d*x^2)), x, 8, -((2*b^2 + 7*a*b*c - 3*a^2*c^2)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(15*a^2*d^2) + ((b - 3*a*c)*x*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(15*a*d^2) + (x^3*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*d) + (sqrt(c)*(2*b^2 + 7*a*b*c - 3*a^2*c^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*a^2*d^(5/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (c^(3/2)*(b - 3*a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*a*d^(5/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^2*sqrt(a + b/(c + d*x^2)), x, 7, ((b - a*c)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*a*d) + (x*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*d) - (sqrt(c)*(b - a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*a*d^(3/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (c^(3/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*d^(3/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^0*sqrt(a + b/(c + d*x^2)), x, 6, x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)) - (sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(d)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(d)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [sqrt(a + b/(c + d*x^2))/x^2, x, 8, (d*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c - ((c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(c*x) - (sqrt(d)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (a*sqrt(c)*sqrt(d)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/((b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [sqrt(a + b/(c + d*x^2))/x^4, x, 8, -((2*b + a*c)*d^2*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c^2*(b + a*c)) - ((c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c*x^3) + ((2*b + a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c^2*(b + a*c)*x) + ((2*b + a*c)*d^(3/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*c^(3/2)*(b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (a*d^(3/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*sqrt(c)*(b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [sqrt(a + b/(c + d*x^2))/x^6, x, 9, ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^3*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(15*c^3*(b + a*c)^2) - ((c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*c*x^5) + ((4*b + 3*a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(15*c^2*(b + a*c)*x^3) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(15*c^3*(b + a*c)^2*x) - ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d^(5/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*c^(5/2)*(b + a*c)^2*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (a*(4*b + 3*a*c)*d^(5/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*c^(3/2)*(b + a*c)^2*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]

    @test_int [x^5*(a + b/(c + d*x^2))^(3/2), x, 8, -((b*c^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/d^3) - ((5*b^2 + 60*a*b*c - 24*a^2*c^2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(48*a*d^3) - ((b + 12*a*c)*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(24*d^3) + ((c + d*x^2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(5/2))/(6*a*d^3) - (b*(b^2 + 12*a*b*c - 24*a^2*c^2)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(16*a^(3/2)*d^3)]
    @test_int [x^3*(a + b/(c + d*x^2))^(3/2), x, 7, (b*c*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/d^2 + ((5*b - 4*a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*d^2) + (a*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*d^2) + (3*b*(b - 4*a*c)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(8*sqrt(a)*d^2)]
    @test_int [x^1*(a + b/(c + d*x^2))^(3/2), x, 6, (-3*b*sqrt(a + b/(c + d*x^2)))/(2*d) + ((c + d*x^2)*(a + b/(c + d*x^2))^(3/2))/(2*d) + (3*sqrt(a)*b*atanh(sqrt(a + b/(c + d*x^2))/sqrt(a)))/(2*d)]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^1, x, 7, (b*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c + a^(3/2)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)) - ((b + a*c)^(3/2)*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/c^(3/2)]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^3, x, 6, (-3*b*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(2*c^2) - ((c + d*x^2)*((b + a*c + a*d*x^2)/(c + d*x^2))^(3/2))/(2*c*x^2) + (3*b*sqrt(b + a*c)*d*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(2*c^(5/2))]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^5, x, 7, (b*d^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c^3 + ((9*b + 4*a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*c^3*x^2) - ((b + a*c)*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*c^3*x^4) - (3*b*(5*b + 4*a*c)*d^2*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(8*c^(7/2)*sqrt(b + a*c))]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^7, x, 8, -((b*d^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c^4) - ((79*b^2 + 108*a*b*c + 24*a^2*c^2)*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(48*c^4*(b + a*c)*x^2) + ((11*b + 12*a*c)*d*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(24*c^4*x^4) - ((c + d*x^2)^3*((b + a*c + a*d*x^2)/(c + d*x^2))^(5/2))/(6*c^2*(b + a*c)*x^6) + (b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*d^3*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(16*c^(9/2)*(b + a*c)^(3/2))]

    @test_int [x^4*(a + b/(c + d*x^2))^(3/2), x, 9, ((b^2 - 14*a*b*c + a^2*c^2)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*a*d^2) + ((7*b - a*c)*x*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*d^2) + (6*a*x^3*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*d) - (x^3*(b + a*c + a*d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/d - (sqrt(c)*(b^2 - 14*a*b*c + a^2*c^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*a*d^(5/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (c^(3/2)*(7*b - a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*d^(5/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^2*(a + b/(c + d*x^2))^(3/2), x, 8, ((7*b - a*c)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*d) + (4*a*x*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*d) - (x*(b + a*c + a*d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/d - (sqrt(c)*(7*b - a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*d^(3/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (sqrt(c)*(3*b - a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*d^(3/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^0*(a + b/(c + d*x^2))^(3/2), x, 7, (b*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c - ((b - a*c)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c + ((b - a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(c)*sqrt(d)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (a*sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(d)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^2, x, 8, (b*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(c*x) + ((2*b + a*c)*d*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/c^2 - ((2*b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(c^2*x) - ((2*b + a*c)*sqrt(d)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(c^(3/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (a*sqrt(d)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(sqrt(c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^4, x, 9, (b*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(c*x^3) - ((8*b + a*c)*d^2*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c^3) - ((4*b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c^2*x^3) + ((8*b + a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(3*c^3*x) + ((8*b + a*c)*d^(3/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*c^(5/2)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (a*(4*b + a*c)*d^(3/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*c^(3/2)*(b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [(a + b/(c + d*x^2))^(3/2)/x^6, x, 10, (b*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(c*x^5) + ((16*b^2 + 16*a*b*c + a^2*c^2)*d^3*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*c^4*(b + a*c)) - ((6*b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*c^2*x^5) + ((8*b + a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*c^3*x^3) - ((16*b^2 + 16*a*b*c + a^2*c^2)*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(5*c^4*(b + a*c)*x) - ((16*b^2 + 16*a*b*c + a^2*c^2)*d^(5/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*c^(7/2)*(b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (a*(8*b + a*c)*d^(5/2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*c^(5/2)*(b + a*c)*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^5/sqrt(a + b/(c + d*x^2)), x, 7, ((5*b^2 + 12*a*b*c + 8*a^2*c^2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(16*a^3*d^3) - ((5*b + 8*a*c)*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(24*a^2*d^3) + (x^2*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(6*a*d^2) - (b*(5*b^2 + 12*a*b*c + 8*a^2*c^2)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(16*a^(7/2)*d^3)]
    @test_int [x^3/sqrt(a + b/(c + d*x^2)), x, 6, -((3*b + 4*a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*a^2*d^2) + ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*a*d^2) + (b*(3*b + 4*a*c)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(8*a^(5/2)*d^2)]
    @test_int [x^1/sqrt(a + b/(c + d*x^2)), x, 5, ((c + d*x^2)*sqrt(a + b/(c + d*x^2)))/(2*a*d) - (b*atanh(sqrt(a + b/(c + d*x^2))/sqrt(a)))/(2*a^(3/2)*d)]
    @test_int [1/(x^1*sqrt(a + b/(c + d*x^2))), x, 6, atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a))/sqrt(a) - (sqrt(c)*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/sqrt(b + a*c)]
    @test_int [1/(x^3*sqrt(a + b/(c + d*x^2))), x, 5, -((c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(2*(b + a*c)*x^2) - (b*d*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(2*sqrt(c)*(b + a*c)^(3/2))]
    @test_int [1/(x^5*sqrt(a + b/(c + d*x^2))), x, 6, ((b + 4*a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*c*(b + a*c)^2*x^2) - ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*c*(b + a*c)*x^4) + (b*(b + 4*a*c)*d^2*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(8*c^(3/2)*(b + a*c)^(5/2))]

    @test_int [x^4/sqrt(a + b/(c + d*x^2)), x, 8, -((4*b + 3*a*c)*x*(b + a*c + a*d*x^2))/(15*a^2*d^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + (x^3*(b + a*c + a*d*x^2))/(5*a*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + ((8*b^2 + 13*a*b*c + 3*a^2*c^2)*x*(b + a*c + a*d*x^2))/(15*a^3*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*(8*b^2 + 13*a*b*c + 3*a^2*c^2)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*a^3*d^(5/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (c^(3/2)*(4*b + 3*a*c)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(15*a^2*d^(5/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^2/sqrt(a + b/(c + d*x^2)), x, 7, (x*(b + a*c + a*d*x^2))/(3*a*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((2*b + a*c)*x*(b + a*c + a*d*x^2))/(3*a^2*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + (sqrt(c)*(2*b + a*c)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*a^2*d^(3/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (c^(3/2)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*a*d^(3/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^0/sqrt(a + b/(c + d*x^2)), x, 6, (x*(b + a*c + a*d*x^2))/(a*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(a*sqrt(d)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (c^(3/2)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/((b + a*c)*sqrt(d)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [1/(x^2*sqrt(a + b/(c + d*x^2))), x, 8, -((b + a*c + a*d*x^2)/((b + a*c)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + (d*x*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*sqrt(d)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/((b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (sqrt(c)*sqrt(d)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/((b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [1/(x^4*sqrt(a + b/(c + d*x^2))), x, 8, -(b + a*c + a*d*x^2)/(3*(b + a*c)*x^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((b - a*c)*d*(b + a*c + a*d*x^2))/(3*c*(b + a*c)^2*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + ((b - a*c)*d^2*x*(b + a*c + a*d*x^2))/(3*c*(b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((b - a*c)*d^(3/2)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*sqrt(c)*(b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (a*sqrt(c)*d^(3/2)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*(b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]

    @test_int [x^5/(a + b/(c + d*x^2))^(3/2), x, 8, -(((b + a*c)^2*(c + d*x^2)^3)/(a*b^2*d^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(16*a^4*d^3) - ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*(c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(24*a^3*b*d^3) + ((7*b^2 + 12*a*b*c + 6*a^2*c^2)*(c + d*x^2)^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(6*a^2*b^2*d^3) - (b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(16*a^(9/2)*d^3)]
    @test_int [x^3/(a + b/(c + d*x^2))^(3/2), x, 7, -((b*(b + a*c))/(a^3*d^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) - ((7*b + 4*a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*a^3*d^2) + ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*a^2*d^2) + (3*b*(5*b + 4*a*c)*atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a)))/(8*a^(7/2)*d^2)]
    @test_int [x^1/(a + b/(c + d*x^2))^(3/2), x, 6, (3*b)/(2*a^2*d*sqrt(a + b/(c + d*x^2))) + (c + d*x^2)/(2*a*d*sqrt(a + b/(c + d*x^2))) - (3*b*atanh(sqrt(a + b/(c + d*x^2))/sqrt(a)))/(2*a^(5/2)*d)]
    @test_int [1/(x^1*(a + b/(c + d*x^2))^(3/2)), x, 7, -(b/(a*(b + a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + atanh(sqrt((b + a*c + a*d*x^2)/(c + d*x^2))/sqrt(a))/a^(3/2) - (c^(3/2)*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(b + a*c)^(3/2)]
    @test_int [1/(x^3*(a + b/(c + d*x^2))^(3/2)), x, 6, (3*b*d)/(2*(b + a*c)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (c + d*x^2)/(2*(b + a*c)*x^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (3*b*sqrt(c)*d*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(2*(b + a*c)^(5/2))]
    @test_int [1/(x^5*(a + b/(c + d*x^2))^(3/2)), x, 7, -((a*b*d^2)/((b + a*c)^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) - ((3*b - 4*a*c)*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(8*(b + a*c)^3*x^2) - ((c + d*x^2)^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/(4*(b + a*c)^2*x^4) - (3*b*(b - 4*a*c)*d^2*atanh((sqrt(c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))/sqrt(b + a*c)))/(8*sqrt(c)*(b + a*c)^(7/2))]

    @test_int [x^4/(a + b/(c + d*x^2))^(3/2), x, 9, -((x^3*(c + d*x^2))/(a*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) - ((8*b + a*c)*x*(b + a*c + a*d*x^2))/(5*a^3*d^2*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + (6*x^3*(b + a*c + a*d*x^2))/(5*a^2*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + ((16*b^2 + 16*a*b*c + a^2*c^2)*x*(b + a*c + a*d*x^2))/(5*a^4*d^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*(16*b^2 + 16*a*b*c + a^2*c^2)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*a^4*d^(5/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (c^(3/2)*(8*b + a*c)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(5*a^3*d^(5/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^2/(a + b/(c + d*x^2))^(3/2), x, 8, -((x*(c + d*x^2))/(a*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + (4*x*(b + a*c + a*d*x^2))/(3*a^2*d*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((8*b + a*c)*x*(b + a*c + a*d*x^2))/(3*a^3*d*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + (sqrt(c)*(8*b + a*c)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*a^3*d^(3/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) - (c^(3/2)*(4*b + a*c)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*a^2*(b + a*c)*d^(3/2)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [x^0/(a + b/(c + d*x^2))^(3/2), x, 7, -((b*x)/(a*(b + a*c)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + ((2*b + a*c)*x*(b + a*c + a*d*x^2))/(a^2*(b + a*c)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*(2*b + a*c)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(a^2*(b + a*c)*sqrt(d)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (c^(3/2)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(a*(b + a*c)*sqrt(d)*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [1/(x^2*(a + b/(c + d*x^2))^(3/2)), x, 8, -(b/(a*(b + a*c)*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + ((b - a*c)*(b + a*c + a*d*x^2))/(a*(b + a*c)^2*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((b - a*c)*d*x*(b + a*c + a*d*x^2))/(a*(b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + (sqrt(c)*(b - a*c)*sqrt(d)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(a*(b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (c^(3/2)*sqrt(d)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/((b + a*c)^2*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]
    @test_int [1/(x^4*(a + b/(c + d*x^2))^(3/2)), x, 9, -(b/(a*(b + a*c)*x^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2)))) + ((3*b - a*c)*(b + a*c + a*d*x^2))/(3*a*(b + a*c)^2*x^3*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - ((7*b - a*c)*d*(b + a*c + a*d*x^2))/(3*(b + a*c)^3*x*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) + ((7*b - a*c)*d^2*x*(b + a*c + a*d*x^2))/(3*(b + a*c)^3*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))) - (sqrt(c)*(7*b - a*c)*d^(3/2)*(b + a*c + a*d*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*(b + a*c)^3*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2)))) + (sqrt(c)*(3*b - a*c)*d^(3/2)*(b + a*c + a*d*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), b/(b + a*c)))/(3*(b + a*c)^3*(c + d*x^2)*sqrt((b + a*c + a*d*x^2)/(c + d*x^2))*sqrt((c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(a*x^n)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*u*(a*x^n)^p=#

    #= Integrands*of*the*form*sqrt(a*x^n)/sqrt(1+x^5)*where*n*mod*10 = 3 =#
    @test_int [sqrt(a*x^23)/sqrt(1 + x^5), x, 6, -((3*sqrt(a*x^23)*sqrt(1 + x^5))/(20*x^9)) + (sqrt(a*x^23)*sqrt(1 + x^5))/(10*x^4) + (3*sqrt(a*x^23)*asinh(x^(5/2)))/(20*x^(23/2))]
    @test_int [sqrt(a*x^13)/sqrt(1 + x^5), x, 5, (sqrt(a*x^13)*sqrt(1 + x^5))/(5*x^4) - (sqrt(a*x^13)*asinh(x^(5/2)))/(5*x^(13/2))]
    @test_int [sqrt(a*x^3)/sqrt(1 + x^5), x, 4, (2*sqrt(a*x^3)*asinh(x^(5/2)))/(5*x^(3/2))]
    @test_int [sqrt(a/x^7)/sqrt(1 + x^5), x, 2, (-(2/5))*sqrt(a/x^7)*x*sqrt(1 + x^5)]
    @test_int [sqrt(a/x^17)/sqrt(1 + x^5), x, 3, (-(2/15))*sqrt(a/x^17)*x*sqrt(1 + x^5) + (4/15)*sqrt(a/x^17)*x^6*sqrt(1 + x^5)]

    @test_int [sqrt(a*x^6)/(x*(1 - x^4)), x, 4, -((sqrt(a*x^6)*atan(x))/(2*x^3)) + (sqrt(a*x^6)*atanh(x))/(2*x^3)]
    @test_int [sqrt(a*x^6)/(x - x^5), x, 5, -((sqrt(a*x^6)*atan(x))/(2*x^3)) + (sqrt(a*x^6)*atanh(x))/(2*x^3)]
    @test_int [(a*x^6)^(3/2)/(x*(1 - x^4)), x, 6, -((a*sqrt(a*x^6))/x^2) - (1/5)*a*x^2*sqrt(a*x^6) + (a*sqrt(a*x^6)*atan(x))/(2*x^3) + (a*sqrt(a*x^6)*atanh(x))/(2*x^3)]

    @test_int [1/(1 - x^4) - sqrt(a*x^6)/(x*(1 - x^4)), x, 8, atan(x)/2 + (sqrt(a*x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(a*x^6)*atanh(x))/(2*x^3)]
    @test_int [1/(1 - x^4) - sqrt(a*x^6)/(x - x^5), x, 9, atan(x)/2 + (sqrt(a*x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(a*x^6)*atanh(x))/(2*x^3)]

    @test_int [sqrt(a*x^3)/(x - x^3), x, 6, -((sqrt(a*x^3)*atan(sqrt(x)))/x^(3/2)) + (sqrt(a*x^3)*atanh(sqrt(x)))/x^(3/2)]

    @test_int [sqrt(a*x^4)/sqrt(1 + x^2), x, 3, (sqrt(a*x^4)*sqrt(1 + x^2))/(2*x) - (sqrt(a*x^4)*asinh(x))/(2*x^2)]
    @test_int [sqrt(a*x^3)/sqrt(1 + x^2), x, 4, (2*sqrt(a*x^3)*sqrt(1 + x^2))/(3*x) - (sqrt(a*x^3)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.F(2*atan(sqrt(x)), 1/2))/(3*x^(3/2)*sqrt(1 + x^2))]
    @test_int [sqrt(a*x^2)/sqrt(1 + x^2), x, 2, (sqrt(a*x^2)*sqrt(1 + x^2))/x]
    @test_int [sqrt(a*x^1)/sqrt(1 + x^2), x, 4, (2*sqrt(a*x)*sqrt(1 + x^2))/(1 + x) - (2*sqrt(a)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.E(2*atan(sqrt(a*x)/sqrt(a)), 1/2))/sqrt(1 + x^2) + (sqrt(a)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.F(2*atan(sqrt(a*x)/sqrt(a)), 1/2))/sqrt(1 + x^2)]
    @test_int [sqrt(a/x^1)/sqrt(1 + x^2), x, 3, (sqrt(a/x)*sqrt(x)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.F(2*atan(sqrt(x)), 1/2))/sqrt(1 + x^2)]
    @test_int [sqrt(a/x^2)/sqrt(1 + x^2), x, 4, (-sqrt(a/x^2))*x*atanh(sqrt(1 + x^2))]
    @test_int [sqrt(a/x^3)/sqrt(1 + x^2), x, 6, -2*sqrt(a/x^3)*x*sqrt(1 + x^2) + (2*sqrt(a/x^3)*x^2*sqrt(1 + x^2))/(1 + x) - (2*sqrt(a/x^3)*x^(3/2)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.E(2*atan(sqrt(x)), 1/2))/sqrt(1 + x^2) + (sqrt(a/x^3)*x^(3/2)*(1 + x)*sqrt((1 + x^2)/(1 + x)^2)*Elliptic.F(2*atan(sqrt(x)), 1/2))/sqrt(1 + x^2)]
    @test_int [sqrt(a/x^4)/sqrt(1 + x^2), x, 2, (-sqrt(a/x^4))*x*sqrt(1 + x^2)]

    @test_int [sqrt(a*x^4)/sqrt(1 + x^3), x, 2, (2*sqrt(a*x^4)*sqrt(1 + x^3))/(3*x^2)]
    @test_int [sqrt(a*x^3)/sqrt(1 + x^3), x, 5, ((1 + sqrt(3))*sqrt(a*x^3)*sqrt(1 + x^3))/(x*(1 + (1 + sqrt(3))*x)) - (3^(1/4)*sqrt(a*x^3)*(1 + x)*sqrt((1 - x + x^2)/(1 + (1 + sqrt(3))*x)^2)*Elliptic.E(acos((1 + (1 - sqrt(3))*x)/(1 + (1 + sqrt(3))*x)), (1/4)*(2 + sqrt(3))))/(x*sqrt((x*(1 + x))/(1 + (1 + sqrt(3))*x)^2)*sqrt(1 + x^3)) - ((1 - sqrt(3))*sqrt(a*x^3)*(1 + x)*sqrt((1 - x + x^2)/(1 + (1 + sqrt(3))*x)^2)*Elliptic.F(acos((1 + (1 - sqrt(3))*x)/(1 + (1 + sqrt(3))*x)), (1/4)*(2 + sqrt(3))))/(2*3^(1/4)*x*sqrt((x*(1 + x))/(1 + (1 + sqrt(3))*x)^2)*sqrt(1 + x^3))]
    @test_int [sqrt(a*x^2)/sqrt(1 + x^3), x, 4, (2*sqrt(a*x^2)*sqrt(1 + x^3))/(x*(1 + sqrt(3) + x)) - (3^(1/4)*sqrt(2 - sqrt(3))*sqrt(a*x^2)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(x*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (2*sqrt(2)*sqrt(a*x^2)*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*x*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]
    @test_int [sqrt(a*x^1)/sqrt(1 + x^3), x, 3, (2/3)*sqrt(a)*asinh((a*x)^(3/2)/a^(3/2))]
    @test_int [sqrt(a/x^1)/sqrt(1 + x^3), x, 3, (sqrt(a/x)*x*(1 + x)*sqrt((1 - x + x^2)/(1 + (1 + sqrt(3))*x)^2)*Elliptic.F(acos((1 + (1 - sqrt(3))*x)/(1 + (1 + sqrt(3))*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*sqrt((x*(1 + x))/(1 + (1 + sqrt(3))*x)^2)*sqrt(1 + x^3))]
    @test_int [sqrt(a/x^2)/sqrt(1 + x^3), x, 4, (-(2/3))*sqrt(a/x^2)*x*atanh(sqrt(1 + x^3))]
    @test_int [sqrt(a/x^3)/sqrt(1 + x^3), x, 6, -2*sqrt(a/x^3)*x*sqrt(1 + x^3) + (2*(1 + sqrt(3))*sqrt(a/x^3)*x^2*sqrt(1 + x^3))/(1 + (1 + sqrt(3))*x) - (2*3^(1/4)*sqrt(a/x^3)*x^2*(1 + x)*sqrt((1 - x + x^2)/(1 + (1 + sqrt(3))*x)^2)*Elliptic.E(acos((1 + (1 - sqrt(3))*x)/(1 + (1 + sqrt(3))*x)), (1/4)*(2 + sqrt(3))))/(sqrt((x*(1 + x))/(1 + (1 + sqrt(3))*x)^2)*sqrt(1 + x^3)) - ((1 - sqrt(3))*sqrt(a/x^3)*x^2*(1 + x)*sqrt((1 - x + x^2)/(1 + (1 + sqrt(3))*x)^2)*Elliptic.F(acos((1 + (1 - sqrt(3))*x)/(1 + (1 + sqrt(3))*x)), (1/4)*(2 + sqrt(3))))/(3^(1/4)*sqrt((x*(1 + x))/(1 + (1 + sqrt(3))*x)^2)*sqrt(1 + x^3))]
    @test_int [sqrt(a/x^4)/sqrt(1 + x^3), x, 5, (-sqrt(a/x^4))*x*sqrt(1 + x^3) + (sqrt(a/x^4)*x^2*sqrt(1 + x^3))/(1 + sqrt(3) + x) - (3^(1/4)*sqrt(2 - sqrt(3))*sqrt(a/x^4)*x^2*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.E(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(2*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3)) + (sqrt(2)*sqrt(a/x^4)*x^2*(1 + x)*sqrt((1 - x + x^2)/(1 + sqrt(3) + x)^2)*Elliptic.F(asin((1 - sqrt(3) + x)/(1 + sqrt(3) + x)), -7 - 4*sqrt(3)))/(3^(1/4)*sqrt((1 + x)/(1 + sqrt(3) + x)^2)*sqrt(1 + x^3))]

    @test_int [sqrt(a*x^(2*n))/sqrt(1 + x^n), x, 2, (x*sqrt(a*x^(2*n))*HypergeometricFunctions._₂F₁(1/2, 1 + 1/n, 2 + 1/n, -x^n))/(1 + n)]
    @test_int [sqrt(a*x^n)/sqrt(1 + x^n), x, 2, (2*x*sqrt(a*x^n)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + 2/n), (1/2)*(3 + 2/n), -x^n))/(2 + n)]
    @test_int [sqrt(a*x^(n/2))/sqrt(1 + x^n), x, 2, (4*x*sqrt(a*x^(n/2))*HypergeometricFunctions._₂F₁(1/2, (1/4)*(1 + 4/n), (1/4)*(5 + 4/n), -x^n))/(4 + n)]

    @test_int [sqrt(a*x^(2*n))/sqrt(1 + x^n) + (2*sqrt(a*x^(2*n)))/(x^n*((2 + n)*sqrt(1 + x^n))), x, -5, (2*x^(1 - n)*sqrt(a*x^(2*n))*sqrt(1 + x^n))/(2 + n)]

    @test_int [sqrt(a*x)/(sqrt(d + e*x)*sqrt(e + f*x)), x, 2, (2*sqrt(-e^2 + d*f)*sqrt(a*x)*sqrt((e*(e + f*x))/(e^2 - d*f))*Elliptic.E(asin((sqrt(f)*sqrt(d + e*x))/sqrt(-e^2 + d*f)), 1 - e^2/(d*f)))/(e*sqrt(f)*sqrt(-((e*x)/d))*sqrt(e + f*x))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a*x^n)^p*(b*x^m)^q ...=#

    @test_int [(a*x^m)^r, x, 2, (x*(a*x^m)^r)/(1 + m*r)]
    @test_int [(a*x^m)^r*(b*x^n)^s, x, 3, (x*(a*x^m)^r*(b*x^n)^s)/(1 + m*r + n*s)]
    @test_int [(a*x^m)^r*(b*x^n)^s*(c*x^p)^t, x, 4, (x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t)/(1 + m*r + n*s + p*t)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(sqrt(a+b*x) + sqrt(c+d*x))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(sqrt(a+b*x) + sqrt(c+b*x))^p=#

    #= ::Subsubsection:: =#
    #=p>0=#

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^2/(sqrt(a + b*x) + sqrt(c + b*x)), x, 5, (2*a^2*(a + b*x)^(3/2))/(3*b^3*(a - c)) - (4*a*(a + b*x)^(5/2))/(5*b^3*(a - c)) + (2*(a + b*x)^(7/2))/(7*b^3*(a - c)) - (2*c^2*(c + b*x)^(3/2))/(3*b^3*(a - c)) + (4*c*(c + b*x)^(5/2))/(5*b^3*(a - c)) - (2*(c + b*x)^(7/2))/(7*b^3*(a - c))]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(c + b*x)), x, 5, -((2*a*(a + b*x)^(3/2))/(3*b^2*(a - c))) + (2*(a + b*x)^(5/2))/(5*b^2*(a - c)) + (2*c*(c + b*x)^(3/2))/(3*b^2*(a - c)) - (2*(c + b*x)^(5/2))/(5*b^2*(a - c))]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(c + b*x)), x, 2, (2*(a + b*x)^(3/2))/(3*b*(a - c)) - (2*(c + b*x)^(3/2))/(3*b*(a - c))]
    @test_int [1/(x^1*(sqrt(a + b*x) + sqrt(c + b*x))), x, 7, (2*sqrt(a + b*x))/(a - c) - (2*sqrt(c + b*x))/(a - c) - (2*sqrt(a)*atanh(sqrt(a + b*x)/sqrt(a)))/(a - c) + (2*sqrt(c)*atanh(sqrt(c + b*x)/sqrt(c)))/(a - c)]
    @test_int [1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))), x, 7, -(sqrt(a + b*x)/((a - c)*x)) + sqrt(c + b*x)/((a - c)*x) - (b*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(a - c)) + (b*atanh(sqrt(c + b*x)/sqrt(c)))/((a - c)*sqrt(c))]

    @test_int [x^2/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, 9, ((a + c)*x^3)/(3*(a - c)^2) + (b*x^4)/(2*(a - c)^2) - ((4*a*c - 5*(a + c)^2)*sqrt(a + b*x)*sqrt(c + b*x))/(32*b^3*(a - c)) + ((4*a*c - 5*(a + c)^2)*(a + b*x)^(3/2)*sqrt(c + b*x))/(16*b^3*(a - c)^2) + (5*(a + c)*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(12*b^3*(a - c)^2) - (x*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(2*b^2*(a - c)^2) - ((4*a*c - 5*(a + c)^2)*atanh(sqrt(a + b*x)/sqrt(c + b*x)))/(32*b^3)]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, 8, ((a + c)*x^2)/(2*(a - c)^2) + (2*b*x^3)/(3*(a - c)^2) - ((a + c)*sqrt(a + b*x)*sqrt(c + b*x))/(4*b^2*(a - c)) + ((a + c)*(a + b*x)^(3/2)*sqrt(c + b*x))/(2*b^2*(a - c)^2) - (2*(a + b*x)^(3/2)*(c + b*x)^(3/2))/(3*b^2*(a - c)^2) - ((a + c)*atanh(sqrt(a + b*x)/sqrt(c + b*x)))/(4*b^2)]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(c + b*x))^2, x, 7, (a - c)^2/(8*b*(sqrt(a + b*x) + sqrt(c + b*x))^4) + atanh(sqrt(a + b*x)/sqrt(c + b*x))/(2*b), ((a + c)*x)/(a - c)^2 + (b*x^2)/(a - c)^2 + (sqrt(a + b*x)*sqrt(c + b*x))/(2*b*(a - c)) - ((a + b*x)^(3/2)*sqrt(c + b*x))/(b*(a - c)^2) + atanh(sqrt(a + b*x)/sqrt(c + b*x))/(2*b)]
    @test_int [1/(x^1*(sqrt(a + b*x) + sqrt(c + b*x))^2), x, 9, (2*b*x)/(a - c)^2 - (2*sqrt(a + b*x)*sqrt(c + b*x))/(a - c)^2 - (2*(a + c)*atanh(sqrt(a + b*x)/sqrt(c + b*x)))/(a - c)^2 + (4*sqrt(a)*sqrt(c)*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + b*x))))/(a - c)^2 + ((a + c)*log(x))/(a - c)^2]
    @test_int [1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))^2), x, 9, -((a + c)/((a - c)^2*x)) + (2*sqrt(a + b*x)*sqrt(c + b*x))/((a - c)^2*x) - (4*b*atanh(sqrt(a + b*x)/sqrt(c + b*x)))/(a - c)^2 + (2*b*(a + c)*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(a)*sqrt(c + b*x))))/(sqrt(a)*(a - c)^2*sqrt(c)) + (2*b*log(x))/(a - c)^2]

    @test_int [x^2/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, 10, -((8*a^3*(a + b*x)^(3/2))/(3*b^3*(a - c)^3)) + (2*a^2*(a + 3*c)*(a + b*x)^(3/2))/(3*b^3*(a - c)^3) + (24*a^2*(a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (4*a*(a + 3*c)*(a + b*x)^(5/2))/(5*b^3*(a - c)^3) - (24*a*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(7/2))/(7*b^3*(a - c)^3) + (8*(a + b*x)^(9/2))/(9*b^3*(a - c)^3) + (8*c^3*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (2*c^2*(3*a + c)*(c + b*x)^(3/2))/(3*b^3*(a - c)^3) - (24*c^2*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (4*c*(3*a + c)*(c + b*x)^(5/2))/(5*b^3*(a - c)^3) + (24*c*(c + b*x)^(7/2))/(7*b^3*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(7/2))/(7*b^3*(a - c)^3) - (8*(c + b*x)^(9/2))/(9*b^3*(a - c)^3)]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, 10, (8*a^2*(a + b*x)^(3/2))/(3*b^2*(a - c)^3) - (2*a*(a + 3*c)*(a + b*x)^(3/2))/(3*b^2*(a - c)^3) - (16*a*(a + b*x)^(5/2))/(5*b^2*(a - c)^3) + (2*(a + 3*c)*(a + b*x)^(5/2))/(5*b^2*(a - c)^3) + (8*(a + b*x)^(7/2))/(7*b^2*(a - c)^3) - (8*c^2*(c + b*x)^(3/2))/(3*b^2*(a - c)^3) + (2*c*(3*a + c)*(c + b*x)^(3/2))/(3*b^2*(a - c)^3) + (16*c*(c + b*x)^(5/2))/(5*b^2*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(5/2))/(5*b^2*(a - c)^3) - (8*(c + b*x)^(7/2))/(7*b^2*(a - c)^3)]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(c + b*x))^3, x, 6, (a - c)^2/(10*b*(sqrt(a + b*x) + sqrt(c + b*x))^5) - 1/(2*b*(sqrt(a + b*x) + sqrt(c + b*x))), -((8*a*(a + b*x)^(3/2))/(3*b*(a - c)^3)) + (2*(a + 3*c)*(a + b*x)^(3/2))/(3*b*(a - c)^3) + (8*(a + b*x)^(5/2))/(5*b*(a - c)^3) + (8*c*(c + b*x)^(3/2))/(3*b*(a - c)^3) - (2*(3*a + c)*(c + b*x)^(3/2))/(3*b*(a - c)^3) - (8*(c + b*x)^(5/2))/(5*b*(a - c)^3)]
    @test_int [1/(x^1*(sqrt(a + b*x) + sqrt(c + b*x))^3), x, 8, (2*(a + 3*c)*sqrt(a + b*x))/(a - c)^3 + (8*(a + b*x)^(3/2))/(3*(a - c)^3) - (2*(3*a + c)*sqrt(c + b*x))/(a - c)^3 - (8*(c + b*x)^(3/2))/(3*(a - c)^3) - (2*sqrt(a)*(a + 3*c)*atanh(sqrt(a + b*x)/sqrt(a)))/(a - c)^3 + (2*sqrt(c)*(3*a + c)*atanh(sqrt(c + b*x)/sqrt(c)))/(a - c)^3]
    @test_int [1/(x^2*(sqrt(a + b*x) + sqrt(c + b*x))^3), x, 14, (8*b*sqrt(a + b*x))/(a - c)^3 - ((a + 3*c)*sqrt(a + b*x))/((a - c)^3*x) - (8*b*sqrt(c + b*x))/(a - c)^3 + ((3*a + c)*sqrt(c + b*x))/((a - c)^3*x) - (3*b*(3*a + c)*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(a - c)^3) - (3*b*(a + 3*c)*atanh(sqrt(c + b*x)/sqrt(c)))/(sqrt(c)*(-a + c)^3), (8*b*sqrt(a + b*x))/(a - c)^3 - ((a + 3*c)*sqrt(a + b*x))/((a - c)^3*x) - (8*b*sqrt(c + b*x))/(a - c)^3 + ((3*a + c)*sqrt(c + b*x))/((a - c)^3*x) - (8*sqrt(a)*b*atanh(sqrt(a + b*x)/sqrt(a)))/(a - c)^3 - (b*(a + 3*c)*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(a - c)^3) + (8*b*sqrt(c)*atanh(sqrt(c + b*x)/sqrt(c)))/(a - c)^3 + (b*(3*a + c)*atanh(sqrt(c + b*x)/sqrt(c)))/((a - c)^3*sqrt(c))]

    @test_int [1/(sqrt(x) + sqrt(1 + x)), x, 3, -((2*x^(3/2))/3) + (2/3)*(1 + x)^(3/2)]
    @test_int [1/(sqrt(x) + sqrt(-1 + x)), x, 3, (-(2/3))*(-1 + x)^(3/2) + (2*x^(3/2))/3]

    @test_int [1/(sqrt(-1 + x) + sqrt(1 + x)), x, 2, (-(1/3))*(-1 + x)^(3/2) + (1/3)*(1 + x)^(3/2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(sqrt(a+b*x) + sqrt(a+c*x))^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*(sqrt(1 - x) + sqrt(1 + x))^2, x, 5, x^4/2 - (2/3)*(1 - x^2)^(3/2) + (2/5)*(1 - x^2)^(5/2)]
    @test_int [x^2*(sqrt(1 - x) + sqrt(1 + x))^2, x, 5, (2*x^3)/3 - (1/4)*x*sqrt(1 - x^2) + (1/2)*x^3*sqrt(1 - x^2) + asin(x)/4]
    @test_int [x^1*(sqrt(1 - x) + sqrt(1 + x))^2, x, 3, x^2 - (2/3)*(1 - x^2)^(3/2)]
    @test_int [x^0*(sqrt(1 - x) + sqrt(1 + x))^2, x, 4, 2*x + x*sqrt(1 - x^2) + asin(x)]
    @test_int [(sqrt(1 - x) + sqrt(1 + x))^2/x^1, x, 6, 2*sqrt(1 - x^2) - 2*atanh(sqrt(1 - x^2)) + 2*log(x)]
    @test_int [(sqrt(1 - x) + sqrt(1 + x))^2/x^2, x, 4, -(2/x) - (2*sqrt(1 - x^2))/x - 2*asin(x)]
    @test_int [(sqrt(1 - x) + sqrt(1 + x))^2/x^3, x, 6, -(1/x^2) - sqrt(1 - x^2)/x^2 + atanh(sqrt(1 - x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3/(sqrt(a + b*x) + sqrt(a + c*x)), x, 5, (2*a^2*(a + b*x)^(3/2))/(3*b^3*(b - c)) - (4*a*(a + b*x)^(5/2))/(5*b^3*(b - c)) + (2*(a + b*x)^(7/2))/(7*b^3*(b - c)) - (2*a^2*(a + c*x)^(3/2))/(3*(b - c)*c^3) + (4*a*(a + c*x)^(5/2))/(5*(b - c)*c^3) - (2*(a + c*x)^(7/2))/(7*(b - c)*c^3)]
    @test_int [x^2/(sqrt(a + b*x) + sqrt(a + c*x)), x, 5, -((2*a*(a + b*x)^(3/2))/(3*b^2*(b - c))) + (2*(a + b*x)^(5/2))/(5*b^2*(b - c)) + (2*a*(a + c*x)^(3/2))/(3*(b - c)*c^2) - (2*(a + c*x)^(5/2))/(5*(b - c)*c^2)]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(a + c*x)), x, 3, (2*(a + b*x)^(3/2))/(3*b*(b - c)) - (2*(a + c*x)^(3/2))/(3*(b - c)*c)]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(a + c*x)), x, 8, (2*sqrt(a + b*x))/(b - c) - (2*sqrt(a + c*x))/(b - c) - (2*sqrt(a)*atanh(sqrt(a + b*x)/sqrt(a)))/(b - c) + (2*sqrt(a)*atanh(sqrt(a + c*x)/sqrt(a)))/(b - c)]
    @test_int [1/(x^1*(sqrt(a + b*x) + sqrt(a + c*x))), x, 7, -(sqrt(a + b*x)/((b - c)*x)) + sqrt(a + c*x)/((b - c)*x) - (b*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)) + (c*atanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c))]
    @test_int [1/(x^2*(sqrt(a + b*x) + sqrt(a + c*x))), x, 9, -(sqrt(a + b*x)/(2*(b - c)*x^2)) - (b*sqrt(a + b*x))/(4*a*(b - c)*x) + sqrt(a + c*x)/(2*(b - c)*x^2) + (c*sqrt(a + c*x))/(4*a*(b - c)*x) + (b^2*atanh(sqrt(a + b*x)/sqrt(a)))/(4*a^(3/2)*(b - c)) - (c^2*atanh(sqrt(a + c*x)/sqrt(a)))/(4*a^(3/2)*(b - c))]

    @test_int [x^3/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 8, (a*x^2)/(b - c)^2 + ((b + c)*x^3)/(3*(b - c)^2) + (a^2*(b + c)*sqrt(a + b*x)*sqrt(a + c*x))/(4*b^2*(b - c)*c^2) + (a*(b + c)*(a + b*x)^(3/2)*sqrt(a + c*x))/(2*b^2*(b - c)^2*c) - (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*b*(b - c)^2*c) - (a^3*(b + c)*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(4*b^(5/2)*c^(5/2))]
    @test_int [x^2/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 7, (2*a*x)/(b - c)^2 + ((b + c)*x^2)/(2*(b - c)^2) - (a*sqrt(a + b*x)*sqrt(a + c*x))/(2*b*(b - c)*c) - ((a + b*x)^(3/2)*sqrt(a + c*x))/(b*(b - c)^2) + (a^2*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(2*b^(3/2)*c^(3/2))]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 9, ((b + c)*x)/(b - c)^2 - (2*sqrt(a + b*x)*sqrt(a + c*x))/(b - c)^2 + (4*a*atanh(sqrt(a + b*x)/sqrt(a + c*x)))/(b - c)^2 - (2*a*(b + c)*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(sqrt(b)*(b - c)^2*sqrt(c)) + (2*a*log(x))/(b - c)^2]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(a + c*x))^2, x, 9, -((2*a)/((b - c)^2*x)) + (2*sqrt(a + b*x)*sqrt(a + c*x))/((b - c)^2*x) + (2*(b + c)*atanh(sqrt(a + b*x)/sqrt(a + c*x)))/(b - c)^2 - (4*sqrt(b)*sqrt(c)*atanh((sqrt(c)*sqrt(a + b*x))/(sqrt(b)*sqrt(a + c*x))))/(b - c)^2 + ((b + c)*log(x))/(b - c)^2]
    @test_int [1/(x^1*(sqrt(a + b*x) + sqrt(a + c*x))^2), x, 6, -(a/((b - c)^2*x^2)) - (b + c)/((b - c)^2*x) + (sqrt(a + b*x)*sqrt(a + c*x))/(2*a*(b - c)*x) + (sqrt(a + b*x)*(a + c*x)^(3/2))/(a*(b - c)^2*x^2) - atanh(sqrt(a + b*x)/sqrt(a + c*x))/(2*a)]
    @test_int [1/(x^2*(sqrt(a + b*x) + sqrt(a + c*x))^2), x, 7, -((2*a)/(3*(b - c)^2*x^3)) - (b + c)/(2*(b - c)^2*x^2) - ((b + c)*sqrt(a + b*x)*sqrt(a + c*x))/(4*a^2*(b - c)*x) - ((b + c)*sqrt(a + b*x)*(a + c*x)^(3/2))/(2*a^2*(b - c)^2*x^2) + (2*(a + b*x)^(3/2)*(a + c*x)^(3/2))/(3*a^2*(b - c)^2*x^3) + ((b + c)*atanh(sqrt(a + b*x)/sqrt(a + c*x)))/(4*a^2)]

    @test_int [x^4/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 10, -((8*a^2*(a + b*x)^(3/2))/(3*b^2*(b - c)^3)) + (2*a^2*(b + 3*c)*(a + b*x)^(3/2))/(3*b^3*(b - c)^3) + (8*a*(a + b*x)^(5/2))/(5*b^2*(b - c)^3) - (4*a*(b + 3*c)*(a + b*x)^(5/2))/(5*b^3*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(7/2))/(7*b^3*(b - c)^3) + (8*a^2*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*a^2*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c)^3*c^3) - (8*a*(a + c*x)^(5/2))/(5*(b - c)^3*c^2) + (4*a*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^3) - (2*(3*b + c)*(a + c*x)^(7/2))/(7*(b - c)^3*c^3)]
    @test_int [x^3/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 6, (8*a*(a + b*x)^(3/2))/(3*b*(b - c)^3) - (2*a*(b + 3*c)*(a + b*x)^(3/2))/(3*b^2*(b - c)^3) + (2*(b + 3*c)*(a + b*x)^(5/2))/(5*b^2*(b - c)^3) - (8*a*(a + c*x)^(3/2))/(3*(b - c)^3*c) + (2*a*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c)^3*c^2) - (2*(3*b + c)*(a + c*x)^(5/2))/(5*(b - c)^3*c^2)]
    @test_int [x^2/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 8, (8*a*sqrt(a + b*x))/(b - c)^3 + (2*(b + 3*c)*(a + b*x)^(3/2))/(3*b*(b - c)^3) - (8*a*sqrt(a + c*x))/(b - c)^3 - (2*(3*b + c)*(a + c*x)^(3/2))/(3*(b - c)^3*c) - (8*a^(3/2)*atanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 + (8*a^(3/2)*atanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3]
    @test_int [x^1/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 14, (2*(b + 3*c)*sqrt(a + b*x))/(b - c)^3 - (4*a*sqrt(a + b*x))/((b - c)^3*x) - (2*(3*b + c)*sqrt(a + c*x))/(b - c)^3 + (4*a*sqrt(a + c*x))/((b - c)^3*x) - (6*sqrt(a)*(b + c)*atanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 + (6*sqrt(a)*(b + c)*atanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3, (2*(b + 3*c)*sqrt(a + b*x))/(b - c)^3 - (4*a*sqrt(a + b*x))/((b - c)^3*x) - (2*(3*b + c)*sqrt(a + c*x))/(b - c)^3 + (4*a*sqrt(a + c*x))/((b - c)^3*x) - (4*sqrt(a)*b*atanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 - (2*sqrt(a)*(b + 3*c)*atanh(sqrt(a + b*x)/sqrt(a)))/(b - c)^3 + (4*sqrt(a)*c*atanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3 + (2*sqrt(a)*(3*b + c)*atanh(sqrt(a + c*x)/sqrt(a)))/(b - c)^3]
    @test_int [x^0/(sqrt(a + b*x) + sqrt(a + c*x))^3, x, 16, -((2*a*sqrt(a + b*x))/((b - c)^3*x^2)) - ((2*b + 3*c)*sqrt(a + b*x))/((b - c)^3*x) + (2*a*sqrt(a + c*x))/((b - c)^3*x^2) + ((3*b + 2*c)*sqrt(a + c*x))/((b - c)^3*x) - (3*b*c*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)^3) + (3*b*c*atanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c)^3), -((2*a*sqrt(a + b*x))/((b - c)^3*x^2)) - (b*sqrt(a + b*x))/((b - c)^3*x) - ((b + 3*c)*sqrt(a + b*x))/((b - c)^3*x) + (2*a*sqrt(a + c*x))/((b - c)^3*x^2) + (c*sqrt(a + c*x))/((b - c)^3*x) + ((3*b + c)*sqrt(a + c*x))/((b - c)^3*x) + (b^2*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)^3) - (b*(b + 3*c)*atanh(sqrt(a + b*x)/sqrt(a)))/(sqrt(a)*(b - c)^3) - (c^2*atanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c)^3) + (c*(3*b + c)*atanh(sqrt(a + c*x)/sqrt(a)))/(sqrt(a)*(b - c)^3)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*(sqrt(a+b*x) + sqrt(a+c*x))^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [sqrt(1 - x)*(sqrt(1 - x) + sqrt(1 + x)), x, 4, x - x^2/2 + (1/2)*x*sqrt(1 - x^2) + asin(x)/2]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(-sqrt(a+b*x) - sqrt(a+c*x))^n*(sqrt(a+b*x) + sqrt(a+c*x))^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x)), x, 6, -(x^4/2) + (2/3)*(1 - x^2)^(3/2) - (2/5)*(1 - x^2)^(5/2)]
    @test_int [x^2*(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x)), x, 6, -((2*x^3)/3) + (1/4)*x*sqrt(1 - x^2) - (1/2)*x^3*sqrt(1 - x^2) - asin(x)/4]
    @test_int [x^1*(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x)), x, 4, -x^2 + (2/3)*(1 - x^2)^(3/2)]
    @test_int [x^0*(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x)), x, 5, -2*x - x*sqrt(1 - x^2) - asin(x)]
    @test_int [(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x))/x^1, x, 7, -2*sqrt(1 - x^2) + 2*atanh(sqrt(1 - x^2)) - 2*log(x)]
    @test_int [(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x))/x^2, x, 5, 2/x + (2*sqrt(1 - x^2))/x + 2*asin(x)]
    @test_int [(-sqrt(1 - x) - sqrt(1 + x))*(sqrt(1 - x) + sqrt(1 + x))/x^3, x, 7, 1/x^2 + sqrt(1 - x^2)/x^2 - atanh(sqrt(1 - x^2))]

    #= ::Subsubsection:: =#
    #=p<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(sqrt(a+b*x) - sqrt(a+c*x))^n*(sqrt(a+b*x) + sqrt(a+c*x))^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [(sqrt(1 - x) + sqrt(1 + x))/(-sqrt(1 - x) + sqrt(1 + x)), x, 15, sqrt(1 - x^2) - atanh(sqrt(1 - x^2)) + log(x)]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(-sqrt(-1 + x) + sqrt(1 + x))/(sqrt(-1 + x) + sqrt(1 + x)), x, 9, x^2/2 - (1/2)*sqrt(-1 + x)*x*sqrt(1 + x) + acosh(x)/2]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g+h*x+i*x^2)^m*(d+e*x+f*sqrt(a+b*x+c*x^2))^n=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x+f*sqrt(a+c*x^2))^n*when*e^2-c*f^2=0=#

    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^n, x, 4, (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(1 + n)/(2*e*(1 + n)) + (a*f^2*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/d))/(2*d^2*e*(1 + n))]

    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^3, x, 3, -((a*d^3*f^2)/(2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) + (a*d*f^2*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/e + (a*f^2*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^2)/(4*e) + (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^4/(8*e) + (3*a*d^2*f^2*log(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e)]
    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^2, x, 3, -((a*d^2*f^2)/(2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) + (a*f^2*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e) + (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^3/(6*e) + (a*d*f^2*log(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/e]
    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^1, x, 4, d*x + (e*x^2)/2 + (1/2)*f*x*sqrt(a + (e^2*x^2)/f^2) + (a*f^2*atanh((e*x)/(f*sqrt(a + (e^2*x^2)/f^2))))/(2*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^1, x, 3, -((a*f^2)/(2*d*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) - (a*f^2*log(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d^2*e) + ((1 + (a*f^2)/d^2)*log(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^2, x, 3, -((a*f^2)/(2*d^2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) - (1 + (a*f^2)/d^2)/(2*e*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))) - (a*f^2*log(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(d^3*e) + (a*f^2*log(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(d^3*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^3, x, 3, -((a*f^2)/(2*d^3*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) - (1 + (a*f^2)/d^2)/(4*e*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^2) - (a*f^2)/(d^3*e*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))) - (3*a*f^2*log(e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d^4*e) + (3*a*f^2*log(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d^4*e)]

    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(5/2), x, 6, (2*a*d*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/e - (a*d^2*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2))) + (a*f^2*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(3/2))/(3*e) + (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(7/2)/(7*e) - (5*a*d^(3/2)*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*e)]
    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(3/2), x, 6, (a*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/e - (a*d*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2))) + (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(5/2)/(5*e) - (3*a*sqrt(d)*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*e)]
    @test_int [(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(1/2), x, 6, -((a*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2)))) + (d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(3/2)/(3*e) - (a*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*sqrt(d)*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(1/2), x, 5, sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/e - (a*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2))) + (a*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*d^(3/2)*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(3/2), x, 5, -((1 + (a*f^2)/d^2)/(e*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))) - (a*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d^2*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2))) + (3*a*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*d^(5/2)*e)]
    @test_int [1/(d + e*x + f*sqrt(a + e^2/f^2*x^2))^(5/2), x, 6, -((1 + (a*f^2)/d^2)/(3*e*(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))^(3/2))) - (2*a*f^2)/(d^3*e*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))) - (a*f^2*sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2)))/(2*d^3*e*(e*x + f*sqrt(a + (e^2*x^2)/f^2))) + (5*a*f^2*atanh(sqrt(d + e*x + f*sqrt(a + (e^2*x^2)/f^2))/sqrt(d)))/(2*d^(7/2)*e)]

    @test_int [sqrt(x - sqrt(-4 + x^2)), x, 3, 4/sqrt(x - sqrt(-4 + x^2)) + (1/3)*(x - sqrt(-4 + x^2))^(3/2)]
    @test_int [sqrt(a*x + b*sqrt(c + a^2*x^2/b^2)), x, 3, -((b^2*c)/(a*sqrt(a*x + b*sqrt(c + (a^2*x^2)/b^2)))) + (a*x + b*sqrt(c + (a^2*x^2)/b^2))^(3/2)/(3*a)]

    @test_int [sqrt(1 + sqrt(1 - x^2)), x, 1, -((2*x^3)/(3*(1 + sqrt(1 - x^2))^(3/2))) + (2*x)/sqrt(1 + sqrt(1 - x^2))]
    @test_int [sqrt(1 + sqrt(1 + x^2)), x, 1, (2*x^3)/(3*(1 + sqrt(1 + x^2))^(3/2)) + (2*x)/sqrt(1 + sqrt(1 + x^2))]
    @test_int [sqrt(5 + sqrt(25 + x^2)), x, 1, (2*x^3)/(3*(5 + sqrt(25 + x^2))^(3/2)) + (10*x)/sqrt(5 + sqrt(25 + x^2))]
    @test_int [sqrt(a + b*sqrt(a^2/b^2 + c*x^2)), x, 1, (2*b^2*c*x^3)/(3*(a + b*sqrt(a^2/b^2 + c*x^2))^(3/2)) + (2*a*x)/sqrt(a + b*sqrt(a^2/b^2 + c*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(d+e*x+f*sqrt(a+b*x+c*x^2))^n*when*e^2-c*f^2=0=#

    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^n, x, 4, (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(1 + n)/(2*e*(1 + n)) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(1 + n)*HypergeometricFunctions._₂F₁(2, 1 + n, 2 + n, (2*e*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(2*d*e - b*f^2)))/(2*e*(2*d*e - b*f^2)^2*(1 + n))]

    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^3, x, 3, (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*(e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(8*e^4) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^2)/(16*e^3) + (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^4/(8*e) - (f^2*(2*d*e - b*f^2)^3*(4*a*e^2 - b^2*f^2))/(32*e^5*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (3*f^2*(2*d*e - b*f^2)^2*(4*a*e^2 - b^2*f^2)*log(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))/(32*e^5)]
    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^2, x, 3, (f^2*(4*a*e^2 - b^2*f^2)*(e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(8*e^3) + (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^3/(6*e) - (f^2*(2*d*e - b*f^2)^2*(4*a*e^2 - b^2*f^2))/(16*e^4*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*log(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))/(8*e^4)]
    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^1, x, 4, d*x + (e*x^2)/2 + (f*(b*f^2 + 2*e^2*x)*sqrt(a + b*x + (e^2*x^2)/f^2))/(4*e^2) + (f^2*(4*a*e^2 - b^2*f^2)*atanh((b*f^2 + 2*e^2*x)/(2*e*f*sqrt(a + b*x + (e^2*x^2)/f^2))))/(8*e^3)]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^1, x, 3, -((f^2*(4*a*e^2 - b^2*f^2))/(2*e*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))) + (2*(d^2*e - b*d*f^2 + a*e*f^2)*log(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(2*d*e - b*f^2)^2 - (f^2*(4*a*e^2 - b^2*f^2)*log(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))/(2*e*(2*d*e - b*f^2)^2)]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^2, x, 3, -((2*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))) - (f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (2*f^2*(4*a*e^2 - b^2*f^2)*log(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(2*d*e - b*f^2)^3 - (2*f^2*(4*a*e^2 - b^2*f^2)*log(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))/(2*d*e - b*f^2)^3]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^3, x, 3, -((d^2*e - b*d*f^2 + a*e*f^2)/((2*d*e - b*f^2)^2*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^2)) - (2*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))) - (2*e*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (6*e*f^2*(4*a*e^2 - b^2*f^2)*log(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(2*d*e - b*f^2)^4 - (6*e*f^2*(4*a*e^2 - b^2*f^2)*log(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2))))/(2*d*e - b*f^2)^4]

    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(5/2), x, 6, (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(4*e^4) + (f^2*(4*a*e^2 - b^2*f^2)*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(3/2))/(12*e^3) + (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(7/2)/(7*e) - (f^2*(2*d*e - b*f^2)^2*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(16*e^4*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) - (5*f^2*(2*d*e - b*f^2)^(3/2)*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(16*sqrt(2)*e^(9/2))]
    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(3/2), x, 6, (f^2*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(4*e^3) + (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(5/2)/(5*e) - (f^2*(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(8*e^3*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) - (3*f^2*sqrt(2*d*e - b*f^2)*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(8*sqrt(2)*e^(7/2))]
    @test_int [(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(1/2), x, 6, (d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(3/2)/(3*e) - (f^2*(4*a - (b^2*f^2)/e^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(4*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) - (f^2*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(4*sqrt(2)*e^(5/2)*sqrt(2*d*e - b*f^2))]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(1/2), x, 5, sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))/e - (f^2*(4*a*e - (b^2*f^2)/e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/(2*(2*d*e - b*f^2)*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (f^2*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(2*sqrt(2)*e^(3/2)*(2*d*e - b*f^2)^(3/2))]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(3/2), x, 5, -((4*(d^2*e - b*d*f^2 + a*e*f^2))/((2*d*e - b*f^2)^2*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))) - (f^2*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/((2*d*e - b*f^2)^2*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (3*f^2*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(sqrt(2)*sqrt(e)*(2*d*e - b*f^2)^(5/2))]
    @test_int [1/(d + e*x + f*sqrt(a + b*x + e^2/f^2*x^2))^(5/2), x, 6, -((4*(d^2*e - b*d*f^2 + a*e*f^2))/(3*(2*d*e - b*f^2)^2*(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))^(3/2))) - (4*f^2*(4*a*e^2 - b^2*f^2))/((2*d*e - b*f^2)^3*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2))) - (2*e*f^2*(4*a*e^2 - b^2*f^2)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/((2*d*e - b*f^2)^3*(b*f^2 + 2*e*(e*x + f*sqrt(a + (x*(b*f^2 + e^2*x))/f^2)))) + (5*sqrt(2)*sqrt(e)*f^2*(4*a*e^2 - b^2*f^2)*atanh((sqrt(2)*sqrt(e)*sqrt(d + e*x + f*sqrt(a + b*x + (e^2*x^2)/f^2)))/sqrt(2*d*e - b*f^2)))/(2*d*e - b*f^2)^(7/2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+c*x^2)^m*(d*x+e*sqrt(a+c*x^2))^n=#

    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^2, x, 3, -((a^5*(x + sqrt(a + x^2))^(-5 + n))/(32*(5 - n))) - (5*a^4*(x + sqrt(a + x^2))^(-3 + n))/(32*(3 - n)) - (5*a^3*(x + sqrt(a + x^2))^(-1 + n))/(16*(1 - n)) + (5*a^2*(x + sqrt(a + x^2))^(1 + n))/(16*(1 + n)) + (5*a*(x + sqrt(a + x^2))^(3 + n))/(32*(3 + n)) + (x + sqrt(a + x^2))^(5 + n)/(32*(5 + n))]
    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^1, x, 3, -((a^3*(x + sqrt(a + x^2))^(-3 + n))/(8*(3 - n))) - (3*a^2*(x + sqrt(a + x^2))^(-1 + n))/(8*(1 - n)) + (3*a*(x + sqrt(a + x^2))^(1 + n))/(8*(1 + n)) + (x + sqrt(a + x^2))^(3 + n)/(8*(3 + n))]
    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^0, x, 3, -((a*(x + sqrt(a + x^2))^(-1 + n))/(2*(1 - n))) + (x + sqrt(a + x^2))^(1 + n)/(2*(1 + n))]
    @test_int [(x + sqrt(a + x^2))^n/(a + x^2)^1, x, 2, (2*(x + sqrt(a + x^2))^(1 + n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -((x + sqrt(a + x^2))^2/a)))/(a*(1 + n))]
    @test_int [(x + sqrt(a + x^2))^n/(a + x^2)^2, x, 2, (8*(x + sqrt(a + x^2))^(3 + n)*HypergeometricFunctions._₂F₁(3, (3 + n)/2, (5 + n)/2, -((x + sqrt(a + x^2))^2/a)))/(a^3*(3 + n))]

    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^2, x, 3, -((a^5*(x - sqrt(a + x^2))^(-5 + n))/(32*(5 - n))) - (5*a^4*(x - sqrt(a + x^2))^(-3 + n))/(32*(3 - n)) - (5*a^3*(x - sqrt(a + x^2))^(-1 + n))/(16*(1 - n)) + (5*a^2*(x - sqrt(a + x^2))^(1 + n))/(16*(1 + n)) + (5*a*(x - sqrt(a + x^2))^(3 + n))/(32*(3 + n)) + (x - sqrt(a + x^2))^(5 + n)/(32*(5 + n))]
    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^1, x, 3, -((a^3*(x - sqrt(a + x^2))^(-3 + n))/(8*(3 - n))) - (3*a^2*(x - sqrt(a + x^2))^(-1 + n))/(8*(1 - n)) + (3*a*(x - sqrt(a + x^2))^(1 + n))/(8*(1 + n)) + (x - sqrt(a + x^2))^(3 + n)/(8*(3 + n))]
    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^0, x, 3, -((a*(x - sqrt(a + x^2))^(-1 + n))/(2*(1 - n))) + (x - sqrt(a + x^2))^(1 + n)/(2*(1 + n))]
    @test_int [(x - sqrt(a + x^2))^n/(a + x^2)^1, x, 2, (2*(x - sqrt(a + x^2))^(1 + n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, -((x - sqrt(a + x^2))^2/a)))/(a*(1 + n))]
    @test_int [(x - sqrt(a + x^2))^n/(a + x^2)^2, x, 2, (8*(x - sqrt(a + x^2))^(3 + n)*HypergeometricFunctions._₂F₁(3, (3 + n)/2, (5 + n)/2, -((x - sqrt(a + x^2))^2/a)))/(a^3*(3 + n))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+c*x^2)^(m/2)*(d*x+e*sqrt(a+c*x^2))^n=#

    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^(5/2), x, 3, -((a^6*(x + sqrt(a + x^2))^(-6 + n))/(64*(6 - n))) - (3*a^5*(x + sqrt(a + x^2))^(-4 + n))/(32*(4 - n)) - (15*a^4*(x + sqrt(a + x^2))^(-2 + n))/(64*(2 - n)) + (5*a^3*(x + sqrt(a + x^2))^n)/(16*n) + (15*a^2*(x + sqrt(a + x^2))^(2 + n))/(64*(2 + n)) + (3*a*(x + sqrt(a + x^2))^(4 + n))/(32*(4 + n)) + (x + sqrt(a + x^2))^(6 + n)/(64*(6 + n))]
    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^(3/2), x, 3, -((a^4*(x + sqrt(a + x^2))^(-4 + n))/(16*(4 - n))) - (a^3*(x + sqrt(a + x^2))^(-2 + n))/(4*(2 - n)) + (3*a^2*(x + sqrt(a + x^2))^n)/(8*n) + (a*(x + sqrt(a + x^2))^(2 + n))/(4*(2 + n)) + (x + sqrt(a + x^2))^(4 + n)/(16*(4 + n))]
    @test_int [(x + sqrt(a + x^2))^n*(a + x^2)^(1/2), x, 3, -((a^2*(x + sqrt(a + x^2))^(-2 + n))/(4*(2 - n))) + (a*(x + sqrt(a + x^2))^n)/(2*n) + (x + sqrt(a + x^2))^(2 + n)/(4*(2 + n))]
    @test_int [(x + sqrt(a + x^2))^n/(a + x^2)^(1/2), x, 2, (x + sqrt(a + x^2))^n/n]
    @test_int [(x + sqrt(a + x^2))^n/(a + x^2)^(3/2), x, 2, (4*(x + sqrt(a + x^2))^(2 + n)*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, -((x + sqrt(a + x^2))^2/a)))/(a^2*(2 + n))]
    @test_int [(x + sqrt(a + x^2))^n/(a + x^2)^(5/2), x, 2, (16*(x + sqrt(a + x^2))^(4 + n)*HypergeometricFunctions._₂F₁(4, (4 + n)/2, (6 + n)/2, -((x + sqrt(a + x^2))^2/a)))/(a^4*(4 + n))]

    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^(5/2), x, 3, (a^6*(x - sqrt(a + x^2))^(-6 + n))/(64*(6 - n)) + (3*a^5*(x - sqrt(a + x^2))^(-4 + n))/(32*(4 - n)) + (15*a^4*(x - sqrt(a + x^2))^(-2 + n))/(64*(2 - n)) - (5*a^3*(x - sqrt(a + x^2))^n)/(16*n) - (15*a^2*(x - sqrt(a + x^2))^(2 + n))/(64*(2 + n)) - (3*a*(x - sqrt(a + x^2))^(4 + n))/(32*(4 + n)) - (x - sqrt(a + x^2))^(6 + n)/(64*(6 + n))]
    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^(3/2), x, 3, (a^4*(x - sqrt(a + x^2))^(-4 + n))/(16*(4 - n)) + (a^3*(x - sqrt(a + x^2))^(-2 + n))/(4*(2 - n)) - (3*a^2*(x - sqrt(a + x^2))^n)/(8*n) - (a*(x - sqrt(a + x^2))^(2 + n))/(4*(2 + n)) - (x - sqrt(a + x^2))^(4 + n)/(16*(4 + n))]
    @test_int [(x - sqrt(a + x^2))^n*(a + x^2)^(1/2), x, 3, (a^2*(x - sqrt(a + x^2))^(-2 + n))/(4*(2 - n)) - (a*(x - sqrt(a + x^2))^n)/(2*n) - (x - sqrt(a + x^2))^(2 + n)/(4*(2 + n))]
    @test_int [(x - sqrt(a + x^2))^n/(a + x^2)^(1/2), x, 2, -((x - sqrt(a + x^2))^n/n)]
    @test_int [(x - sqrt(a + x^2))^n/(a + x^2)^(3/2), x, 2, -((4*(x - sqrt(a + x^2))^(2 + n)*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, -((x - sqrt(a + x^2))^2/a)))/(a^2*(2 + n)))]
    @test_int [(x - sqrt(a + x^2))^n/(a + x^2)^(5/2), x, 2, -((16*(x - sqrt(a + x^2))^(4 + n)*HypergeometricFunctions._₂F₁(4, (4 + n)/2, (6 + n)/2, -((x - sqrt(a + x^2))^2/a)))/(a^4*(4 + n)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g+h*x+i*x^2)^m*(d+e*x+f*sqrt(a+b*x+c*x^2))^n=#

    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^2, x, 4, ((d^2 - a*f^2)^5*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-5 + n))/(32*e*f^4*(5 - n)) - (5*(d^2 - a*f^2)^4*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-3 + n))/(32*e*f^4*(3 - n)) + (5*(d^2 - a*f^2)^3*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-1 + n))/(16*e*f^4*(1 - n)) + (5*(d^2 - a*f^2)^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n))/(16*e*f^4*(1 + n)) - (5*(d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(3 + n))/(32*e*f^4*(3 + n)) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(5 + n)/(32*e*f^4*(5 + n))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^1, x, 4, ((d^2 - a*f^2)^3*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-3 + n))/(8*e*f^2*(3 - n)) - (3*(d^2 - a*f^2)^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-1 + n))/(8*e*f^2*(1 - n)) - (3*(d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n))/(8*e*f^2*(1 + n)) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(3 + n)/(8*e*f^2*(3 + n))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^0, x, 4, ((d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-1 + n))/(2*e*(1 - n)) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n)/(2*e*(1 + n))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^1, x, 2, -((2*f^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^2/(d^2 - a*f^2)))/(e*(d^2 - a*f^2)*(1 + n)))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^2, x, 3, -((8*f^4*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(3 + n)*HypergeometricFunctions._₂F₁(3, (3 + n)/2, (5 + n)/2, (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^2/(d^2 - a*f^2)))/(e*(d^2 - a*f^2)^3*(3 + n)))]

    @test_int [(d + e*x + f*sqrt((a*f^2 + e*x*(2*d + e*x))/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^0, x, 5, ((d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-1 + n))/(2*e*(1 - n)) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n)/(2*e*(1 + n))]
    @test_int [(d + e*x + f*sqrt((a*f^2 + e*x*(2*d + e*x))/f^2))^n/(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^1, x, 3, -((2*f^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(1 + n)*HypergeometricFunctions._₂F₁(1, (1 + n)/2, (3 + n)/2, (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^2/(d^2 - a*f^2)))/(e*(d^2 - a*f^2)*(1 + n)))]

    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^(3/2), x, 4, -(((d^2 - a*f^2)^4*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-4 + n))/(16*e*f^3*(4 - n))) + ((d^2 - a*f^2)^3*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-2 + n))/(4*e*f^3*(2 - n)) + (3*(d^2 - a*f^2)^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(8*e*f^3*n) - ((d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(2 + n))/(4*e*f^3*(2 + n)) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(4 + n)/(16*e*f^3*(4 + n))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^(1/2), x, 4, -(((d^2 - a*f^2)^2*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-2 + n))/(4*e*f*(2 - n))) - ((d^2 - a*f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(2*e*f*n) + (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(2 + n)/(4*e*f*(2 + n))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^(1/2), x, 3, (f*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(e*n)]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a + 2*d*e*x/f^2 + e^2*x^2/f^2)^(3/2), x, 3, (4*f^3*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(2 + n)*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^2/(d^2 - a*f^2)))/(e*(d^2 - a*f^2)^2*(2 + n))]

    @test_int [(d + e*x + f*sqrt((a*f^2 + e*x*(2*d + e*x))/f^2))^n/((a*f^2 + e*x*(2*d + e*x))/f^2)^(1/2), x, 4, (f*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(e*n)]

    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n*(a*g + 2*d*e*g*x/f^2 + e^2*g*x^2/f^2)^(1/2), x, 5, -(((d^2 - a*f^2)^2*sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(-2 + n))/(4*e*f*(2 - n)*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))) - ((d^2 - a*f^2)*sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(2*e*f*n*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)) + (sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(2 + n))/(4*e*f*(2 + n)*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a*g + 2*d*e*g*x/f^2 + e^2*g*x^2/f^2)^(1/2), x, 4, (f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(e*n*sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2))]
    @test_int [(d + e*x + f*sqrt(a + 2*d*e*x/f^2 + e^2*x^2/f^2))^n/(a*g + 2*d*e*g*x/f^2 + e^2*g*x^2/f^2)^(3/2), x, 4, (4*f^3*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^(2 + n)*HypergeometricFunctions._₂F₁(2, (2 + n)/2, (4 + n)/2, (d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^2/(d^2 - a*f^2)))/(e*(d^2 - a*f^2)^2*g*(2 + n)*sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2))]

    @test_int [(d + e*x + f*sqrt((a*f^2 + e*x*(2*d + e*x))/f^2))^n/((a*f^2*g + e*g*x*(2*d + e*x))/f^2)^(1/2), x, 5, (f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)*(d + e*x + f*sqrt(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2))^n)/(e*n*sqrt(a*g + (2*d*e*g*x)/f^2 + (e^2*g*x^2)/f^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*x)^m*(c+d*x^2)^p*(e+f*x^2)^q=#

    @test_int [1/((a + b*x)*sqrt(c + d*x^2)*sqrt(e + f*x^2)), x, 7, -((b*atanh((sqrt(b^2*e + a^2*f)*sqrt(c + d*x^2))/(sqrt(b^2*c + a^2*d)*sqrt(e + f*x^2))))/(sqrt(b^2*c + a^2*d)*sqrt(b^2*e + a^2*f))) + (sqrt(-c)*sqrt(1 + (d*x^2)/c)*sqrt(1 + (f*x^2)/e)*Elliptic.Pi(-((b^2*c)/(a^2*d)), asin((sqrt(d)*x)/sqrt(-c)), (c*f)/(d*e)))/(a*sqrt(d)*sqrt(c + d*x^2)*sqrt(e + f*x^2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(A+B*x^n) / (a+b*x^(2*(m+1))+c*x^n+d*x^(2*n))=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(A+B*x^n) / (a+b*x^2+c*x^n+d*x^(2*n))=#

    @test_int [(e - 2*f*x^2)/(e^2 + 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4), x, 4, -(log(e - 2*sqrt(-d)*sqrt(f)*x + 2*f*x^2)/(4*sqrt(-d)*sqrt(f))) + log(e + 2*sqrt(-d)*sqrt(f)*x + 2*f*x^2)/(4*sqrt(-d)*sqrt(f))]
    @test_int [(e - 2*f*x^2)/(e^2 - 4*d*f*x^2 + 4*e*f*x^2 + 4*f^2*x^4), x, 4, -(log(e - 2*sqrt(d)*sqrt(f)*x + 2*f*x^2)/(4*sqrt(d)*sqrt(f))) + log(e + 2*sqrt(d)*sqrt(f)*x + 2*f*x^2)/(4*sqrt(d)*sqrt(f))]

    @test_int [(e - 4*f*x^3)/(e^2 + 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6), x, 2, atan((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]
    @test_int [(e - 4*f*x^3)/(e^2 - 4*d*f*x^2 + 4*e*f*x^3 + 4*f^2*x^6), x, 2, atanh((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]

    @test_int [(e - 2*f*(-1 + n)*x^n)/(e^2 + 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 2, atan((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^n))/(2*sqrt(d)*sqrt(f))]
    @test_int [(e - 2*f*(-1 + n)*x^n)/(e^2 - 4*d*f*x^2 + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 2, atanh((2*sqrt(d)*sqrt(f)*x)/(e + 2*f*x^n))/(2*sqrt(d)*sqrt(f))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(A+B*x^n) / (a+b*x^(2*(m+1))+c*x^n+d*x^(2*n))=#

    @test_int [x/(e^2 + 4*e*f*x^2 + 4*d*f*x^4 + 4*f^2*x^4), x, 4, atan((sqrt(f)*(e + 2*(d + f)*x^2))/(sqrt(d)*e))/(4*sqrt(d)*e*sqrt(f))]
    @test_int [x/(e^2 + 4*e*f*x^2 - 4*d*f*x^4 + 4*f^2*x^4), x, 4, -(atanh((sqrt(f)*(e - 2*(d - f)*x^2))/(sqrt(d)*e))/(4*sqrt(d)*e*sqrt(f)))]

    @test_int [(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^6), x, 2, atan((2*sqrt(d)*sqrt(f)*x^3)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))]
    @test_int [(x^2*(3*e + 2*f*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^6), x, 2, atanh((2*sqrt(d)*sqrt(f)*x^3)/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f))]

    @test_int [(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^(2 + 2*m)), x, 2, atan((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f)), atan((2*sqrt(d)*sqrt(f)*(1 - m^2)*x^(1 + m))/((1 - m)*(1 + m)*(e + 2*f*x^2)))/(2*sqrt(d)*sqrt(f))]
    @test_int [(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 - 4*d*f*x^(2 + 2*m)), x, 2, atanh((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^2))/(2*sqrt(d)*sqrt(f)), atanh((2*sqrt(d)*sqrt(f)*(1 - m^2)*x^(1 + m))/((1 - m)*(1 + m)*(e + 2*f*x^2)))/(2*sqrt(d)*sqrt(f))]

    @test_int [(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 + 4*d*f*x^4 + 4*f^2*x^6), x, 2, atan((2*sqrt(d)*sqrt(f)*x^2)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]
    @test_int [(x*(2*e - 2*f*x^3))/(e^2 + 4*e*f*x^3 - 4*d*f*x^4 + 4*f^2*x^6), x, 2, atanh((2*sqrt(d)*sqrt(f)*x^2)/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]

    @test_int [x^2/(e^2 + 4*e*f*x^3 + 4*d*f*x^6 + 4*f^2*x^6), x, 4, atan((sqrt(f)*(e + 2*(d + f)*x^3))/(sqrt(d)*e))/(6*sqrt(d)*e*sqrt(f))]
    @test_int [x^2/(e^2 + 4*e*f*x^3 - 4*d*f*x^6 + 4*f^2*x^6), x, 4, -(atanh((sqrt(f)*(e - 2*(d - f)*x^3))/(sqrt(d)*e))/(6*sqrt(d)*e*sqrt(f)))]

    @test_int [(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 + 4*d*f*x^(2 + 2*m)), x, 2, atan((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]
    @test_int [(x^m*(e*(1 + m) + 2*f*(-2 + m)*x^3))/(e^2 + 4*e*f*x^3 + 4*f^2*x^6 - 4*d*f*x^(2 + 2*m)), x, 2, atanh((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^3))/(2*sqrt(d)*sqrt(f))]

    @test_int [(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 2, atan((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^n))/(2*sqrt(d)*sqrt(f))]
    @test_int [(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 - 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)), x, 2, atanh((2*sqrt(d)*sqrt(f)*x^(1 + m))/(e + 2*f*x^n))/(2*sqrt(d)*sqrt(f))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u / (c+d*x^n+e*sqrt(a+b*x^n))=#

    @test_int [x^5/(a*c + b*c*x^2 + d*sqrt(a + b*x^2)), x, 4, -(((2*a*c^2 - d^2)*x^2)/(2*b^2*c^3)) + (d*(2*a*c^2 - d^2)*sqrt(a + b*x^2))/(b^3*c^4) - (d*(a + b*x^2)^(3/2))/(3*b^3*c^2) + (a + b*x^2)^2/(4*b^3*c) + ((a*c^2 - d^2)^2*log(d + c*sqrt(a + b*x^2)))/(b^3*c^5)]
    @test_int [x^3/(a*c + b*c*x^2 + d*sqrt(a + b*x^2)), x, 4, x^2/(2*b*c) - (d*sqrt(a + b*x^2))/(b^2*c^2) - ((a*c^2 - d^2)*log(d + c*sqrt(a + b*x^2)))/(b^2*c^3)]
    @test_int [x^1/(a*c + b*c*x^2 + d*sqrt(a + b*x^2)), x, 3, log(d + c*sqrt(a + b*x^2))/(b*c)]
    @test_int [1/(x^1*(a*c + b*c*x^2 + d*sqrt(a + b*x^2))), x, 7, (d*atanh(sqrt(a + b*x^2)/sqrt(a)))/(sqrt(a)*(a*c^2 - d^2)) + (c*log(x))/(a*c^2 - d^2) - (c*log(d + c*sqrt(a + b*x^2)))/(a*c^2 - d^2)]
    @test_int [1/(x^3*(a*c + b*c*x^2 + d*sqrt(a + b*x^2))), x, 8, -((a*c - d*sqrt(a + b*x^2))/(2*a*(a*c^2 - d^2)*x^2)) - (b*d*(3*a*c^2 - d^2)*atanh(sqrt(a + b*x^2)/sqrt(a)))/(2*a^(3/2)*(a*c^2 - d^2)^2) - (b*c^3*log(x))/(a*c^2 - d^2)^2 + (b*c^3*log(d + c*sqrt(a + b*x^2)))/(a*c^2 - d^2)^2]

    @test_int [x^2/(a*c + b*c*x^2 + d*sqrt(a + b*x^2)), x, 8, x/(b*c) - (sqrt(a*c^2 - d^2)*atan((sqrt(b)*c*x)/sqrt(a*c^2 - d^2)))/(b^(3/2)*c^2) + (sqrt(a*c^2 - d^2)*atan((sqrt(b)*d*x)/(sqrt(a*c^2 - d^2)*sqrt(a + b*x^2))))/(b^(3/2)*c^2) - (d*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(b^(3/2)*c^2)]
    @test_int [x^0/(a*c + b*c*x^2 + d*sqrt(a + b*x^2)), x, 4, atan((sqrt(b)*c*x)/sqrt(a*c^2 - d^2))/(sqrt(b)*sqrt(a*c^2 - d^2)) - atan((sqrt(b)*d*x)/(sqrt(a*c^2 - d^2)*sqrt(a + b*x^2)))/(sqrt(b)*sqrt(a*c^2 - d^2))]
    @test_int [1/(x^2*(a*c + b*c*x^2 + d*sqrt(a + b*x^2))), x, 7, -(c/((a*c^2 - d^2)*x)) + (d*sqrt(a + b*x^2))/(a*(a*c^2 - d^2)*x) - (sqrt(b)*c^2*atan((sqrt(b)*c*x)/sqrt(a*c^2 - d^2)))/(a*c^2 - d^2)^(3/2) + (sqrt(b)*c^2*atan((sqrt(b)*d*x)/(sqrt(a*c^2 - d^2)*sqrt(a + b*x^2))))/(a*c^2 - d^2)^(3/2)]

    @test_int [x^8/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 4, -(((2*a*c^2 - d^2)*x^3)/(3*b^2*c^3)) + (2*d*(2*a*c^2 - d^2)*sqrt(a + b*x^3))/(3*b^3*c^4) - (2*d*(a + b*x^3)^(3/2))/(9*b^3*c^2) + (a + b*x^3)^2/(6*b^3*c) + (2*(a*c^2 - d^2)^2*log(d + c*sqrt(a + b*x^3)))/(3*b^3*c^5)]
    @test_int [x^5/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 4, x^3/(3*b*c) - (2*d*sqrt(a + b*x^3))/(3*b^2*c^2) - (2*(a*c^2 - d^2)*log(d + c*sqrt(a + b*x^3)))/(3*b^2*c^3)]
    @test_int [x^2/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 3, (2*log(d + c*sqrt(a + b*x^3)))/(3*b*c)]
    @test_int [1/(x^1*(a*c + b*c*x^3 + d*sqrt(a + b*x^3))), x, 7, (2*d*atanh(sqrt(a + b*x^3)/sqrt(a)))/(3*sqrt(a)*(a*c^2 - d^2)) + (c*log(x))/(a*c^2 - d^2) - (2*c*log(d + c*sqrt(a + b*x^3)))/(3*(a*c^2 - d^2))]
    @test_int [1/(x^4*(a*c + b*c*x^3 + d*sqrt(a + b*x^3))), x, 8, -((a*c - d*sqrt(a + b*x^3))/(3*a*(a*c^2 - d^2)*x^3)) - (b*d*(3*a*c^2 - d^2)*atanh(sqrt(a + b*x^3)/sqrt(a)))/(3*a^(3/2)*(a*c^2 - d^2)^2) - (b*c^3*log(x))/(a*c^2 - d^2)^2 + (2*b*c^3*log(d + c*sqrt(a + b*x^3)))/(3*(a*c^2 - d^2)^2)]

    @test_int [x^3/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 10, x/(b*c) - (d*x^4*sqrt(1 + (b*x^3)/a)*AppellF1(4/3, 1/2, 1, 7/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))))/(4*(a*c^2 - d^2)*sqrt(a + b*x^3)) + ((a*c^2 - d^2)^(1/3)*atan((1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/sqrt(3)))/(sqrt(3)*b^(4/3)*c^(5/3)) - ((a*c^2 - d^2)^(1/3)*log((a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x))/(3*b^(4/3)*c^(5/3)) + ((a*c^2 - d^2)^(1/3)*log((a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2))/(6*b^(4/3)*c^(5/3))]
    @test_int [x^1/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 9, -((d*x^2*sqrt(1 + (b*x^3)/a)*AppellF1(2/3, 1/2, 1, 5/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))))/(2*(a*c^2 - d^2)*sqrt(a + b*x^3))) - atan((1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/sqrt(3))/(sqrt(3)*b^(2/3)*c^(1/3)*(a*c^2 - d^2)^(1/3)) - log((a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x)/(3*b^(2/3)*c^(1/3)*(a*c^2 - d^2)^(1/3)) + log((a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2)/(6*b^(2/3)*c^(1/3)*(a*c^2 - d^2)^(1/3))]
    @test_int [x^0/(a*c + b*c*x^3 + d*sqrt(a + b*x^3)), x, 9, -((d*x*sqrt(1 + (b*x^3)/a)*AppellF1(1/3, 1/2, 1, 4/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))))/((a*c^2 - d^2)*sqrt(a + b*x^3))) - (c^(1/3)*atan((1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)*(a*c^2 - d^2)^(2/3)) + (c^(1/3)*log((a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x))/(3*b^(1/3)*(a*c^2 - d^2)^(2/3)) - (c^(1/3)*log((a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2))/(6*b^(1/3)*(a*c^2 - d^2)^(2/3))]
    @test_int [1/(x^2*(a*c + b*c*x^3 + d*sqrt(a + b*x^3))), x, 10, -(c/((a*c^2 - d^2)*x)) + (d*sqrt(1 + (b*x^3)/a)*AppellF1(-(1/3), 1/2, 1, 2/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))))/((a*c^2 - d^2)*x*sqrt(a + b*x^3)) + (b^(1/3)*c^(5/3)*atan((1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/sqrt(3)))/(sqrt(3)*(a*c^2 - d^2)^(4/3)) + (b^(1/3)*c^(5/3)*log((a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x))/(3*(a*c^2 - d^2)^(4/3)) - (b^(1/3)*c^(5/3)*log((a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2))/(6*(a*c^2 - d^2)^(4/3))]
    @test_int [1/(x^3*(a*c + b*c*x^3 + d*sqrt(a + b*x^3))), x, 10, -(c/(2*(a*c^2 - d^2)*x^2)) + (d*sqrt(1 + (b*x^3)/a)*AppellF1(-(2/3), 1/2, 1, 1/3, -((b*x^3)/a), -((b*c^2*x^3)/(a*c^2 - d^2))))/(2*(a*c^2 - d^2)*x^2*sqrt(a + b*x^3)) + (b^(2/3)*c^(7/3)*atan((1 - (2*b^(1/3)*c^(2/3)*x)/(a*c^2 - d^2)^(1/3))/sqrt(3)))/(sqrt(3)*(a*c^2 - d^2)^(5/3)) - (b^(2/3)*c^(7/3)*log((a*c^2 - d^2)^(1/3) + b^(1/3)*c^(2/3)*x))/(3*(a*c^2 - d^2)^(5/3)) + (b^(2/3)*c^(7/3)*log((a*c^2 - d^2)^(2/3) - b^(1/3)*c^(2/3)*(a*c^2 - d^2)^(1/3)*x + b^(2/3)*c^(4/3)*x^2))/(6*(a*c^2 - d^2)^(5/3))]

    @test_int [1/(a*c + b*c*x^n + d*sqrt(a + b*x^n)), x, 4, -((d*x*sqrt(1 + (b*x^n)/a)*AppellF1(1/n, 1/2, 1, 1 + 1/n, -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))))/((a*c^2 - d^2)*sqrt(a + b*x^n))) + (c*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*c^2*x^n)/(a*c^2 - d^2))))/(a*c^2 - d^2)]
    @test_int [x^m/(a*c + b*c*x^n + d*sqrt(a + b*x^n)), x, 4, -((d*x^(1 + m)*sqrt(1 + (b*x^n)/a)*AppellF1((1 + m)/n, 1/2, 1, (1 + m + n)/n, -((b*x^n)/a), -((b*c^2*x^n)/(a*c^2 - d^2))))/((a*c^2 - d^2)*(1 + m)*sqrt(a + b*x^n))) + (c*x^(1 + m)*HypergeometricFunctions._₂F₁(1, (1 + m)/n, (1 + m + n)/n, -((b*c^2*x^n)/(a*c^2 - d^2))))/((a*c^2 - d^2)*(1 + m))]
    @test_int [x^(n-1)/(a*c + b*c*x^n + d*sqrt(a + b*x^n)), x, 3, (2*log(d + c*sqrt(a + b*x^n)))/(b*c*n)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(a*x^m+b*x^n+...)^p=#

    @test_int [1/(sqrt(x) + 4*x^(3/2)), x, 3, atan(2*sqrt(x))]
    @test_int [1/(sqrt(x) - x^(5/2)), x, 5, atan(sqrt(x)) + atanh(sqrt(x))]
    @test_int [1/(-x^(1/4) + sqrt(x)), x, 4, 4*x^(1/4) + 2*sqrt(x) + 4*log(1 - x^(1/4))]
    @test_int [1/(x^(1/3) + sqrt(x)), x, 4, 6*x^(1/6) - 3*x^(1/3) + 2*sqrt(x) - 6*log(1 + x^(1/6))]
    @test_int [1/(x^(1/4) + sqrt(x)), x, 4, -4*x^(1/4) + 2*sqrt(x) + 4*log(1 + x^(1/4))]
    @test_int [1/(-x^(1/3) + x^(2/3)), x, 4, 3*x^(1/3) + 3*log(1 - x^(1/3))]
    @test_int [1/(x^(-1/4) + sqrt(x)), x, 9, 2*sqrt(x) + (4*atan((1 - 2*x^(1/4))/sqrt(3)))/sqrt(3) + (4/3)*log(1 + x^(1/4)) - (2/3)*log(1 - x^(1/4) + sqrt(x))]
    @test_int [1/(x^(1/4) + x^(1/3)), x, 4, -12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*sqrt(x) - (12*x^(7/12))/7 + (3*x^(2/3))/2 + 12*log(1 + x^(1/12))]
    @test_int [1/(x^(-1/3) + x^(-1/4)), x, 4, 12*x^(1/12) - 6*x^(1/6) + 4*x^(1/4) - 3*x^(1/3) + (12*x^(5/12))/5 - 2*sqrt(x) + (12*x^(7/12))/7 - (3*x^(2/3))/2 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(11/12))/11 - x + (12*x^(13/12))/13 - (6*x^(7/6))/7 + (4*x^(5/4))/5 - 12*log(1 + x^(1/12))]
    @test_int [1/(-x^(-1/3) + sqrt(x)), x, 9, 2*sqrt(x) + (3/5)*sqrt(2*(5 - sqrt(5)))*atan((1 - sqrt(5) + 4*x^(1/6))/sqrt(2*(5 + sqrt(5)))) - (3/5)*sqrt(2*(5 + sqrt(5)))*atan((1/2)*sqrt((1/10)*(5 + sqrt(5)))*(1 + sqrt(5) + 4*x^(1/6))) + (6/5)*log(1 - x^(1/6)) - (3/10)*(1 + sqrt(5))*log(2 + x^(1/6) - sqrt(5)*x^(1/6) + 2*x^(1/3)) - (3/10)*(1 - sqrt(5))*log(2 + x^(1/6) + sqrt(5)*x^(1/6) + 2*x^(1/3))]

    @test_int [sqrt(x)/(x + x^2), x, 3, 2*atan(sqrt(x))]
    @test_int [x/(4*sqrt(x) + x), x, 4, -8*sqrt(x) + x + 32*log(4 + sqrt(x))]
    @test_int [sqrt(x)/(x^(1/3) + x), x, 13, 2*sqrt(x) + (3*atan(1 - sqrt(2)*x^(1/6)))/sqrt(2) - (3*atan(1 + sqrt(2)*x^(1/6)))/sqrt(2) - (3*log(1 - sqrt(2)*x^(1/6) + x^(1/3)))/(2*sqrt(2)) + (3*log(1 + sqrt(2)*x^(1/6) + x^(1/3)))/(2*sqrt(2))]
    @test_int [x^(1/3)/(x^(1/4) + sqrt(x)), x, 10, -12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*sqrt(3)*atan((1 - 2*x^(1/12))/sqrt(3)) + 6*log(1 + x^(1/12)) - 2*log(1 + x^(1/4))]
    @test_int [sqrt(x)/(x^(1/4) + x^(1/3)), x, 4, -12*x^(1/12) + 6*x^(1/6) - 4*x^(1/4) + 3*x^(1/3) - (12*x^(5/12))/5 + 2*sqrt(x) - (12*x^(7/12))/7 + (3*x^(2/3))/2 - (4*x^(3/4))/3 + (6*x^(5/6))/5 - (12*x^(11/12))/11 + x - (12*x^(13/12))/13 + (6*x^(7/6))/7 + 12*log(1 + x^(1/12))]
    @test_int [sqrt(x)/(-x^(-1/3) + sqrt(x)), x, 10, 6*x^(1/6) + x - (3/5)*sqrt(2*(5 + sqrt(5)))*atan((1 - sqrt(5) + 4*x^(1/6))/sqrt(2*(5 + sqrt(5)))) - (3/5)*sqrt(2*(5 - sqrt(5)))*atan((1/2)*sqrt((1/10)*(5 + sqrt(5)))*(1 + sqrt(5) + 4*x^(1/6))) + (6/5)*log(1 - x^(1/6)) - (3/10)*(1 - sqrt(5))*log(2 + x^(1/6) - sqrt(5)*x^(1/6) + 2*x^(1/3)) - (3/10)*(1 + sqrt(5))*log(2 + x^(1/6) + sqrt(5)*x^(1/6) + 2*x^(1/3))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*(a + b*x^n)^p*when*n<0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*u*(a + b/x)^p=#

    @test_int [x^m*sqrt(b - a/x)/sqrt(a - b*x), x, 3, (2*sqrt(b - a/x)*x^(1 + m))/((1 + 2*m)*sqrt(a - b*x))]

    @test_int [x^2*sqrt(b - a/x)/sqrt(a - b*x), x, 3, (2*sqrt(b - a/x)*x^3)/(5*sqrt(a - b*x))]
    @test_int [x^1*sqrt(b - a/x)/sqrt(a - b*x), x, 3, (2*sqrt(b - a/x)*x^2)/(3*sqrt(a - b*x))]
    @test_int [x^0*sqrt(b - a/x)/sqrt(a - b*x), x, 3, (2*sqrt(b - a/x)*x)/sqrt(a - b*x)]
    @test_int [sqrt(b - a/x)/(x^1*sqrt(a - b*x)), x, 3, -((2*sqrt(b - a/x))/sqrt(a - b*x))]
    @test_int [sqrt(b - a/x)/(x^2*sqrt(a - b*x)), x, 3, -((2*sqrt(b - a/x))/(3*x*sqrt(a - b*x)))]

    @test_int [(a + b/x)^m*(c + d*x)^n, x, 4, ((a + b/x)^m*x*(c + d*x)^n*AppellF1(1 - m, -m, -n, 2 - m, -((a*x)/b), -((d*x)/c)))/((1 + (a*x)/b)^m*(1 + d*x/c)^n*(1 - m))]

    @test_int [(a + b/x)^m*(c + d*x)^2, x, 5, (d*(6*a*c - b*d*(2 - m))*(a + b/x)^(1 + m)*x^2)/(6*a^2) + (d^2*(a + b/x)^(1 + m)*x^3)/(3*a) - (b*(6*a^2*c^2 - 6*a*b*c*d*(1 - m) + b^2*d^2*(2 - 3*m + m^2))*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, 1 + b/(a*x)))/(6*a^4*(1 + m))]
    @test_int [(a + b/x)^m*(c + d*x)^1, x, 4, (d*(a + b/x)^(1 + m)*x^2)/(2*a) - (b*(2*a*c - b*d*(1 - m))*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, 1 + b/(a*x)))/(2*a^3*(1 + m))]
    @test_int [(a + b/x)^m*(c + d*x)^0, x, 2, -((b*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, 1 + b/(a*x)))/(a^2*(1 + m)))]
    @test_int [(a + b/x)^m/(c + d*x)^1, x, 5, -((c*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (c*(a + b/x))/(a*c - b*d)))/(d*(a*c - b*d)*(1 + m))) + ((a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, 1 + b/(a*x)))/(a*d*(1 + m))]
    @test_int [(a + b/x)^m/(c + d*x)^2, x, 3, -((b*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, (c*(a + b/x))/(a*c - b*d)))/((a*c - b*d)^2*(1 + m)))]
    @test_int [(a + b/x)^m/(c + d*x)^3, x, 4, -((d*(a + b/x)^(1 + m))/(2*c*(a*c - b*d)*(d + c/x)^2)) - (b*(2*a*c - b*d*(1 + m))*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, (c*(a + b/x))/(a*c - b*d)))/(2*c*(a*c - b*d)^3*(1 + m))]
    @test_int [(a + b/x)^m/(c + d*x)^4, x, 5, (d^2*(a + b/x)^(1 + m))/(3*c^2*(a*c - b*d)*(d + c/x)^3) - (d*(6*a*c - b*d*(4 + m))*(a + b/x)^(1 + m))/(6*c^2*(a*c - b*d)^2*(d + c/x)^2) - (b*(6*a^2*c^2 - 6*a*b*c*d*(1 + m) + b^2*d^2*(2 + 3*m + m^2))*(a + b/x)^(1 + m)*HypergeometricFunctions._₂F₁(2, 1 + m, 2 + m, (c*(a + b/x))/(a*c - b*d)))/(6*c^2*(a*c - b*d)^4*(1 + m))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*u*(a + b/x^2)^p=#

    @test_int [x^m*sqrt(b - a/x^2)/sqrt(a - b*x^2), x, 3, (sqrt(b - a/x^2)*x^(1 + m))/(m*sqrt(a - b*x^2))]

    @test_int [x^2*sqrt(b - a/x^2)/sqrt(a - b*x^2), x, 3, (sqrt(b - a/x^2)*x^3)/(2*sqrt(a - b*x^2))]
    @test_int [x^1*sqrt(b - a/x^2)/sqrt(a - b*x^2), x, 3, (sqrt(b - a/x^2)*x^2)/sqrt(a - b*x^2)]
    @test_int [x^0*sqrt(b - a/x^2)/sqrt(a - b*x^2), x, 3, (sqrt(b - a/x^2)*x*log(x))/sqrt(a - b*x^2)]
    @test_int [sqrt(b - a/x^2)/(x^1*sqrt(a - b*x^2)), x, 3, -(sqrt(b - a/x^2)/sqrt(a - b*x^2))]
    @test_int [sqrt(b - a/x^2)/(x^2*sqrt(a - b*x^2)), x, 3, -(sqrt(b - a/x^2)/(2*x*sqrt(a - b*x^2)))]

    @test_int [(c + d*x)^(3/2)/sqrt(a + b/x^2), x, 8, (2*c*sqrt(c + d*x)*(b + a*x^2))/(5*a*sqrt(a + b/x^2)*x) + (2*(c + d*x)^(3/2)*(b + a*x^2))/(5*a*sqrt(a + b/x^2)*x) + (2*sqrt(b)*(a*c^2 - 3*b*d^2)*sqrt(c + d*x)*sqrt(1 + (a*x^2)/b)*Elliptic.E(asin(sqrt(1 - (sqrt(-a)*x)/sqrt(b))/sqrt(2)), -((2*sqrt(-a)*sqrt(b)*d)/(a*c - sqrt(-a)*sqrt(b)*d))))/(5*(-a)^(3/2)*d*sqrt(a + b/x^2)*x*sqrt((a*(c + d*x))/(a*c - sqrt(-a)*sqrt(b)*d))) - (2*sqrt(b)*c*(a*c^2 + b*d^2)*sqrt((a*(c + d*x))/(a*c - sqrt(-a)*sqrt(b)*d))*sqrt(1 + (a*x^2)/b)*Elliptic.F(asin(sqrt(1 - (sqrt(-a)*x)/sqrt(b))/sqrt(2)), -((2*sqrt(-a)*sqrt(b)*d)/(a*c - sqrt(-a)*sqrt(b)*d))))/(5*(-a)^(3/2)*d*sqrt(a + b/x^2)*x*sqrt(c + d*x))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*y'(x)*F(y(x))=#

    @test_int [(-1 + x^3)/(-4*x + x^4)^(2/3), x, 1, (3/4)*(-4*x + x^4)^(1/3)]
    @test_int [(2 - x^2)*(6*x - x^3)^(1/4), x, 1, (4/15)*(6*x - x^3)^(5/4)]
    @test_int [(1 + x^4)*sqrt(5*x + x^5), x, 1, (2/15)*(5*x + x^5)^(3/2)]
    @test_int [(2 + 5*x^4)*sqrt(2*x + x^5), x, 1, (2/3)*(2*x + x^5)^(3/2)]

    @test_int [(x + 3*x^2)/sqrt(x^2 + 2*x^3), x, 1, sqrt(x^2 + 2*x^3)]

    @test_int [(2 + (1 - 5*x)^(1/3))/(3 + (1 - 5*x)^(1/3)), x, 4, (-(9/5))*(1 - 5*x)^(1/3) + (3/10)*(1 - 5*x)^(2/3) + x + (27/5)*log(3 + (1 - 5*x)^(1/3))]

    @test_int [(1 + sqrt(x))/(-1 + sqrt(x)), x, 3, 4*sqrt(x) + x + 4*log(1 - sqrt(x))]
    @test_int [(1 - sqrt(2 + 3*x))/(1 + sqrt(2 + 3*x)), x, 4, - x + (4/3)*sqrt(2 + 3*x) - (4/3)*log(1 + sqrt(2 + 3*x))]
    @test_int [(-1 + sqrt(a + b*x))/(1 + sqrt(a + b*x)), x, 4, x - (4*sqrt(a + b*x))/b + (4*log(1 + sqrt(a + b*x)))/b]

    @test_int [(a + b*n*x^(n-1))/(a*x + b*x^n), x, 5, log(a*x + b*x^n), n*log(x) + log(b + a*x^(1 - n))]
    @test_int [(a + b*n*x^(-1 + n))/(x^n*(b + a*x^(1 - n))), x, 4, n*log(x) + log(b + a*x^(1 - n))]

    @test_int [x^1*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(2*a*d + (3*b*d + 3*a*e + b*d*m + a*e*n)*x + (4*c*d + 4*b*e + 4*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (5*c*e + 5*b*f + 5*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (6*c*f + 6*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(7 + 2*m + 3*n)*x^5), x, 1, x^2*(a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n)]
    @test_int [x^0*(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(  a*d + (2*b*d + 2*a*e + b*d*m + a*e*n)*x + (3*c*d + 3*b*e + 3*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (4*c*e + 4*b*f + 4*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (5*c*f + 5*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(6 + 2*m + 3*n)*x^5), x, 1, x^1*(a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n)]
    @test_int [(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n*(b*d + a*e + b*d*m + a*e*n + (2*c*d + 2*b*e + 2*a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x + (3*c*e + 3*b*f + 3*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^2 + (4*c*f + 4*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^3 + c*g*(5 + 2*m + 3*n)*x^4), x, 1, (a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n)]
    @test_int [(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n/x^2*((-a)*d + (b*d*m + a*e*n)*x + (c*d + b*e + a*f + 2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (2*c*e + 2*b*f + 2*a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (3*c*f + 3*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(4 + 2*m + 3*n)*x^5), x, -2, ((a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n))/x]
    @test_int [(a + b*x + c*x^2)^m*(d + e*x + f*x^2 + g*x^3)^n/x^3*(-2*a*d + ((-b)*d - a*e + b*d*m + a*e*n)*x + (2*c*d*m + b*e*m + b*e*n + 2*a*f*n)*x^2 + (c*e + b*f + a*g + 2*c*e*m + b*f*m + c*e*n + 2*b*f*n + 3*a*g*n)*x^3 + (2*c*f + 2*b*g + 2*c*f*m + b*g*m + 2*c*f*n + 3*b*g*n)*x^4 + c*g*(3 + 2*m + 3*n)*x^5), x, -2, ((a + b*x + c*x^2)^(1 + m)*(d + e*x + f*x^2 + g*x^3)^(1 + n))/x^2]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*(c+d*x)^n)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a+b*(c+d*x)^(1/2))^p=#

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [x^3*(a + b*sqrt(c + d*x))^2, x, 4, -((a^2*c^3*x)/d^3) - (4*a*b*c^3*(c + d*x)^(3/2))/(3*d^4) + (c^2*(3*a^2 - b^2*c)*(c + d*x)^2)/(2*d^4) + (12*a*b*c^2*(c + d*x)^(5/2))/(5*d^4) - (c*(a^2 - b^2*c)*(c + d*x)^3)/d^4 - (12*a*b*c*(c + d*x)^(7/2))/(7*d^4) + ((a^2 - 3*b^2*c)*(c + d*x)^4)/(4*d^4) + (4*a*b*(c + d*x)^(9/2))/(9*d^4) + (b^2*(c + d*x)^5)/(5*d^4)]
    @test_int [x^2*(a + b*sqrt(c + d*x))^2, x, 4, (a^2*c^2*x)/d^2 + (4*a*b*c^2*(c + d*x)^(3/2))/(3*d^3) - (c*(2*a^2 - b^2*c)*(c + d*x)^2)/(2*d^3) - (8*a*b*c*(c + d*x)^(5/2))/(5*d^3) + ((a^2 - 2*b^2*c)*(c + d*x)^3)/(3*d^3) + (4*a*b*(c + d*x)^(7/2))/(7*d^3) + (b^2*(c + d*x)^4)/(4*d^3)]
    @test_int [x^1*(a + b*sqrt(c + d*x))^2, x, 4, -((a^2*c*x)/d) - (4*a*b*c*(c + d*x)^(3/2))/(3*d^2) + ((a^2 - b^2*c)*(c + d*x)^2)/(2*d^2) + (4*a*b*(c + d*x)^(5/2))/(5*d^2) + (b^2*(c + d*x)^3)/(3*d^2)]
    @test_int [x^0*(a + b*sqrt(c + d*x))^2, x, 4, a^2*x + (4*a*b*(c + d*x)^(3/2))/(3*d) + (b^2*(c + d*x)^2)/(2*d)]
    @test_int [(a + b*sqrt(c + d*x))^2/x^1, x, 7, b^2*d*x + 4*a*b*sqrt(c + d*x) - 4*a*b*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c)) + (a^2 + b^2*c)*log(x)]
    @test_int [(a + b*sqrt(c + d*x))^2/x^2, x, 6, -((a + b*sqrt(c + d*x))^2/x) - (2*a*b*d*atanh(sqrt(c + d*x)/sqrt(c)))/sqrt(c) + b^2*d*log(x)]
    @test_int [(a + b*sqrt(c + d*x))^2/x^3, x, 6, -((b*d*(b*c + a*sqrt(c + d*x)))/(2*c*x)) - (a + b*sqrt(c + d*x))^2/(2*x^2) + (a*b*d^2*atanh(sqrt(c + d*x)/sqrt(c)))/(2*c^(3/2))]

    @test_int [x^3*sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)^3*(a + b*sqrt(c + d*x))^(3/2))/(3*b^8*d^4)) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(5/2))/(5*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(7/2))/(7*b^8*d^4) + (4*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*sqrt(c + d*x))^(9/2))/(9*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(11/2))/(11*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(13/2))/(13*b^8*d^4) - (28*a*(a + b*sqrt(c + d*x))^(15/2))/(15*b^8*d^4) + (4*(a + b*sqrt(c + d*x))^(17/2))/(17*b^8*d^4)]
    @test_int [x^2*sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)^2*(a + b*sqrt(c + d*x))^(3/2))/(3*b^6*d^3)) + (4*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*sqrt(c + d*x))^(5/2))/(5*b^6*d^3) - (8*a*(5*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(7/2))/(7*b^6*d^3) + (8*(5*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(9/2))/(9*b^6*d^3) - (20*a*(a + b*sqrt(c + d*x))^(11/2))/(11*b^6*d^3) + (4*(a + b*sqrt(c + d*x))^(13/2))/(13*b^6*d^3)]
    @test_int [x^1*sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(3/2))/(3*b^4*d^2)) + (4*(3*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(5/2))/(5*b^4*d^2) - (12*a*(a + b*sqrt(c + d*x))^(7/2))/(7*b^4*d^2) + (4*(a + b*sqrt(c + d*x))^(9/2))/(9*b^4*d^2)]
    @test_int [x^0*sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a + b*sqrt(c + d*x))^(3/2))/(3*b^2*d)) + (4*(a + b*sqrt(c + d*x))^(5/2))/(5*b^2*d)]
    @test_int [sqrt(a + b*sqrt(c + d*x))/x^1, x, 7, 4*sqrt(a + b*sqrt(c + d*x)) - 2*sqrt(a - b*sqrt(c))*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))) - 2*sqrt(a + b*sqrt(c))*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c)))]
    @test_int [sqrt(a + b*sqrt(c + d*x))/x^2, x, 8, -(sqrt(a + b*sqrt(c + d*x))/x) + (b*d*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))))/(2*sqrt(a - b*sqrt(c))*sqrt(c)) - (b*d*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))))/(2*sqrt(a + b*sqrt(c))*sqrt(c))]
    @test_int [sqrt(a + b*sqrt(c + d*x))/x^3, x, 9, -(sqrt(a + b*sqrt(c + d*x))/(2*x^2)) + (b*d*(b*c - a*sqrt(c + d*x))*sqrt(a + b*sqrt(c + d*x)))/(8*c*(a^2 - b^2*c)*x) - (b*(2*a - 3*b*sqrt(c))*d^2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))))/(16*(a - b*sqrt(c))^(3/2)*c^(3/2)) + (b*(2*a + 3*b*sqrt(c))*d^2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))))/(16*(a + b*sqrt(c))^(3/2)*c^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [x^3/(a + b*sqrt(c + d*x)), x, 4, -((a*(a^4 - 3*a^2*b^2*c + 3*b^4*c^2)*x)/(b^6*d^3)) + (2*(a^2 - b^2*c)^3*sqrt(c + d*x))/(b^7*d^4) + (2*(a^4 - 3*a^2*b^2*c + 3*b^4*c^2)*(c + d*x)^(3/2))/(3*b^5*d^4) - (a*(a^2 - 3*b^2*c)*(c + d*x)^2)/(2*b^4*d^4) + (2*(a^2 - 3*b^2*c)*(c + d*x)^(5/2))/(5*b^3*d^4) - (a*(c + d*x)^3)/(3*b^2*d^4) + (2*(c + d*x)^(7/2))/(7*b*d^4) - (2*a*(a^2 - b^2*c)^3*log(a + b*sqrt(c + d*x)))/(b^8*d^4)]
    @test_int [x^2/(a + b*sqrt(c + d*x)), x, 4, -((a*(a^2 - 2*b^2*c)*x)/(b^4*d^2)) + (2*(a^2 - b^2*c)^2*sqrt(c + d*x))/(b^5*d^3) + (2*(a^2 - 2*b^2*c)*(c + d*x)^(3/2))/(3*b^3*d^3) - (a*(c + d*x)^2)/(2*b^2*d^3) + (2*(c + d*x)^(5/2))/(5*b*d^3) - (2*a*(a^2 - b^2*c)^2*log(a + b*sqrt(c + d*x)))/(b^6*d^3)]
    @test_int [x^1/(a + b*sqrt(c + d*x)), x, 4, -((a*x)/(b^2*d)) + (2*(a^2 - b^2*c)*sqrt(c + d*x))/(b^3*d^2) + (2*(c + d*x)^(3/2))/(3*b*d^2) - (2*a*(a^2 - b^2*c)*log(a + b*sqrt(c + d*x)))/(b^4*d^2)]
    @test_int [x^0/(a + b*sqrt(c + d*x)), x, 4, (2*sqrt(c + d*x))/(b*d) - (2*a*log(a + b*sqrt(c + d*x)))/(b^2*d)]
    @test_int [1/(x^1*(a + b*sqrt(c + d*x))), x, 7, (2*b*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c)))/(a^2 - b^2*c) + (a*log(x))/(a^2 - b^2*c) - (2*a*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)]
    @test_int [1/(x^2*(a + b*sqrt(c + d*x))), x, 8, -((a - b*sqrt(c + d*x))/((a^2 - b^2*c)*x)) + (b*(a^2 + b^2*c)*d*atanh(sqrt(c + d*x)/sqrt(c)))/(sqrt(c)*(a^2 - b^2*c)^2) + (a*b^2*d*log(x))/(a^2 - b^2*c)^2 - (2*a*b^2*d*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)^2]
    @test_int [1/(x^3*(a + b*sqrt(c + d*x))), x, 9, -((a - b*sqrt(c + d*x))/(2*(a^2 - b^2*c)*x^2)) - (b*d*(4*a*b*c - (a^2 + 3*b^2*c)*sqrt(c + d*x)))/(4*c*(a^2 - b^2*c)^2*x) - (b*(a^4 - 6*a^2*b^2*c - 3*b^4*c^2)*d^2*atanh(sqrt(c + d*x)/sqrt(c)))/(4*c^(3/2)*(a^2 - b^2*c)^3) + (a*b^4*d^2*log(x))/(a^2 - b^2*c)^3 - (2*a*b^4*d^2*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)^3]

    @test_int [x^3/(a + b*sqrt(c + d*x))^2, x, 4, ((5*a^4 - 9*a^2*b^2*c + 3*b^4*c^2)*x)/(b^6*d^3) - (12*a*(a^2 - b^2*c)^2*sqrt(c + d*x))/(b^7*d^4) - (4*a*(2*a^2 - 3*b^2*c)*(c + d*x)^(3/2))/(3*b^5*d^4) + (3*(a^2 - b^2*c)*(c + d*x)^2)/(2*b^4*d^4) - (4*a*(c + d*x)^(5/2))/(5*b^3*d^4) + (c + d*x)^3/(3*b^2*d^4) + (2*a*(a^2 - b^2*c)^3)/(b^8*d^4*(a + b*sqrt(c + d*x))) + (2*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*log(a + b*sqrt(c + d*x)))/(b^8*d^4)]
    @test_int [x^2/(a + b*sqrt(c + d*x))^2, x, 4, ((3*a^2 - 2*b^2*c)*x)/(b^4*d^2) - (8*a*(a^2 - b^2*c)*sqrt(c + d*x))/(b^5*d^3) - (4*a*(c + d*x)^(3/2))/(3*b^3*d^3) + (c + d*x)^2/(2*b^2*d^3) + (2*a*(a^2 - b^2*c)^2)/(b^6*d^3*(a + b*sqrt(c + d*x))) + (2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*log(a + b*sqrt(c + d*x)))/(b^6*d^3)]
    @test_int [x^1/(a + b*sqrt(c + d*x))^2, x, 4, x/(b^2*d) - (4*a*sqrt(c + d*x))/(b^3*d^2) + (2*a*(a^2 - b^2*c))/(b^4*d^2*(a + b*sqrt(c + d*x))) + (2*(3*a^2 - b^2*c)*log(a + b*sqrt(c + d*x)))/(b^4*d^2)]
    @test_int [x^0/(a + b*sqrt(c + d*x))^2, x, 4, (2*a)/(b^2*d*(a + b*sqrt(c + d*x))) + (2*log(a + b*sqrt(c + d*x)))/(b^2*d)]
    @test_int [1/(x^1*(a + b*sqrt(c + d*x))^2), x, 7, (2*a)/((a^2 - b^2*c)*(a + b*sqrt(c + d*x))) + (4*a*b*sqrt(c)*atanh(sqrt(c + d*x)/sqrt(c)))/(a^2 - b^2*c)^2 + ((a^2 + b^2*c)*log(x))/(a^2 - b^2*c)^2 - (2*(a^2 + b^2*c)*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)^2]
    @test_int [1/(x^2*(a + b*sqrt(c + d*x))^2), x, 8, (4*a*b^2*d)/((a^2 - b^2*c)^2*(a + b*sqrt(c + d*x))) - (a - b*sqrt(c + d*x))/((a^2 - b^2*c)*x*(a + b*sqrt(c + d*x))) + (2*a*b*(a^2 + 3*b^2*c)*d*atanh(sqrt(c + d*x)/sqrt(c)))/(sqrt(c)*(a^2 - b^2*c)^3) + (b^2*(3*a^2 + b^2*c)*d*log(x))/(a^2 - b^2*c)^3 - (2*b^2*(3*a^2 + b^2*c)*d*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)^3]
    @test_int [1/(x^3*(a + b*sqrt(c + d*x))^2), x, 9, (a*b^2*(a^2 + 11*b^2*c)*d^2)/(2*c*(a^2 - b^2*c)^3*(a + b*sqrt(c + d*x))) - (a - b*sqrt(c + d*x))/(2*(a^2 - b^2*c)*x^2*(a + b*sqrt(c + d*x))) - (b*d*(3*a*b*c - (a^2 + 2*b^2*c)*sqrt(c + d*x)))/(2*c*(a^2 - b^2*c)^2*x*(a + b*sqrt(c + d*x))) - (a*b*(a^4 - 10*a^2*b^2*c - 15*b^4*c^2)*d^2*atanh(sqrt(c + d*x)/sqrt(c)))/(2*c^(3/2)*(a^2 - b^2*c)^4) + (b^4*(5*a^2 + b^2*c)*d^2*log(x))/(a^2 - b^2*c)^4 - (2*b^4*(5*a^2 + b^2*c)*d^2*log(a + b*sqrt(c + d*x)))/(a^2 - b^2*c)^4]

    @test_int [x^3/sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)^3*sqrt(a + b*sqrt(c + d*x)))/(b^8*d^4)) + (4*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(3/2))/(3*b^8*d^4) - (12*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(5/2))/(5*b^8*d^4) + (4*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*sqrt(c + d*x))^(7/2))/(7*b^8*d^4) - (20*a*(7*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(9/2))/(9*b^8*d^4) + (12*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(11/2))/(11*b^8*d^4) - (28*a*(a + b*sqrt(c + d*x))^(13/2))/(13*b^8*d^4) + (4*(a + b*sqrt(c + d*x))^(15/2))/(15*b^8*d^4)]
    @test_int [x^2/sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)^2*sqrt(a + b*sqrt(c + d*x)))/(b^6*d^3)) + (4*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*sqrt(c + d*x))^(3/2))/(3*b^6*d^3) - (8*a*(5*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(5/2))/(5*b^6*d^3) + (8*(5*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(7/2))/(7*b^6*d^3) - (20*a*(a + b*sqrt(c + d*x))^(9/2))/(9*b^6*d^3) + (4*(a + b*sqrt(c + d*x))^(11/2))/(11*b^6*d^3)]
    @test_int [x^1/sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*(a^2 - b^2*c)*sqrt(a + b*sqrt(c + d*x)))/(b^4*d^2)) + (4*(3*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(3/2))/(3*b^4*d^2) - (12*a*(a + b*sqrt(c + d*x))^(5/2))/(5*b^4*d^2) + (4*(a + b*sqrt(c + d*x))^(7/2))/(7*b^4*d^2)]
    @test_int [x^0/sqrt(a + b*sqrt(c + d*x)), x, 4, -((4*a*sqrt(a + b*sqrt(c + d*x)))/(b^2*d)) + (4*(a + b*sqrt(c + d*x))^(3/2))/(3*b^2*d)]
    @test_int [1/(x^1*sqrt(a + b*sqrt(c + d*x))), x, 6, -((2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))))/sqrt(a - b*sqrt(c))) - (2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))))/sqrt(a + b*sqrt(c))]
    @test_int [1/(x^2*sqrt(a + b*sqrt(c + d*x))), x, 7, -(((a - b*sqrt(c + d*x))*sqrt(a + b*sqrt(c + d*x)))/((a^2 - b^2*c)*x)) - (b*d*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))))/(2*(a - b*sqrt(c))^(3/2)*sqrt(c)) + (b*d*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))))/(2*(a + b*sqrt(c))^(3/2)*sqrt(c))]
    @test_int [1/(x^3*sqrt(a + b*sqrt(c + d*x))), x, 8, -(((a - b*sqrt(c + d*x))*sqrt(a + b*sqrt(c + d*x)))/(2*(a^2 - b^2*c)*x^2)) - (b*d*sqrt(a + b*sqrt(c + d*x))*(6*a*b*c - (a^2 + 5*b^2*c)*sqrt(c + d*x)))/(8*c*(a^2 - b^2*c)^2*x) + (b*(2*a - 5*b*sqrt(c))*d^2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a - b*sqrt(c))))/(16*(a - b*sqrt(c))^(5/2)*c^(3/2)) - (b*(2*a + 5*b*sqrt(c))*d^2*atanh(sqrt(a + b*sqrt(c + d*x))/sqrt(a + b*sqrt(c))))/(16*(a + b*sqrt(c))^(5/2)*c^(3/2))]

    #= ::Subsubsection::Closed:: =#
    #=p*symbolic=#

    @test_int [x^3*(a + b*sqrt(c + d*x))^p, x, 4, -((2*a*(a^2 - b^2*c)^3*(a + b*sqrt(c + d*x))^(1 + p))/(b^8*d^4*(1 + p))) + (2*(a^2 - b^2*c)^2*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(2 + p))/(b^8*d^4*(2 + p)) - (6*a*(7*a^2 - 3*b^2*c)*(a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(3 + p))/(b^8*d^4*(3 + p)) + (2*(35*a^4 - 30*a^2*b^2*c + 3*b^4*c^2)*(a + b*sqrt(c + d*x))^(4 + p))/(b^8*d^4*(4 + p)) - (10*a*(7*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(5 + p))/(b^8*d^4*(5 + p)) + (6*(7*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(6 + p))/(b^8*d^4*(6 + p)) - (14*a*(a + b*sqrt(c + d*x))^(7 + p))/(b^8*d^4*(7 + p)) + (2*(a + b*sqrt(c + d*x))^(8 + p))/(b^8*d^4*(8 + p))]
    @test_int [x^2*(a + b*sqrt(c + d*x))^p, x, 4, -((2*a*(a^2 - b^2*c)^2*(a + b*sqrt(c + d*x))^(1 + p))/(b^6*d^3*(1 + p))) + (2*(5*a^4 - 6*a^2*b^2*c + b^4*c^2)*(a + b*sqrt(c + d*x))^(2 + p))/(b^6*d^3*(2 + p)) - (4*a*(5*a^2 - 3*b^2*c)*(a + b*sqrt(c + d*x))^(3 + p))/(b^6*d^3*(3 + p)) + (4*(5*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(4 + p))/(b^6*d^3*(4 + p)) - (10*a*(a + b*sqrt(c + d*x))^(5 + p))/(b^6*d^3*(5 + p)) + (2*(a + b*sqrt(c + d*x))^(6 + p))/(b^6*d^3*(6 + p))]
    @test_int [x^1*(a + b*sqrt(c + d*x))^p, x, 4, -((2*a*(a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(1 + p))/(b^4*d^2*(1 + p))) + (2*(3*a^2 - b^2*c)*(a + b*sqrt(c + d*x))^(2 + p))/(b^4*d^2*(2 + p)) - (6*a*(a + b*sqrt(c + d*x))^(3 + p))/(b^4*d^2*(3 + p)) + (2*(a + b*sqrt(c + d*x))^(4 + p))/(b^4*d^2*(4 + p))]
    @test_int [x^0*(a + b*sqrt(c + d*x))^p, x, 4, -((2*a*(a + b*sqrt(c + d*x))^(1 + p))/(b^2*d*(1 + p))) + (2*(a + b*sqrt(c + d*x))^(2 + p))/(b^2*d*(2 + p))]
    @test_int [(a + b*sqrt(c + d*x))^p/x^1, x, 6, -(((a + b*sqrt(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (a + b*sqrt(c + d*x))/(a - b*sqrt(c))))/((a - b*sqrt(c))*(1 + p))) - ((a + b*sqrt(c + d*x))^(1 + p)*HypergeometricFunctions._₂F₁(1, 1 + p, 2 + p, (a + b*sqrt(c + d*x))/(a + b*sqrt(c))))/((a + b*sqrt(c))*(1 + p))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*F((c*x)^n)=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + b*(c*x)^n)^(p/2) / x=#

    @test_int [(a + b*(c*x)^n)^(5/2)/x, x, 8, (2*a^2*sqrt(a + b*(c*x)^n))/n + (2*a*(a + b*(c*x)^n)^(3/2))/(3*n) + (2*(a + b*(c*x)^n)^(5/2))/(5*n) - (2*a^(5/2)*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [(a + b*(c*x)^n)^(3/2)/x, x, 7, (2*a*sqrt(a + b*(c*x)^n))/n + (2*(a + b*(c*x)^n)^(3/2))/(3*n) - (2*a^(3/2)*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [(a + b*(c*x)^n)^(1/2)/x, x, 6, (2*sqrt(a + b*(c*x)^n))/n - (2*sqrt(a)*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [1/(x*(a + b*(c*x)^n)^(1/2)), x, 5, -((2*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/(sqrt(a)*n))]
    @test_int [1/(x*(a + b*(c*x)^n)^(3/2)), x, 6, 2/(a*n*sqrt(a + b*(c*x)^n)) - (2*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/(a^(3/2)*n)]
    @test_int [1/(x*(a + b*(c*x)^n)^(5/2)), x, 7, 2/(3*a*n*(a + b*(c*x)^n)^(3/2)) + 2/(a^2*n*sqrt(a + b*(c*x)^n)) - (2*atanh(sqrt(a + b*(c*x)^n)/sqrt(a)))/(a^(5/2)*n)]

    @test_int [(-a + b*(c*x)^n)^(5/2)/x, x, 8, (2*a^2*sqrt(-a + b*(c*x)^n))/n - (2*a*(-a + b*(c*x)^n)^(3/2))/(3*n) + (2*(-a + b*(c*x)^n)^(5/2))/(5*n) - (2*a^(5/2)*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [(-a + b*(c*x)^n)^(3/2)/x, x, 7, -((2*a*sqrt(-a + b*(c*x)^n))/n) + (2*(-a + b*(c*x)^n)^(3/2))/(3*n) + (2*a^(3/2)*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [(-a + b*(c*x)^n)^(1/2)/x, x, 6, (2*sqrt(-a + b*(c*x)^n))/n - (2*sqrt(a)*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/n]
    @test_int [1/(x*(-a + b*(c*x)^n)^(1/2)), x, 5, (2*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/(sqrt(a)*n)]
    @test_int [1/(x*(-a + b*(c*x)^n)^(3/2)), x, 6, -(2/(a*n*sqrt(-a + b*(c*x)^n))) - (2*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/(a^(3/2)*n)]
    @test_int [1/(x*(-a + b*(c*x)^n)^(5/2)), x, 7, -(2/(3*a*n*(-a + b*(c*x)^n)^(3/2))) + 2/(a^2*n*sqrt(-a + b*(c*x)^n)) + (2*atan(sqrt(-a + b*(c*x)^n)/sqrt(a)))/(a^(5/2)*n)]

    #= ::Subsection::Closed:: =#
    #=F(x^n) / x=#

    @test_int [1/(x*sqrt(a + b*x)), x, 2, -((2*atanh(sqrt(a + b*x)/sqrt(a)))/sqrt(a))]
    @test_int [1/(x*sqrt(a + b*(c*x)^m)), x, 5, -((2*atanh(sqrt(a + b*(c*x)^m)/sqrt(a)))/(sqrt(a)*m))]
    @test_int [1/(x*sqrt(a + b*(c*(d*x)^m)^n)), x, 6, -((2*atanh(sqrt(a + b*(c*(d*x)^m)^n)/sqrt(a)))/(sqrt(a)*m*n))]
    @test_int [1/(x*sqrt(a + b*(c*(d*(e*x)^m)^n)^p)), x, 7, -((2*atanh(sqrt(a + b*(c*(d*(e*x)^m)^n)^p)/sqrt(a)))/(sqrt(a)*m*n*p))]
    @test_int [1/(x*sqrt(a + b*(c*(d*(e*(f*x)^m)^n)^p)^q)), x, 8, -((2*atanh(sqrt(a + b*(c*(d*(e*(f*x)^m)^n)^p)^q)/sqrt(a)))/(sqrt(a)*m*n*p*q))]

    @test_int [sqrt(-1 + 1/x^2)*(-1 + x^2)^3/x, x, 8, (35/16)*sqrt(-1 + 1/x^2) - (35/48)*(-1 + 1/x^2)^(3/2)*x^2 - (7/24)*(-1 + 1/x^2)^(5/2)*x^4 - (1/6)*(-1 + 1/x^2)^(7/2)*x^6 - (35/16)*atan(sqrt(-1 + 1/x^2))]
    @test_int [sqrt(-1 + 1/x^2)*(-1 + x^2)^2/x, x, 7, (-(15/8))*sqrt(-1 + 1/x^2) + (5/8)*(-1 + 1/x^2)^(3/2)*x^2 + (1/4)*(-1 + 1/x^2)^(5/2)*x^4 + (15/8)*atan(sqrt(-1 + 1/x^2))]
    @test_int [sqrt(-1 + 1/x^2)*(-1 + x^2)^1/x, x, 6, (3/2)*sqrt(-1 + 1/x^2) - (1/2)*(-1 + 1/x^2)^(3/2)*x^2 - (3/2)*atan(sqrt(-1 + 1/x^2))]
    @test_int [sqrt(-1 + 1/x^2)/(x*(-1 + x^2)^1), x, 2, sqrt(-1 + 1/x^2)]
    @test_int [sqrt(-1 + 1/x^2)/(x*(-1 + x^2)^2), x, 4, 1/sqrt(-1 + 1/x^2) - sqrt(-1 + 1/x^2)]
    @test_int [sqrt(-1 + 1/x^2)/(x*(-1 + x^2)^3), x, 4, -(1/(3*(-1 + 1/x^2)^(3/2))) - 2/sqrt(-1 + 1/x^2) + sqrt(-1 + 1/x^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*F(x^n)=#

    @test_int [x*sqrt(1 + 1/x^2)/(1 + x^2)^2, x, 2, 1/sqrt(1 + x^(-2))]
    @test_int [1/(sqrt(1 + x^(-2))*x*(1 + x^2)), x, 2, 1/sqrt(1 + x^(-2))]

    @test_int [x/(a + b*x^2 + sqrt(a + b*x^2)), x, 3, log(1 + sqrt(a + b*x^2))/b]
    @test_int [x/(x^2 - (x^2)^(1/3)), x, 3, (3/4)*log(1 - (x^2)^(2/3))]
    @test_int [x*(1 + x^2)^3*sqrt(2 + 2*x^2 + x^4), x, 3, (-(1/15))*(2 + 2*x^2 + x^4)^(3/2) + (1/10)*(1 + x^2)^2*(2 + 2*x^2 + x^4)^(3/2)]

    @test_int [x^5*sqrt(1 - x^3)*(1 + x^9)^2, x, 3, (-(8/9))*(1 - x^3)^(3/2) + (32/15)*(1 - x^3)^(5/2) - (22/7)*(1 - x^3)^(7/2) + (86/27)*(1 - x^3)^(9/2) - (74/33)*(1 - x^3)^(11/2) + (14/13)*(1 - x^3)^(13/2) - (14/45)*(1 - x^3)^(15/2) + (2/51)*(1 - x^3)^(17/2)]

    #= Note: Use*the*substitution*u=x^2*instead*of*algebraic*expansion. =#
    @test_int [x/(a + b*x^2)^(3/2) + x/((1 + x^2)*sqrt(a + b*x^2)), x, 5, -(1/(b*sqrt(a + b*x^2))) - atanh(sqrt(a + b*x^2)/sqrt(a - b))/sqrt(a - b)]
    @test_int [(x*(1 + a + x^2 + b*x^2))/((1 + x^2)*(a + b*x^2)^(3/2)), x, 5, -(1/(b*sqrt(a + b*x^2))) - atanh(sqrt(a + b*x^2)/sqrt(a - b))/sqrt(a - b)]

    @test_int [x/(a + b*x^2)^(5/2) + x/(a + b*x^2)^(3/2) + x/((1 + x^2)*sqrt(a + b*x^2)), x, 6, -(1/(3*b*(a + b*x^2)^(3/2))) - 1/(b*sqrt(a + b*x^2)) - atanh(sqrt(a + b*x^2)/sqrt(a - b))/sqrt(a - b)]
    @test_int [(x*(1 + a + a^2 + x^2 + a*x^2 + b*x^2 + 2*a*b*x^2 + b*x^4 + b^2*x^4))/((1 + x^2)*(a + b*x^2)^(5/2)) , x, 9, -(1/(3*b*(a + b*x^2)^(3/2))) - 1/(b*sqrt(a + b*x^2)) - atanh(sqrt(a + b*x^2)/sqrt(a - b))/sqrt(a - b)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*F((a + b*x)^(1/n), x)=#

    @test_int [1/sqrt(sqrt(x) + x), x, 4, 2*sqrt(sqrt(x) + x) - 2*atanh(sqrt(x)/sqrt(sqrt(x) + x))]
    @test_int [sqrt(sqrt(x) + x), x, 6, (-(1/4))*sqrt(sqrt(x) + x) + (1/6)*sqrt(x)*sqrt(sqrt(x) + x) + (2/3)*x*sqrt(sqrt(x) + x) + (1/4)*atanh(sqrt(x)/sqrt(sqrt(x) + x))]

    @test_int [sqrt(-x)*(sqrt(-x) + x), x, 2, (2/5)*(-x)^(5/2) - x^2/2]
    @test_int [(5 + x^(1/4))/(-6 + x), x, 8, 4*x^(1/4) - 2*6^(1/4)*atan(x^(1/4)/6^(1/4)) - 2*6^(1/4)*atanh(x^(1/4)/6^(1/4)) + 5*log(6 - x)]

    @test_int [1/(4 - x + sqrt(4 - x)), x, 2, -2*log(1 + sqrt(4 - x))]
    @test_int [1/(1 + x - sqrt(2 + x)), x, 4, (1/5)*(5 - sqrt(5))*log(1 - sqrt(5) - 2*sqrt(2 + x)) + (1/5)*(5 + sqrt(5))*log(1 + sqrt(5) - 2*sqrt(2 + x))]
    @test_int [1/(4 + x + sqrt(1 + x)), x, 5, (-2*atan((1 + 2*sqrt(1 + x))/sqrt(11)))/sqrt(11) + log(4 + x + sqrt(1 + x))]

    @test_int [1/(x - sqrt(1 + x)), x, 4, (1/5)*(5 - sqrt(5))*log(1 - sqrt(5) - 2*sqrt(1 + x)) + (1/5)*(5 + sqrt(5))*log(1 + sqrt(5) - 2*sqrt(1 + x))]
    @test_int [1/(x - sqrt(2 + x)), x, 4, (4/3)*log(2 - sqrt(2 + x)) + (2/3)*log(1 + sqrt(2 + x))]
    @test_int [1/(x - sqrt(1 - x)), x, 4, (1/5)*(5 - sqrt(5))*log(1 - sqrt(5) + 2*sqrt(1 - x)) + (1/5)*(5 + sqrt(5))*log(1 + sqrt(5) + 2*sqrt(1 - x))]

    @test_int [sqrt(1 + x + sqrt(x)), x, 5, (-(1/4))*(1 + 2*sqrt(x))*sqrt(1 + sqrt(x) + x) + (2/3)*(1 + sqrt(x) + x)^(3/2) - (3/8)*asinh((1 + 2*sqrt(x))/sqrt(3))]
    @test_int [sqrt(1 + x + sqrt(1 + x)), x, 6, (2/3)*(1 + x + sqrt(1 + x))^(3/2) - (1/4)*sqrt(1 + x + sqrt(1 + x))*(1 + 2*sqrt(1 + x)) + (1/4)*atanh(sqrt(1 + x)/sqrt(1 + x + sqrt(1 + x)))]

    @test_int [sqrt(x + sqrt(-1 + x)), x, 5, (-(1/4))*(1 + 2*sqrt(-1 + x))*sqrt(sqrt(-1 + x) + x) + (2/3)*(sqrt(-1 + x) + x)^(3/2) - (3/8)*asinh((1 + 2*sqrt(-1 + x))/sqrt(3))]
    @test_int [sqrt(2*x + sqrt(-1 + 2*x)), x, 5, (1/3)*(2*x + sqrt(-1 + 2*x))^(3/2) - (1/8)*sqrt(2*x + sqrt(-1 + 2*x))*(1 + 2*sqrt(-1 + 2*x)) - (3/16)*asinh((1 + 2*sqrt(-1 + 2*x))/sqrt(3))]
    @test_int [sqrt(3*x + sqrt(-7 + 8*x)), x, 5, -(((4 + 3*sqrt(-7 + 8*x))*sqrt(21 - 3*(7 - 8*x) + 8*sqrt(-7 + 8*x)))/(36*sqrt(2))) + (21 - 3*(7 - 8*x) + 8*sqrt(-7 + 8*x))^(3/2)/(72*sqrt(2)) - (47*asinh((4 + 3*sqrt(-7 + 8*x))/sqrt(47)))/(36*sqrt(6))]

    @test_int [1/sqrt(x + sqrt(1 + x)), x, 4, 2*sqrt(x + sqrt(1 + x)) - atanh((1 + 2*sqrt(1 + x))/(2*sqrt(x + sqrt(1 + x))))]

    @test_int [(1 + x)/(4 + x + sqrt(-9 + 6*x)), x, 7, x - 2*sqrt(3)*sqrt(-3 + 2*x) + 4*sqrt(6)*atan((3 + sqrt(-9 + 6*x))/(2*sqrt(6))) + 3*log(4 + x + sqrt(3)*sqrt(-3 + 2*x))]
    @test_int [(12 - x)/(4 + x + sqrt(-9 + 6*x)), x, 7, -x + 2*sqrt(3)*sqrt(-3 + 2*x) - 21*sqrt(3/2)*atan((3 + sqrt(-9 + 6*x))/(2*sqrt(6))) + 10*log(4 + x + sqrt(3)*sqrt(-3 + 2*x))]
    @test_int [(-1 + x^3)/(sqrt(x)*(1 + x^2)), x, 8, (2*x^(3/2))/3 + sqrt(2)*atan(1 - sqrt(2)*sqrt(x)) - sqrt(2)*atan(1 + sqrt(2)*sqrt(x))]
    @test_int [1/(2*sqrt(-1 + x)*sqrt(-sqrt(-1 + x) + x)), x, 4, -asinh((1 - 2*sqrt(-1 + x))/sqrt(3))]
    @test_int [(1 + x^(7/2))/(1 - x^2), x, 10, -2*sqrt(x) - (2*x^(5/2))/5 + atan(sqrt(x)) - log(1 - sqrt(x)) + (1/2)*log(1 + x), -2*sqrt(x) - (2*x^(5/2))/5 + atan(sqrt(x)) + atanh(sqrt(x)) + atanh(x)]

    @test_int [(4 + 2*x)/((-1 + 2*x)^(1/3) + sqrt(-1 + 2*x)), x, 3, -x + 18*(-1 + 2*x)^(1/6) - 9*(-1 + 2*x)^(1/3) + 6*sqrt(-1 + 2*x) - (3/4)*(-1 + 2*x)^(2/3) + (3/5)*(-1 + 2*x)^(5/6) + (3/7)*(-1 + 2*x)^(7/6) - (3/8)*(-1 + 2*x)^(4/3) + (1/3)*(-1 + 2*x)^(3/2) - 18*log(1 + (-1 + 2*x)^(1/6))]

    #= Integrands*of*the*form*sqrt(a+b*sqrt(c+d*sqrt(e+f*x))) =#
    @test_int [1/sqrt(2 + sqrt(1 + sqrt(x))), x, 5, -48*sqrt(2 + sqrt(1 + sqrt(x))) + (88/3)*(2 + sqrt(1 + sqrt(x)))^(3/2) - (48/5)*(2 + sqrt(1 + sqrt(x)))^(5/2) + (8/7)*(2 + sqrt(1 + sqrt(x)))^(7/2)]
    @test_int [sqrt(2 + sqrt(4 + sqrt(x))), x, 5, (64/5)*(2 + sqrt(4 + sqrt(x)))^(5/2) - (48/7)*(2 + sqrt(4 + sqrt(x)))^(7/2) + (8/9)*(2 + sqrt(4 + sqrt(x)))^(9/2)]
    @test_int [sqrt(2 - sqrt(4 + sqrt(-9 + 5*x))), x, 5, (64/25)*(2 - sqrt(4 + sqrt(-9 + 5*x)))^(5/2) - (48/35)*(2 - sqrt(4 + sqrt(-9 + 5*x)))^(7/2) + (8/45)*(2 - sqrt(4 + sqrt(-9 + 5*x)))^(9/2)]
    @test_int [1/sqrt(2 + sqrt(1 + sqrt(x))), x, 5, -48*sqrt(2 + sqrt(1 + sqrt(x))) + (88/3)*(2 + sqrt(1 + sqrt(x)))^(3/2) - (48/5)*(2 + sqrt(1 + sqrt(x)))^(5/2) + (8/7)*(2 + sqrt(1 + sqrt(x)))^(7/2)]

    #= Integrands*of*the*form*sqrt(a+b*sqrt(c+d*sqrt(e+f*sqrt(g+h*x)))) =#
    @test_int [sqrt(1 + sqrt(1 + sqrt(1 + sqrt(x)))), x, 6, (-(32/5))*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(5/2) + (48/7)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(7/2) + (112/9)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(9/2) - (320/11)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(11/2) + (288/13)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(13/2) - (112/15)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(15/2) + (16/17)*(1 + sqrt(1 + sqrt(1 + sqrt(x))))^(17/2)]
    @test_int [sqrt(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x)))), x, 5, (-(16/3))*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(3/2) + (136/5)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(5/2) - (480/7)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(7/2) + (304/3)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(9/2) - (760/11)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(11/2) + (300/13)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(13/2) - (56/15)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(15/2) + (4/17)*(2 + sqrt(3 + sqrt(-1 + 2*sqrt(x))))^(17/2)]

    @test_int [x*sqrt(1 + sqrt(1 + sqrt(-1 + x))), x, 5, (16/5)*(1 + sqrt(1 + sqrt(-1 + x)))^(5/2) - (24/7)*(1 + sqrt(1 + sqrt(-1 + x)))^(7/2) + 8*(1 + sqrt(1 + sqrt(-1 + x)))^(9/2) - (160/11)*(1 + sqrt(1 + sqrt(-1 + x)))^(11/2) + (144/13)*(1 + sqrt(1 + sqrt(-1 + x)))^(13/2) - (56/15)*(1 + sqrt(1 + sqrt(-1 + x)))^(15/2) + (8/17)*(1 + sqrt(1 + sqrt(-1 + x)))^(17/2)]

    @test_int [1/(sqrt(-1 + x)*sqrt(-sqrt(-1 + x) + x)), x, 3, -2*asinh((1 - 2*sqrt(-1 + x))/sqrt(3))]
    @test_int [1/sqrt(1 + x + sqrt(-1 + 2*x)), x, 4, 2*sqrt(1 + x + sqrt(-1 + 2*x)) - sqrt(2)*asinh((1 + sqrt(-1 + 2*x))/sqrt(2)), sqrt(2)*sqrt(2 + 2*x + 2*sqrt(-1 + 2*x)) - sqrt(2)*asinh((1 + sqrt(-1 + 2*x))/sqrt(2))]
    @test_int [(q + p*x)/(sqrt(b + a*x)*(f + sqrt(b + a*x))), x, 3, (p*x)/a - (2*f*p*sqrt(b + a*x))/a^2 - (2*(b*p - f^2*p - a*q)*log(f + sqrt(b + a*x)))/a^2]
    @test_int [sqrt(1 - sqrt(x) - x), x, 5, (-(1/4))*(1 + 2*sqrt(x))*sqrt(1 - sqrt(x) - x) - (2/3)*(1 - sqrt(x) - x)^(3/2) - (5/8)*asin((1 + 2*sqrt(x))/sqrt(5))]

    @test_int [(9 + 6*sqrt(x) + x)/(4*sqrt(x) + x), x, 4, 4*sqrt(x) + x + 2*log(4 + sqrt(x))]
    @test_int [(6 - 8*x^(7/2))/(5 - 9*sqrt(x)), x, 8, -((56145628*sqrt(x))/43046721) + (125000*x)/4782969 + (50000*x^(3/2))/1594323 + (2500*x^2)/59049 + (400*x^(5/2))/6561 + (200*x^3)/2187 + (80*x^(7/2))/567 + (2*x^4)/9 - (280728140*log(5 - 9*sqrt(x)))/387420489]

    #= Although*the*following*optimal*antiderivative*contains*the*imaginary*unit, it*is*continuous*for*x*along*the*real*line. =#
    @test_int [(sqrt(1 + x)*(1 + x^3))/(1 + x^2), x, 16, -2*sqrt(1 + x) - (2/3)*(1 + x)^(3/2) + (2/5)*(1 + x)^(5/2) + (1 - im)^(3/2)*atanh(sqrt(1 + x)/sqrt(1 - im)) + (1 + im)^(3/2)*atanh(sqrt(1 + x)/sqrt(1 + im)), -2*sqrt(1 + x) - (2/3)*(1 + x)^(3/2) + (2/5)*(1 + x)^(5/2) - sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) - 2*sqrt(1 + x))/sqrt(2*(-1 + sqrt(2)))) + sqrt(1 + sqrt(2))*atan((sqrt(2*(1 + sqrt(2))) + 2*sqrt(1 + x))/sqrt(2*(-1 + sqrt(2)))) - log(1 + sqrt(2) + x - sqrt(2*(1 + sqrt(2)))*sqrt(1 + x))/(2*sqrt(1 + sqrt(2))) + log(1 + sqrt(2) + x + sqrt(2*(1 + sqrt(2)))*sqrt(1 + x))/(2*sqrt(1 + sqrt(2)))]

    @test_int [sqrt(-1 - sqrt(x) + x)/((-1 + x)*sqrt(x)), x, 9, atan((3 - sqrt(x))/(2*sqrt(-1 - sqrt(x) + x))) - 2*atanh((1 - 2*sqrt(x))/(2*sqrt(-1 - sqrt(x) + x))) - atanh((1 + 3*sqrt(x))/(2*sqrt(-1 - sqrt(x) + x)))]

    @test_int [(1 + 2*sqrt(1 + x))/(x*sqrt(1 + x)*sqrt(x + sqrt(1 + x))), x, 6, -atan((3 + sqrt(1 + x))/(2*sqrt(x + sqrt(1 + x)))) + 3*atanh((1 - 3*sqrt(1 + x))/(2*sqrt(x + sqrt(1 + x))))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*F(((a + b*x)/(c + d*x))^(1/n), x)=#

    #= Following*pairs*of*integrands*are*equal: =#
    @test_int [1/(sqrt(x)*sqrt(1 + x)), x, 2, 2*asinh(sqrt(x))]
    @test_int [sqrt(x/(1 + x))/x, x, 3, 2*asinh(sqrt(x))]

    @test_int [sqrt(x)/sqrt(1 + x), x, 3, sqrt(x)*sqrt(1 + x) - asinh(sqrt(x))]
    @test_int [sqrt(x/(1 + x)), x, 4, sqrt(x)*sqrt(1 + x) - asinh(sqrt(x))]

    @test_int [sqrt(-1 + x)/(x^2*sqrt(1 + x)), x, 3, -((sqrt(-1 + x)*sqrt(1 + x))/x) + atan(sqrt(-1 + x)*sqrt(1 + x))]
    @test_int [sqrt((-1 + x)/(1 + x))/x^2, x, 4, -((sqrt(-1 + x)*sqrt(1 + x))/x) + atan(sqrt(-1 + x)*sqrt(1 + x))]

    @test_int [x^3*sqrt(-1 + x)/sqrt(1 + x), x, 4, (-(3/8))*sqrt(-1 + x)*sqrt(1 + x) + (1/24)*(7 - 2*x)*(-1 + x)^(3/2)*sqrt(1 + x) + (1/4)*(-1 + x)^(3/2)*x^2*sqrt(1 + x) + (3*acosh(x))/8]
    @test_int [x^3*sqrt((-1 + x)/(1 + x)), x, 5, (-(3/8))*sqrt(-1 + x)*sqrt(1 + x) + (1/24)*(7 - 2*x)*(-1 + x)^(3/2)*sqrt(1 + x) + (1/4)*(-1 + x)^(3/2)*x^2*sqrt(1 + x) + (3*acosh(x))/8]

    @test_int [sqrt(-(x/(1 + x)))/x, x, 2, 2*atan(sqrt(-(x/(1 + x))))]
    @test_int [sqrt((1 - x)/(1 + x))/(-1 + x), x, 2, 2*atan(sqrt((1 - x)/(1 + x)))]
    @test_int [sqrt((a + b*x)/(c - b*x))/(a + b*x), x, 3, (2*atan(sqrt((a + b*x)/(c - b*x))))/b]
    @test_int [sqrt((a + b*x)/(c + d*x))/(a + b*x), x, 3, (2*atanh((sqrt(d)*sqrt((a + b*x)/(c + d*x)))/sqrt(b)))/(sqrt(b)*sqrt(d))]

    @test_int [sqrt(-(x/(1 + x))), x, 3, sqrt(-(x/(1 + x)))*(1 + x) - atan(sqrt(-(x/(1 + x))))]
    @test_int [sqrt((1 - x)/(1 + x)), x, 3, sqrt((1 - x)/(1 + x))*(1 + x) - 2*atan(sqrt((1 - x)/(1 + x)))]

    @test_int [sqrt((a + x)/(a - x)), x, 3, -((a - x)*sqrt((a + x)/(a - x))) + 2*a*atan(sqrt((a + x)/(a - x)))]
    @test_int [sqrt((-a + x)/(a + x)), x, 3, sqrt(-((a - x)/(a + x)))*(a + x) - 2*a*atanh(sqrt(-((a - x)/(a + x))))]
    @test_int [sqrt((a + b*x)/(c + d*x)), x, 3, (sqrt((a + b*x)/(c + d*x))*(c + d*x))/d - ((b*c - a*d)*atanh((sqrt(d)*sqrt((a + b*x)/(c + d*x)))/sqrt(b)))/(sqrt(b)*d^(3/2))]
    @test_int [sqrt((-1 + x)/(5 + 3*x)), x, 4, (1/3)*sqrt(-1 + x)*sqrt(5 + 3*x) - (8*asinh((1/2)*sqrt(3/2)*sqrt(-1 + x)))/(3*sqrt(3))]

    @test_int [sqrt((-1 + 5*x)/(1 + 7*x))/x^2, x, 4, -((sqrt(-1 + 5*x)*sqrt(1 + 7*x))/x) - 12*atan(sqrt(1 + 7*x)/sqrt(-1 + 5*x))]

    @test_int [x/(sqrt((1 - x)/(1 + x))*(1 + x)), x, 3, (-sqrt((1 - x)/(1 + x)))*(1 + x)]
    @test_int [x/((1 + x)*sqrt(-1 + 2/(1 + x))), x, 4, -((1 + x)*sqrt(-1 + 2/(1 + x)))]
    @test_int [x/((1 + x)*sqrt((2 + x)/(3 + x))), x, 7, sqrt(2 + x)*sqrt(3 + x) - asinh(sqrt(2 + x)) + 2*sqrt(2)*atanh((sqrt(2)*sqrt(2 + x))/sqrt(3 + x))]

    @test_int [sqrt(1 + 1/x)/(1 + x)^2, x, 2, 2/sqrt(1 + x^(-1))]
    @test_int [sqrt(1 + 1/x)/sqrt(1 - x^2), x, 5, -((sqrt(1 + 1/x)*sqrt(x)*asin(1 - 2*x))/sqrt(1 + x))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*F((a + b*x + c*x^2)^(n/2), x)=#

    #= ::Subsection::Closed:: =#
    #=Euler*substitution #1*for*sqrt(a+b*x+c*x^2)*when*a>0=#

    @test_int [1/(x + sqrt(3 - 2*x - x^2))^1, x, 8, atan((sqrt(3) - sqrt(3 - 2*x - x^2))/x) - (1/2)*log(-((3 - x - sqrt(3)*sqrt(3 - 2*x - x^2))/x^2)) + (1/14)*(7 + sqrt(7))*log(1 + sqrt(3) - sqrt(7) - (sqrt(3)*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x) + (1/14)*(7 - sqrt(7))*log(1 + sqrt(3) + sqrt(7) - (sqrt(3)*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x)]
    @test_int [1/(x + sqrt(3 - 2*x - x^2))^2, x, 5, (2*(4 - sqrt(3) + (3*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x))/(7*(2 - sqrt(3) - (2*(1 + sqrt(3))*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x + (sqrt(3)*(sqrt(3) - sqrt(3 - 2*x - x^2))^2)/x^2)) + (8*atanh((3 - x - sqrt(3)*x - sqrt(3)*sqrt(3 - 2*x - x^2))/(sqrt(7)*x)))/(7*sqrt(7))]
    @test_int [1/(x + sqrt(3 - 2*x - x^2))^3, x, 6, -((4*(9 - 5*sqrt(3) + ((21 + 5*sqrt(3))*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x))/(21*(2 - sqrt(3) - (2*(1 + sqrt(3))*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x + (sqrt(3)*(sqrt(3) - sqrt(3 - 2*x - x^2))^2)/x^2)^2)) + (2*(18 - 43*sqrt(3) - ((18 + 49*sqrt(3))*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x))/(147*(2 - sqrt(3) - (2*(1 + sqrt(3))*(sqrt(3) - sqrt(3 - 2*x - x^2)))/x + (sqrt(3)*(sqrt(3) - sqrt(3 - 2*x - x^2))^2)/x^2)) + (12*atanh((3 - x - sqrt(3)*x - sqrt(3)*sqrt(3 - 2*x - x^2))/(sqrt(7)*x)))/(49*sqrt(7))]

    #= ::Subsection::Closed:: =#
    #=Euler*substitution #2*for*sqrt(a+b*x+c*x^2)*when*c>0=#

    @test_int [1/(x + sqrt(-3 - 2*x + x^2)), x, 3, -(2/(1 - x - sqrt(-3 - 2*x + x^2))) + 2*log(1 - x - sqrt(-3 - 2*x + x^2)) - (3/2)*log(x + sqrt(-3 - 2*x + x^2))]
    @test_int [1/(x + sqrt(-3 - 2*x + x^2))^2, x, 3, -(2/(1 - x - sqrt(-3 - 2*x + x^2))) + 3/(2*(x + sqrt(-3 - 2*x + x^2))) + 4*log(1 - x - sqrt(-3 - 2*x + x^2)) - 4*log(x + sqrt(-3 - 2*x + x^2))]
    @test_int [1/(x + sqrt(-3 - 2*x + x^2))^3, x, 3, -(2/(1 - x - sqrt(-3 - 2*x + x^2))) + 3/(4*(x + sqrt(-3 - 2*x + x^2))^2) + 4/(x + sqrt(-3 - 2*x + x^2)) + 6*log(1 - x - sqrt(-3 - 2*x + x^2)) - 6*log(x + sqrt(-3 - 2*x + x^2))]

    #= ::Subsection::Closed:: =#
    #=Euler*substitution #3*for*sqrt(a+b*x+c*x^2)*when*a<0*and*c<0=#

    @test_int [1/(x + sqrt(-3 - 4*x - x^2))^1, x, 10, -atan(sqrt(-1 - x)/sqrt(3 + x)) - sqrt(2)*atan((1 - (3*sqrt(-1 - x))/sqrt(3 + x))/sqrt(2)) + (1/2)*log(3 + x) + (1/2)*log((3*sqrt(-1 - x) + sqrt(-1 - x)*x + x*sqrt(3 + x))/(3 + x)^(3/2))]
    @test_int [1/(x + sqrt(-3 - 4*x - x^2))^2, x, 5, (1 - sqrt(-1 - x)/sqrt(3 + x))/(1 - (3*(1 + x))/(3 + x) - (2*sqrt(-1 - x))/sqrt(3 + x)) + atan((1 - (3*sqrt(-1 - x))/sqrt(3 + x))/sqrt(2))/sqrt(2)]
    @test_int [1/(x + sqrt(-3 - 4*x - x^2))^3, x, 6, -((13 - (27*sqrt(-1 - x))/sqrt(3 + x))/(18*(1 - (3*(1 + x))/(3 + x) - (2*sqrt(-1 - x))/sqrt(3 + x)))) - (2*(2 - sqrt(-1 - x)/sqrt(3 + x)))/(9*(1 - (3*(1 + x))/(3 + x) - (2*sqrt(-1 - x))/sqrt(3 + x))^2) - (3*atan((1 - (3*sqrt(-1 - x))/sqrt(3 + x))/sqrt(2)))/(2*sqrt(2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*F(a + b*x + c*x^2 + d*x^3 + e*x^4, x)*when*d^3 - 4*c*d*e + 8*b*e^2=0=#

    #= It*would*be*better*to*make*the*substitution*u=x+x^2*than*u=x+1/2, but*that*is*tough*to*know... =#
    @test_int [x^3*(1 + x)^3*(1 + 2*x)*sqrt(1 - x^2 - 2*x^3 - x^4), x, 5, (-(1/15))*(1 - x^2 - 2*x^3 - x^4)^(3/2)*(2 + 3*x^2 + 6*x^3 + 3*x^4), (-(2/15))*(1 - x^2 - 2*x^3 - x^4)^(3/2) - (1/5)*x^2*(1 + x)^2*(1 - x^2 - 2*x^3 - x^4)^(3/2)]
    @test_int [(1 + 2*x)*(x + x^2)^3*sqrt(1 - (x + x^2)^2), x, 6, (-(1/15))*(1 - x^2 - 2*x^3 - x^4)^(3/2)*(2 + 3*x^2 + 6*x^3 + 3*x^4), (-(2/15))*(1 - x^2 - 2*x^3 - x^4)^(3/2) - (1/5)*x^2*(1 + x)^2*(1 - x^2 - 2*x^3 - x^4)^(3/2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(0 + b*x + c*x^2 + d*x^3 + e*x^4)^p*when*d^3 - 4*c*d*e + 8*b*e^2=0=#

    @test_int [(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 7, (2/35)*(13 - 3*(-1 + x)^2)*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/7)*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x) + (16/5)*sqrt(3)*Elliptic.E(asin(1 - x), -(1/3)) - (176/35)*sqrt(3)*Elliptic.F(asin(1 - x), -(1/3))]
    @test_int [(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 6, (1/3)*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (2*Elliptic.E(asin(1 - x), -(1/3)))/sqrt(3) - (4*Elliptic.F(asin(1 - x), -(1/3)))/sqrt(3)]
    @test_int [1/(8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 3, -(Elliptic.F(asin(1 - x), -(1/3))/sqrt(3))]
    @test_int [1/(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 6, ((5 + (-1 + x)^2)*(-1 + x))/(24*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)) + Elliptic.E(asin(1 - x), -(1/3))/(8*sqrt(3)) - Elliptic.F(asin(1 - x), -(1/3))/(4*sqrt(3))]
    @test_int [1/(8*x - 8*x^2 + 4*x^3 - x^4)^(5/2), x, 7, ((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((26 + 7*(-1 + x)^2)*(-1 + x))/(432*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)) + (7*Elliptic.E(asin(1 - x), -(1/3)))/(144*sqrt(3)) - (11*Elliptic.F(asin(1 - x), -(1/3)))/(144*sqrt(3))]

    @test_int [((2 - x)*x*(4 - 2*x + x^2))^(3/2), x, 7, (2/35)*(13 - 3*(-1 + x)^2)*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/7)*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x) + (16/5)*sqrt(3)*Elliptic.E(asin(1 - x), -(1/3)) - (176/35)*sqrt(3)*Elliptic.F(asin(1 - x), -(1/3))]
    @test_int [((2 - x)*x*(4 - 2*x + x^2))^(1/2), x, 6, (1/3)*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (2*Elliptic.E(asin(1 - x), -(1/3)))/sqrt(3) - (4*Elliptic.F(asin(1 - x), -(1/3)))/sqrt(3)]
    @test_int [1/((2 - x)*x*(4 - 2*x + x^2))^(1/2), x, 3, -(Elliptic.F(asin(1 - x), -(1/3))/sqrt(3))]
    @test_int [1/((2 - x)*x*(4 - 2*x + x^2))^(3/2), x, 6, ((5 + (-1 + x)^2)*(-1 + x))/(24*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)) + Elliptic.E(asin(1 - x), -(1/3))/(8*sqrt(3)) - Elliptic.F(asin(1 - x), -(1/3))/(4*sqrt(3))]
    @test_int [1/((2 - x)*x*(4 - 2*x + x^2))^(5/2), x, 7, ((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((26 + 7*(-1 + x)^2)*(-1 + x))/(432*sqrt(3 - 2*(-1 + x)^2 - (-1 + x)^4)) + (7*Elliptic.E(asin(1 - x), -(1/3)))/(144*sqrt(3)) - (11*Elliptic.F(asin(1 - x), -(1/3)))/(144*sqrt(3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + 0*x + c*x^2 + d*x^3 + e*x^4)^p*when*d^3 - 4*c*d*e=0=#

    @test_int [(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2), x, 6, (1/7)*(c/d + x)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2) + (2*c*(c/d + x)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)*(7*c^3 + 20*a*d^2 - 3*c*d^2*(c/d + x)^2))/(35*d^2) - (16*c^3*(c^3 + 8*a*d^2)*(c/d + x)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/(35*d^2*sqrt(c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))) + (16*c^(13/4)*(c^3 + 4*a*d^2)^(3/4)*(c^3 + 8*a*d^2)*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.E(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(35*d^5*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)) + (8*c^(7/4)*(c^3 + 4*a*d^2)^(3/4)*(sqrt(c^3 + 4*a*d^2)*(c^3 + 5*a*d^2) - c^(3/2)*(c^3 + 8*a*d^2))*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.F(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(35*d^5*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))]
    @test_int [(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(1/2), x, 5, (1/3)*(c/d + x)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4) - (2*c^2*(c/d + x)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/(3*sqrt(c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))) + (2*c^(9/4)*(c^3 + 4*a*d^2)^(3/4)*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.E(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(3*d^3*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)) + (c^(3/4)*(c^3 + 4*a*d^2)^(1/4)*(c^3 + 4*a*d^2 - c^(3/2)*sqrt(c^3 + 4*a*d^2))*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.F(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(3*d^3*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))]
    @test_int [1/(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(1/2), x, 2, ((c^3 + 4*a*d^2)^(1/4)*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.F(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(2*c^(1/4)*d*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))]
    @test_int [1/(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(3/2), x, 5, -(((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(8*a*c*(c^3 + 4*a*d^2)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))) - (d^2*(c/d + x)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/(8*a*(c^3 + 4*a*d^2)^(3/2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))) + (c^(1/4)*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.E(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(8*a*d*(c^3 + 4*a*d^2)^(1/4)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)) + ((c^3 + 4*a*d^2 - c^(3/2)*sqrt(c^3 + 4*a*d^2))*sqrt((d^2*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))/((c^3 + 4*a*d^2)*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))^2))*(sqrt(c) + (d^2*(c/d + x)^2)/sqrt(c^3 + 4*a*d^2))*Elliptic.F(2*atan((c + d*x)/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))), (1/2)*(1 + c^(3/2)/sqrt(c^3 + 4*a*d^2))))/(16*a*c^(5/4)*d*(c^3 + 4*a*d^2)^(3/4)*sqrt(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + b*x + 0*x^2 + d*x^3 + e*x^4)^p*when*d^3 + 8*b*e^2=0=#

    #= [(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(3/2), x, 7, ((d/(4*e) + x)*(43*d^4 + 1280*a*e^3 - 144*d^2*e^2*(d/(4*e) + x)^2)*sqrt((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4))/(2240*sqrt(2)*e) + ((d/(4*e) + x)*((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4)^(3/2))/(896*sqrt(2)) + (3*d^2*(d^4 + 512*a*e^3)*(3*d^2 - 2*sqrt(d^4 - 64*a*e^3))*sqrt(3*d^2 + 2*sqrt(d^4 - 64*a*e^3))*sqrt(1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 - 2*sqrt(d^4 - 64*a*e^3)))*sqrt(1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 + 2*sqrt(d^4 - 64*a*e^3)))*Elliptic.E(asin((4*e*(d/(4*e) + x))/sqrt(3*d^2 + 2*sqrt(d^4 - 64*a*e^3))), (3*d^2 + 2*sqrt(d^4 - 64*a*e^3))/(3*d^2 - 2*sqrt(d^4 - 64*a*e^3))))/(1120*sqrt(2)*e^3*sqrt((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4)) + (sqrt(3*d^2 + 2*sqrt(d^4 - 64*a*e^3))*(11*d^8 - 1984*a*d^4*e^3 + 81920*a^2*e^6 + 6*d^2*sqrt(d^4 - 64*a*e^3)*(d^4 + 512*a*e^3))*sqrt(1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 - 2*sqrt(d^4 - 64*a*e^3)))*sqrt(1 - (16*e^2*(d/(4*e) + x)^2)/(3*d^2 + 2*sqrt(d^4 - 64*a*e^3)))*Elliptic.F(asin((4*e*(d/(4*e) + x))/sqrt(3*d^2 + 2*sqrt(d^4 - 64*a*e^3))), (3*d^2 + 2*sqrt(d^4 - 64*a*e^3))/(3*d^2 - 2*sqrt(d^4 - 64*a*e^3))))/(1120*sqrt(2)*e^3*sqrt((5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^4))] =#
    @test_int [(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(1/2), x, 5, (1/3)*(d/(4*e) + x)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4) - (2*d^2*(d/(4*e) + x)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/(sqrt(5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))) + (d^2*(5*d^4 + 256*a*e^3)^(3/4)*sqrt((e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))^2))*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))*Elliptic.E(2*atan((d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)), (1/2)*(1 + (3*d^2)/sqrt(5*d^4 + 256*a*e^3))))/(8*sqrt(2)*e^2*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) + ((5*d^4 + 256*a*e^3)^(1/4)*(5*d^4 + 256*a*e^3 - 3*d^2*sqrt(5*d^4 + 256*a*e^3))*sqrt((e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))^2))*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))*Elliptic.F(2*atan((d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)), (1/2)*(1 + (3*d^2)/sqrt(5*d^4 + 256*a*e^3))))/(48*sqrt(2)*e^2*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))]
    @test_int [1/(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(1/2), x, 2, ((5*d^4 + 256*a*e^3)^(1/4)*sqrt((e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))^2))*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))*Elliptic.F(2*atan((d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)), (1/2)*(1 + (3*d^2)/sqrt(5*d^4 + 256*a*e^3))))/(sqrt(2)*e*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))]
    @test_int [1/(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)^(3/2), x, 5, (4*e*(d/(4*e) + x)*(13*d^4 - 256*a*e^3 - 48*d^2*e^2*(d/(4*e) + x)^2))/((5*d^8 - 64*a*d^4*e^3 - 16384*a^2*e^6)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) + (384*d^2*e^2*(d/(4*e) + x)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(3/2)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))) - (12*sqrt(2)*d^2*sqrt((e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))^2))*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))*Elliptic.E(2*atan((d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)), (1/2)*(1 + (3*d^2)/sqrt(5*d^4 + 256*a*e^3))))/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(1/4)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4)) - (2*sqrt(2)*(5*d^4 + 256*a*e^3 - 3*d^2*sqrt(5*d^4 + 256*a*e^3))*sqrt((e*(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))^2))*(1 + (16*e^2*(d/(4*e) + x)^2)/sqrt(5*d^4 + 256*a*e^3))*Elliptic.F(2*atan((d + 4*e*x)/(5*d^4 + 256*a*e^3)^(1/4)), (1/2)*(1 + (3*d^2)/sqrt(5*d^4 + 256*a*e^3))))/((d^4 - 64*a*e^3)*(5*d^4 + 256*a*e^3)^(3/4)*sqrt(8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(a + b*x + c*x^2 + d*x^3 + e*x^4)^(p/2)*when*d^3 - 4*c*d*e + 8*b*e^2=0=#

    @test_int [(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 8, -((16*(7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(35*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))) + (2/35)*(13 + 5*a - 3*(-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/7)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x) + (16*(7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(35*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (4*(3 + a)*(16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(35*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 7, -((2*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))) + (1/3)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (2*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (2*(3 + a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 3, (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 7, If(13>=8, ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(2*(3 + a)*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(3 + a)*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)), ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(12 + 7*a + a^2)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)))]
    @test_int [1/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2), x, 8, If(13>=8, ((5 + a + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(12*(3 + a)^2*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*(3 + a)^2*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*(3 + a)^2*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(3 + a)*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)), ((5 + a + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(12*(12 + 7*a + a^2)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*(12 + 7*a + a^2)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*(12 + 7*a + a^2)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(3 + a)*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)))]

    @test_int [x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 14, (3/16)*(4 + a)*(1 + (-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4) + (1/8)*(1 + (-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2) - (16*(7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(35*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (2/35)*(13 + 5*a - 3*(-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/7)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x) + (3/16)*(4 + a)^2*atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (16*(7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(35*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (4*(3 + a)*(16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(35*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 12, (1/4)*(1 + (-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4) - (2*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (1/3)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/4)*(4 + a)*atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (2*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (2*(3 + a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 7, (1/2)*atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 10, If(13>=8, (1 + (-1 + x)^2)/(2*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(2*(3 + a)*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(3 + a)*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)), (1 + (-1 + x)^2)/(2*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(12 + 7*a + a^2)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)))]
    @test_int [x/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2), x, 12, If(13>=8, (1 + (-1 + x)^2)/(6*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + (1 + (-1 + x)^2)/(3*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(12*(3 + a)^2*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*(3 + a)^2*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*(3 + a)^2*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(3 + a)*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)), (1 + (-1 + x)^2)/(6*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + (1 + (-1 + x)^2)/(3*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((5 + a + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((104 + 47*a + 5*a^2 + 4*(7 + 2*a)*(-1 + x)^2)*(-1 + x))/(12*(12 + 7*a + a^2)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((7 + 2*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(3*(12 + 7*a + a^2)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((7 + 2*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(3*(12 + 7*a + a^2)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((16 + 5*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(3 + a)*(4 + a)^2*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)))]

    @test_int [x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 15, (3/8)*(4 + a)*(1 + (-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4) + (1/4)*(1 + (-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2) + (4*(140 + 111*a + 21*a^2)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(315*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (2/315)*(2*(80 + 27*a) + 3*(20 + 7*a)*(-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/63)*(15 + 7*(-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x) + (3/8)*(4 + a)^2*atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - (4*(140 + 111*a + 21*a^2)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(315*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (4*(3 + a)*(100 + 33*a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(315*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 13, (1/2)*(1 + (-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4) + (2*(8 + 3*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(15*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (1/15)*(7 + 3*(-1 + x)^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)*(-1 + x) + (1/2)*(4 + a)*atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - (2*(8 + 3*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(15*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (8*(3 + a)*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(15*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x, 11, ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4) + atan((1 + (-1 + x)^2)/sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x, 10, (1 + (-1 + x)^2)/((4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((4 + a)*(2 + (-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(2*(3 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(2*(3 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]
    @test_int [x^2/(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(5/2), x, 13, (1 + (-1 + x)^2)/(3*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + (2*(1 + (-1 + x)^2))/(3*(4 + a)^2*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((4 + a)*(2 + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((29 + 7*a + (13 + 3*a)*(-1 + x)^2)*(-1 + x))/(12*(3 + a)^2*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) - ((13 + 3*a)*(1 - sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*(-1 + x))/(12*(3 + a)^2*(4 + a)*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + ((13 + 3*a)*(1 - sqrt(4 + a))*sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.E(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(3 + a)^2*(4 + a)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)) + (sqrt(1 + sqrt(4 + a))*(1 + (-1 + x)^2/(1 - sqrt(4 + a)))*Elliptic.F(atan((-1 + x)/sqrt(1 + sqrt(4 + a))), -((2*sqrt(4 + a))/(1 - sqrt(4 + a)))))/(12*(12 + 7*a + a^2)*sqrt((1 + (-1 + x)^2/(1 - sqrt(4 + a)))/(1 + (-1 + x)^2/(1 + sqrt(4 + a))))*sqrt(3 + a - 2*(-1 + x)^2 - (-1 + x)^4))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*F(a + b*x + c*x^2 + d*x^3 + e*x^4, x)*when*b^3 - 4*a*b*c + 8*a^2*d=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + b*x + 0*x^2 + d*x^3 + e*x^4)^p*when*b^3 + 8*a^2*d=0=#

    #= [(8 + 8*x - x^3 + 8*x^4)^(1/2), x, 6, 0] =#
    @test_int [1/(8 + 8*x - x^3 + 8*x^4)^(1/2), x, 4, -((x^2*sqrt((261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (sqrt(29)*(4 + x)^2)/x^2)^2)*(87 + (sqrt(29)*(4 + x)^2)/x^2)*Elliptic.F(2*atan((4 + x)/(sqrt(3)*29^(1/4)*x)), (1/58)*(29 + sqrt(29))))/(8*sqrt(3)*29^(1/4)*sqrt(8 + 8*x - x^3 + 8*x^4)))]
    @test_int [1/(8 + 8*x - x^3 + 8*x^4)^(3/2), x, 10, -(((66 - (1 + 4/x)^2)*x^2)/(1008*sqrt(8 + 8*x - x^3 + 8*x^4))) + ((216 - 7*(1 + 4/x)^2)*(1 + 4/x)*x^2)/(12528*sqrt(8 + 8*x - x^3 + 8*x^4)) + (7*(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)*(1 + 4/x)*x^2)/(432*sqrt(29)*sqrt(8 + 8*x - x^3 + 8*x^4)*(87 + (sqrt(29)*(4 + x)^2)/x^2)) - (7*x^2*sqrt((261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (sqrt(29)*(4 + x)^2)/x^2)^2)*(87 + (sqrt(29)*(4 + x)^2)/x^2)*Elliptic.E(2*atan((4 + x)/(sqrt(3)*29^(1/4)*x)), (1/58)*(29 + sqrt(29))))/(144*sqrt(3)*29^(3/4)*sqrt(8 + 8*x - x^3 + 8*x^4)) + ((14 - 5*sqrt(29))*x^2*sqrt((261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (sqrt(29)*(4 + x)^2)/x^2)^2)*(87 + (sqrt(29)*(4 + x)^2)/x^2)*Elliptic.F(2*atan((4 + x)/(sqrt(3)*29^(1/4)*x)), (1/58)*(29 + sqrt(29))))/(576*sqrt(3)*29^(3/4)*sqrt(8 + 8*x - x^3 + 8*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + b*x + c*x^2 + 0*x^3 + e*x^4)^p*when*b^2 - 4*a*c=0=#

    #= [(1 + 4*x + 4*x^2 + 4*x^4)^(1/2), x, 0, 0] =#
    @test_int [1/(1 + 4*x + 4*x^2 + 4*x^4)^(1/2), x, 3, -(((sqrt(5) + (1 + 1/x)^2)*sqrt((5 - 2*(1 + 1/x)^2 + (1 + 1/x)^4)/(sqrt(5) + (1 + 1/x)^2)^2)*x^2*Elliptic.F(2*atan((1 + 1/x)/5^(1/4)), (1/10)*(5 + sqrt(5))))/(2*5^(1/4)*sqrt(1 + 4*x + 4*x^2 + 4*x^4)))]
    @test_int [1/(1 + 4*x + 4*x^2 + 4*x^4)^(3/2), x, 9, -(((3 - (1 + 1/x)^2)*x^2)/sqrt(1 + 4*x + 4*x^2 + 4*x^4)) + ((13 - 9*(1 + 1/x)^2)*(1 + 1/x)*x^2)/(10*sqrt(1 + 4*x + 4*x^2 + 4*x^4)) + (9*(5 - 2*(1 + 1/x)^2 + (1 + 1/x)^4)*(1 + 1/x)*x^2)/(10*(sqrt(5) + (1 + 1/x)^2)*sqrt(1 + 4*x + 4*x^2 + 4*x^4)) - (9*(sqrt(5) + (1 + 1/x)^2)*sqrt((5 - 2*(1 + 1/x)^2 + (1 + 1/x)^4)/(sqrt(5) + (1 + 1/x)^2)^2)*x^2*Elliptic.E(2*atan((1 + 1/x)/5^(1/4)), (1/10)*(5 + sqrt(5))))/(2*5^(3/4)*sqrt(1 + 4*x + 4*x^2 + 4*x^4)) + (3*(3 - sqrt(5))*(sqrt(5) + (1 + 1/x)^2)*sqrt((5 - 2*(1 + 1/x)^2 + (1 + 1/x)^4)/(sqrt(5) + (1 + 1/x)^2)^2)*x^2*Elliptic.F(2*atan((1 + 1/x)/5^(1/4)), (1/10)*(5 + sqrt(5))))/(4*5^(3/4)*sqrt(1 + 4*x + 4*x^2 + 4*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a + b*x + c*x^2 + d*x^3 + e*x^4)^p*when*b^3 - 4*a*b*c + 8*a^2*d=0=#

    #= [(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(1/2), x, 0, 0] =#
    @test_int [1/(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(1/2), x, 4, -(((sqrt(517) + (3 + 4/x)^2)*sqrt((517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(sqrt(517) + (3 + 4/x)^2)^2)*x^2*Elliptic.F(2*atan((4 + 3*x)/(517^(1/4)*x)), (517 + 19*sqrt(517))/1034))/(8*517^(1/4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)))]
    @test_int [1/(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(3/2), x, 10, -(((172 - 7*(3 + 4/x)^2)*x^2)/(208*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4))) + ((50896 - 2455*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(322608*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + (2455*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(322608*(sqrt(517) + (3 + 4/x)^2)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) - (2455*(sqrt(517) + (3 + 4/x)^2)*sqrt((517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(sqrt(517) + (3 + 4/x)^2)^2)*x^2*Elliptic.E(2*atan((4 + 3*x)/(517^(1/4)*x)), (517 + 19*sqrt(517))/1034))/(624*517^(3/4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + ((4910 - 203*sqrt(517))*(sqrt(517) + (3 + 4/x)^2)*sqrt((517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(sqrt(517) + (3 + 4/x)^2)^2)*x^2*Elliptic.F(2*atan((4 + 3*x)/(517^(1/4)*x)), (517 + 19*sqrt(517))/1034))/(2496*517^(3/4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4))]
    @test_int [1/(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(5/2), x, 12, -(((124415 - 6308*(3 + 4/x)^2)*x^2)/(97344*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4))) - ((64489 - 1399*(3 + 4/x)^2)*x^2)/(624*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + ((18932921731 - 1086525994*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(78056941248*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + ((11921698 - 359497*(3 + 4/x)^2)*(3 + 4/x)*x^2)/(483912*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + (543262997*(517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)*(3 + 4/x)*x^2)/(39028470624*(sqrt(517) + (3 + 4/x)^2)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) - (543262997*(sqrt(517) + (3 + 4/x)^2)*sqrt((517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(sqrt(517) + (3 + 4/x)^2)^2)*x^2*Elliptic.E(2*atan((4 + 3*x)/(517^(1/4)*x)), (517 + 19*sqrt(517))/1034))/(75490272*517^(3/4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)) + ((4346103976 - 175318963*sqrt(517))*(sqrt(517) + (3 + 4/x)^2)*sqrt((517 - 38*(3 + 4/x)^2 + (3 + 4/x)^4)/(sqrt(517) + (3 + 4/x)^2)^2)*x^2*Elliptic.F(2*atan((4 + 3*x)/(517^(1/4)*x)), (517 + 19*sqrt(517))/1034))/(1207844352*517^(3/4)*sqrt(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4))]

    #= [(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(1/2), x, 6, 0] =#
    @test_int [1/(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(1/2), x, 4, -((sqrt((613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(sqrt(613) + (6 - x)^2/x^2)^2)*(sqrt(613) + (6 - x)^2/x^2)*x^2*Elliptic.F(2*atan((6 - x)/(613^(1/4)*x)), (613 + 91*sqrt(613))/1226))/(12*613^(1/4)*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)))]
    @test_int [1/(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(3/2), x, 10, -(((176 - 23*(1 - 6/x)^2)*x^2)/(51759*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4))) + ((45401 - 3722*(1 - 6/x)^2)*(1 - 6/x)*x^2)/(31728267*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)) + (3722*(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)*(1 - 6/x)*x^2)/(31728267*(sqrt(613) + (6 - x)^2/x^2)*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)) + (3722*sqrt((613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(sqrt(613) + (6 - x)^2/x^2)^2)*(sqrt(613) + (6 - x)^2/x^2)*x^2*Elliptic.E(2*atan((6 - x)/(613^(1/4)*x)), (613 + 91*sqrt(613))/1226))/(51759*613^(3/4)*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)) - ((7444 - 145*sqrt(613))*sqrt((613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(sqrt(613) + (6 - x)^2/x^2)^2)*(sqrt(613) + (6 - x)^2/x^2)*x^2*Elliptic.F(2*atan((6 - x)/(613^(1/4)*x)), (613 + 91*sqrt(613))/1226))/(207036*613^(3/4)*sqrt(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4))]

    #= ::Section::Closed:: =#
    #=Integrands*requiring*algebraic*expansion=#

    @test_int [(2*sqrt(3 - x) + 3/sqrt(1 + x))^2/x, x, 12, -4*x + 12*asin((1 - x)/2) - 24*sqrt(3)*atanh((sqrt(3)*sqrt(1 + x))/sqrt(3 - x)) + 21*log(x) - 9*log(1 + x)]

    @test_int [(-1 + x + x^2)/(1 + sqrt(1 + x^2)), x, 14, -(1/x) - x + sqrt(1 + x^2) + sqrt(1 + x^2)/x + (1/2)*x*sqrt(1 + x^2) - asinh(x)/2 - log(1 + sqrt(1 + x^2))]
    @test_int [(-1 + x + x^2)/(1 + x + sqrt(1 + x^2)), x, 12, (6*x^2 + 2*x^3 + (4 - 3*x - 2*x^2)*sqrt(1 + x^2) - 3*asinh(x) - 6*log(1 + sqrt(1 + x^2)))/12, x/2 + x^2/2 + x^3/6 - (1/4)*x*sqrt(1 + x^2) - (1/6)*(1 + x^2)^(3/2) + 1/(2*(x + sqrt(1 + x^2))) - asinh(x)/4 + (1/2)*log(x + sqrt(1 + x^2)) - log(1 + x + sqrt(1 + x^2))]

    @test_int [(2*sqrt(-1 + x) + x)/(sqrt(-1 + x)*x), x, 2, 2*sqrt(-1 + x) + 2*log(x)]

    #= Positive*integer*powers*of*monomial*sums =#
    @test_int [(a + b*x^(2/3)+c*sqrt(x))^2, x, 4, a^2*x + (4/3)*a*c*x^(3/2) + (6/5)*a*b*x^(5/3) + (c^2*x^2)/2 + (12/13)*b*c*x^(13/6) + (3/7)*b^2*x^(7/3)]
    @test_int [(a + b*x^(2/3)+c*sqrt(x))^3, x, 4, a^3*x + 2*a^2*c*x^(3/2) + (9/5)*a^2*b*x^(5/3) + (3/2)*a*c^2*x^2 + (36/13)*a*b*c*x^(13/6) + (9/7)*a*b^2*x^(7/3) + (2/5)*c^3*x^(5/2) + (9/8)*b*c^2*x^(8/3) + (18/17)*b^2*c*x^(17/6) + (b^3*x^3)/3]

    @test_int [(-1 + x^2)/(sqrt(a - b + b/x^2)*x^3), x, 5, sqrt(a - b*(1 - 1/x^2))/b + atanh(sqrt(a - b*(1 - 1/x^2))/sqrt(a - b))/sqrt(a - b)]
    @test_int [(-1 + x^2)/(sqrt(a + b*(-1 + 1/x^2))*x^3), x, 6, sqrt(a - b*(1 - 1/x^2))/b + atanh(sqrt(a - b*(1 - 1/x^2))/sqrt(a - b))/sqrt(a - b)]

    @test_int [(1 + x)/((4 + x^2)*sqrt(9 + x^2)), x, 6, atan((sqrt(5)*x)/(2*sqrt(9 + x^2)))/(2*sqrt(5)) - atanh(sqrt(9 + x^2)/sqrt(5))/sqrt(5)]

    #= Checks*to*ensure*that*expansion*occurs*before*substitution*for*fractional*powers*of*linears: =#
    @test_int [x*(1 + sqrt(1 - x^2)), x, 3, x^2/2 - (1/3)*(1 - x^2)^(3/2)]
    @test_int [x*(1 + sqrt(1 - x)*sqrt(1 + x)), x, 3, x^2/2 - (1/3)*(1 - x^2)^(3/2)]

    @test_int [x*(1 + 1/(sqrt(2 + x)*sqrt(3 + x))), x, 5, x^2/2 + sqrt(2 + x)*sqrt(3 + x) - 5*asinh(sqrt(2 + x))]

    @test_int [(x - sqrt(x^6))/(x*(1 - x^4)), x, 9, atan(x)/2 + (sqrt(x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(x^6)*atanh(x))/(2*x^3)]
    @test_int [(1 - sqrt(x^6)/x)/(1 - x^4), x, 9, atan(x)/2 + (sqrt(x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(x^6)*atanh(x))/(2*x^3)]
    @test_int [(x - sqrt(x^6))/(x - x^5), x, 10, atan(x)/2 + (sqrt(x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(x^6)*atanh(x))/(2*x^3)]
    @test_int [x/(x + sqrt(x^6)), x, 11, atan(x)/2 + (sqrt(x^6)*atan(x))/(2*x^3) + atanh(x)/2 - (sqrt(x^6)*atanh(x))/(2*x^3)]

    @test_int [(sqrt(x) - sqrt(x^3))/(x - x^3), x, 12, atan(sqrt(x)) + (sqrt(x^3)*atan(sqrt(x)))/x^(3/2) + atanh(sqrt(x)) - (sqrt(x^3)*atanh(sqrt(x)))/x^(3/2)]
    @test_int [1/(sqrt(x) + sqrt(x^3)), x, 13, atan(sqrt(x)) + (sqrt(x^3)*atan(sqrt(x)))/x^(3/2) + atanh(sqrt(x)) - (sqrt(x^3)*atanh(sqrt(x)))/x^(3/2)]
    @test_int [1/(sqrt(-1 + x) + sqrt((-1 + x)^3)), x, 14, atan(sqrt(-1 + x)) + (sqrt((-1 + x)^3)*atan(sqrt(-1 + x)))/(-1 + x)^(3/2) + atanh(sqrt(-1 + x)) - (sqrt((-1 + x)^3)*atanh(sqrt(-1 + x)))/(-1 + x)^(3/2)]

    #= Following*integrands*are*equal. =#
    @test_int [-3/(4 + 5*x)^2 - (5 + 4*x)/((4 + 5*x)^2*sqrt(1 - x^2)), x, 2, 3/(5*(4 + 5*x)) + sqrt(1 - x^2)/(4 + 5*x)]
    @test_int [(-5 - 4*x - 3*sqrt(1 - x^2))/((4 + 5*x)^2*sqrt(1 - x^2)), x, 8, 3/(5*(4 + 5*x)) + sqrt(1 - x^2)/(4 + 5*x)]
    @test_int [1/((-5 - 4*x)*sqrt(1 - x^2) + 3*(1 - x^2)), x, 16, 3/(5*(4 + 5*x)) + sqrt(1 - x^2)/(4 + 5*x)]
    @test_int [1/(3 - 3*x^2 - 5*sqrt(1 - x^2) - 4*x*sqrt(1 - x^2)), x, 16, 3/(5*(4 + 5*x)) + sqrt(1 - x^2)/(4 + 5*x)]
    @test_int [(-1 + sqrt(1 - x^2))/(sqrt(1 - x^2)*(2 + x - 2*sqrt(1 - x^2))^2), x, 31, 3/(5*(4 + 5*x)) + sqrt(1 - x^2)/(4 + 5*x)]

    @test_int [(a + b*x^(n-1))/(c*x + d*x^n), x, 5, (b*log(x))/d - ((b*c - a*d)*log(d + c*x^(1 - n)))/(c*d*(1 - n))]

    @test_int [sqrt(1 + 2*x^2)/(1 + sqrt(1 + 2*x^2)), x, 6, -(1/(2*x)) + x + sqrt(1 + 2*x^2)/(2*x) - asinh(sqrt(2)*x)/sqrt(2)]
    @test_int [sqrt(-1 + 4*x^2)/(x + sqrt(-1 + 4*x^2)), x, 8, (4*x)/3 - (1/3)*sqrt(-1 + 4*x^2) - atanh(sqrt(3)*x)/(3*sqrt(3)) + atanh(sqrt(3)*sqrt(-1 + 4*x^2))/(3*sqrt(3))]

    @test_int [(a + b*x + c*x^2)/(sqrt(-1 + x^2)*(d + e*x)^3), x, 4, -(((c*d^2 - b*d*e + a*e^2)*sqrt(-1 + x^2))/(2*e*(d^2 - e^2)*(d + e*x)^2)) + ((c*(d^3 - 4*d*e^2) - e*(3*a*d*e - b*(d^2 + 2*e^2)))*sqrt(-1 + x^2))/(2*e*(d^2 - e^2)^2*(d + e*x)) - ((3*b*d*e - a*(2*d^2 + e^2) - c*(d^2 + 2*e^2))*atanh((e + d*x)/(sqrt(d^2 - e^2)*sqrt(-1 + x^2))))/(2*(d^2 - e^2)^(5/2))]

    #= ::Section::Closed:: =#
    #=Integrands*requiring*algebraic*simplification=#

    #= Following*pairs*or*triples*of*integrands*are*equal. =#
    @test_int [(1 + 2*x^8)/(x*(1 + x^8)^(3/2)), x, 4, -(1/(4*sqrt(1 + x^8))) - (1/4)*atanh(sqrt(1 + x^8))]
    @test_int [(sqrt(1 + x^8)*(1 + 2*x^8))/(x + 2*x^9 + x^17), x, 6, -(1/(4*sqrt(1 + x^8))) - (1/4)*atanh(sqrt(1 + x^8))]

    @test_int [1 - 9*x^2 + x/sqrt(1 - 9*x^2), x, 2, x - 3*x^3 - (1/9)*sqrt(1 - 9*x^2)]
    @test_int [(x + (1 - 9*x^2)^(3/2))/sqrt(1 - 9*x^2), x, 3, x - 3*x^3 - (1/9)*sqrt(1 - 9*x^2)]

    @test_int [((-3 + 2*sqrt(x))*(-3*sqrt(x) + x)^(2/3))/sqrt(x), x, 2, (6/5)*(-3*sqrt(x) + x)^(5/3)]
    @test_int [(9 - 9*sqrt(x) + 2*x)/(-3*sqrt(x) + x)^(1/3), x, 3, (6/5)*(-3*sqrt(x) + x)^(5/3)]

    @test_int [1/sqrt(4 - 9*x^2), x, 1, (1/3)*asin((3*x)/2)]
    @test_int [1/(sqrt(2 - 3*x)*sqrt(2 + 3*x)), x, 2, (1/3)*asin((3*x)/2)]
    @test_int [1/sqrt((2 - 3*x)*(2 + 3*x)), x, 2, (1/3)*asin((3*x)/2)]

    @test_int [1/sqrt(15 - 2*x - x^2), x, 2, -asin((1/4)*(-1 - x))]
    @test_int [1/(sqrt(3 - x)*sqrt(5 + x)), x, 3, -asin((1/4)*(-1 - x))]
    @test_int [1/sqrt((3 - x)*(5 + x)), x, 3, -asin((1/4)*(-1 - x))]

    @test_int [1/sqrt(-15 - 8*x - x^2), x, 2, asin(4 + x)]
    @test_int [1/(sqrt(-3 - x)*sqrt(5 + x)), x, 3, asin(4 + x)]
    @test_int [1/sqrt((-(3 + x))*(5 + x)), x, 3, asin(4 + x)]

    @test_int [1 - sqrt(x), x, 1, x - (2*x^(3/2))/3]
    @test_int [(1 - x)/(1 + sqrt(x)), x, 4, x - (2*x^(3/2))/3]

    @test_int [sqrt(1/(1 - x^2)), x, 2, sqrt(1/(1 - x^2))*sqrt(1 - x^2)*asin(x)]
    @test_int [sqrt((1 + x^2)/(1 - x^4)), x, 3, sqrt(1/(1 - x^2))*sqrt(1 - x^2)*asin(x)]

    @test_int [sqrt(1/(-1 + x^2)), x, 2, sqrt(1 - x^2)*sqrt(1/(-1 + x^2))*asin(x)]
    @test_int [sqrt((1 + x^2)/(-1 + x^4)), x, 3, sqrt(1 - x^2)*sqrt(1/(-1 + x^2))*asin(x)]

    #= Following*pairs*of*integrands*are*equal. =#
    @test_int [1/sqrt(1 - x), x, 1, -2*sqrt(1 - x)]
    @test_int [sqrt(1 + x)/sqrt(1 - x^2), x, 2, -2*sqrt(1 - x)]

    @test_int [1/sqrt(1 + x), x, 1, 2*sqrt(1 + x)]
    @test_int [sqrt(1 - x)/sqrt(1 - x^2), x, 2, 2*sqrt(1 + x)]

    @test_int [sqrt(1 - x), x, 1, (-(2/3))*(1 - x)^(3/2)]
    @test_int [sqrt(1 - x^2)/sqrt(1 + x), x, 2, (-(2/3))*(1 - x)^(3/2)]

    @test_int [sqrt(1 + x), x, 1, (2/3)*(1 + x)^(3/2)]
    @test_int [sqrt(1 - x^2)/sqrt(1 - x), x, 2, (2/3)*(1 + x)^(3/2)]

    @test_int [sqrt(2 + 3*x)/sqrt(1 + x), x, 3, sqrt(1 + x)*sqrt(2 + 3*x) - asinh(sqrt(2 + 3*x))/sqrt(3)]
    @test_int [(sqrt(2 + 3*x)*sqrt(1 - x))/sqrt(1 - x^2), x, 4, sqrt(1 + x)*sqrt(2 + 3*x) - asinh(sqrt(2 + 3*x))/sqrt(3)]

    @test_int [(1 + x)^(3/2)/(x*(1 - x)^(3/2)), x, 7, (4*sqrt(1 + x))/sqrt(1 - x) - asin(x) - atanh(sqrt(1 - x)*sqrt(1 + x))]
    @test_int [(1 + x)^3/(x*(1 - x^2)^(3/2)), x, 6, (4*(1 + x))/sqrt(1 - x^2) - asin(x) - atanh(sqrt(1 - x^2))]

    @test_int [(1 + a*x)^(3/2)/(x*(1 - a*x)^(3/2)), x, 7, (4*sqrt(1 + a*x))/sqrt(1 - a*x) - asin(a*x) - atanh(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(1 + a*x)^3/(x*(1 - a^2*x^2)^(3/2)), x, 6, (4*(1 + a*x))/sqrt(1 - a^2*x^2) - asin(a*x) - atanh(sqrt(1 - a^2*x^2))]

    #= Following*pairs*of*integrands*are*equal. =#
    @test_int [1/sqrt(1 - x^2), x, 1, asin(x)]
    @test_int [sqrt(1 + x^2)/sqrt(1 - x^4), x, 2, asin(x)]

    @test_int [1/sqrt(1 + x^2), x, 1, asinh(x)]
    @test_int [sqrt(1 - x^2)/sqrt(1 - x^4), x, 2, asinh(x)]

    @test_int [sqrt(1 - x^2), x, 2, (1/2)*x*sqrt(1 - x^2) + asin(x)/2]
    @test_int [sqrt(1 - x^4)/sqrt(1 + x^2), x, 3, (1/2)*x*sqrt(1 - x^2) + asin(x)/2]

    @test_int [sqrt(1 + x^2), x, 2, (1/2)*x*sqrt(1 + x^2) + asinh(x)/2]
    @test_int [sqrt(1 - x^4)/sqrt(1 - x^2), x, 3, (1/2)*x*sqrt(1 + x^2) + asinh(x)/2]

    @test_int [((a + b + c*x^2)/d)^m, x, 3, (d*x*((a + b)/d + (c*x^2)/d)^(1 + m)*HypergeometricFunctions._₂F₁(1, 3/2 + m, 3/2, -((c*x^2)/(a + b))))/(a + b), (x*((a + b)/d + (c*x^2)/d)^m*HypergeometricFunctions._₂F₁(1/2, -m, 3/2, -((c*x^2)/(a + b))))/(1 + c*x^2/(a + b))^m]

    #= ::Section::Closed:: =#
    #=Integrands*requiring*rationalization*of*denominator=#

    @test_int [1/(x - sqrt(1 + x^2)), x, 4, -(x^2/2) - (1/2)*x*sqrt(1 + x^2) - asinh(x)/2]
    @test_int [1/(x - sqrt(1 - x^2)), x, 7, -(asin(x)/2) - (1/2)*atanh(x/sqrt(1 - x^2)) + (1/4)*log(1 - 2*x^2)]
    @test_int [1/(x - sqrt(1 + 2*x^2)), x, 7, -(sqrt(2)*asinh(sqrt(2)*x)) + atanh(x/sqrt(1 + 2*x^2)) - log(1 + x^2)/2]

    #= Integrands*are*equal.  Denominators*needs*to*be*rationalized*before*expansion. =#
    @test_int [(2*x - x^3 + x^2*sqrt(2 - x^2))/(-2 + 2*x^2), x, 10, -(x^2/4) + (1/4)*x*sqrt(2 - x^2) - (1/2)*atanh(x/sqrt(2 - x^2)) + (1/4)*log(1 - x^2)]
    @test_int [(x*sqrt(2 - x^2))/(x - sqrt(2 - x^2)), x, 12, -(x^2/4) + (1/4)*x*sqrt(2 - x^2) - (1/2)*atanh(x/sqrt(2 - x^2)) + (1/4)*log(1 - x) + (1/4)*log(1 + x)]

    @test_int [x/(-x + sqrt(2*x - x^2)), x, 5, -(x/2) - (1/2)*sqrt(2*x - x^2) + (1/2)*atanh(sqrt(2*x - x^2)) - (1/2)*log(1 - x)]
    @test_int [(x + sqrt(2*x - x^2))/(2 - 2*x), x, 7, -(x/2) - (1/2)*sqrt(2*x - x^2) + (1/2)*atanh(sqrt(2*x - x^2)) - (1/2)*log(1 - x)]
    @test_int [(sqrt(2 - x)*sqrt(x) + x)/(2 - 2*x), x, 9, -(x/2) - (1/2)*sqrt(2*x - x^2) + (1/2)*atanh(sqrt(2*x - x^2)) - (1/2)*log(1 - x)]
    @test_int [sqrt(x)/(sqrt(2 - x) - sqrt(x)), x, 7, (-(1/2))*sqrt(2 - x)*sqrt(x) - x/2 + (1/2)*atanh(sqrt(2 - x)*sqrt(x)) - (1/2)*log(1 - x)]

    #= ::Section::Closed:: =#
    #=Integrands*requiring*piecewise*constant*extraction=#

    @test_int [1/((1 + x)*(-1 + x^2))^(2/3), x, 3, -((3*(1 - x^2))/(2*(-((1 + x)*(1 - x^2)))^(2/3))), -((3*(1 - x)*(1 + x))/(2*(-1 - x + x^2 + x^3)^(2/3)))]

    @test_int [(-1 + x^2)/((1 + x^2)*sqrt(x*(1 + x^2))), x, 2, -((2*x)/sqrt(x*(1 + x^2)))]
    @test_int [(-1 + x^2)/((1 + x^2)*sqrt(x + x^3)), x, 2, -((2*x)/sqrt(x + x^3))]

    @test_int [sqrt((-1 + x^2)^2/(x*(1 + x^2)))/(1 + x^2), x, 2, (2*x*sqrt((1 - x^2)^2/(x*(1 + x^2))))/(1 - x^2)]
    @test_int [sqrt((-1 + x^2)^2/(x + x^3))/(1 + x^2), x, 3, (2*x*sqrt((1 - x^2)^2/(x + x^3)))/(1 - x^2)]

    @test_int [1/(sqrt(a + b/x^2)*sqrt(c + d*x^2)), x, 5, (sqrt(b + a*x^2)*atanh((sqrt(d)*sqrt(b + a*x^2))/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*sqrt(d)*sqrt(a + b/x^2)*x)]

    @test_int [sqrt(-2*x^2 + x^4)/((-1 + x^2)*(2 + x^2)), x, 7, (2*sqrt(-2*x^2 + x^4)*atan((1/2)*sqrt(-2 + x^2)))/(3*x*sqrt(-2 + x^2)) - (sqrt(-2*x^2 + x^4)*atan(sqrt(-2 + x^2)))/(3*x*sqrt(-2 + x^2))]

    @test_int [sqrt(1 - 1/(-1 + x^2)^2)/(2 - x^2), x, 13, ((1 - x^2)*sqrt(1 - 1/(1 - x^2)^2)*atan(sqrt(-2 + x^2)))/(x*sqrt(-2 + x^2)), ((1 - x^2)*sqrt(-2*x^2 + x^4)*sqrt(1 - 1/(1 - x^2)^2)*atan(sqrt(-2 + x^2)))/(x*sqrt(-2 + x^2)*sqrt(-1 + (-1 + x^2)^2))]
    @test_int [sqrt((-2*x^2 + x^4)/(-1 + x^2)^2)/(2 + x^2), x, 8, -((2*(1 - x^2)*sqrt(-((2*x^2 - x^4)/(1 - x^2)^2))*atan((1/2)*sqrt(-2 + x^2)))/(3*x*sqrt(-2 + x^2))) + ((1 - x^2)*sqrt(-((2*x^2 - x^4)/(1 - x^2)^2))*atan(sqrt(-2 + x^2)))/(3*x*sqrt(-2 + x^2))]

    @test_int [(1 + 2*x/(1 + x^2))^(5/2), x, 6, (-(4/3))*(1 - 2*x)*(1 + x)*sqrt(1 + (2*x)/(1 + x^2)) - ((1 - x)*(1 + x)^3*sqrt(1 + (2*x)/(1 + x^2)))/(3*(1 + x^2)) - ((4 + 3*x)*(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2)))/(1 + x) + (5*sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2))*asinh(x))/(1 + x)]
    @test_int [(1 + 2*x/(1 + x^2))^(3/2), x, 6, -((1 - x)*(1 + x)*sqrt(1 + (2*x)/(1 + x^2))) - (x*(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2)))/(1 + x) + (3*sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2))*asinh(x))/(1 + x)]
    @test_int [(1 + 2*x/(1 + x^2))^(1/2), x, 4, ((1 + x^2)*sqrt(1 + (2*x)/(1 + x^2)))/(1 + x) + (sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2))*asinh(x))/(1 + x)]
    @test_int [1/(1 + 2*x/(1 + x^2))^(1/2), x, 7, (1 + x)/sqrt(1 + (2*x)/(1 + x^2)) - ((1 + x)*asinh(x))/(sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2))) - (sqrt(2)*(1 + x)*atanh((1 - x)/(sqrt(2)*sqrt(1 + x^2))))/(sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2)))]
    @test_int [1/(1 + 2*x/(1 + x^2))^(3/2), x, 8, (3*(2 + x))/(2*sqrt(1 + (2*x)/(1 + x^2))) - (1 + x^2)/(2*(1 + x)*sqrt(1 + (2*x)/(1 + x^2))) - (3*(1 + x)*asinh(x))/(sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2))) - (9*(1 + x)*atanh((1 - x)/(sqrt(2)*sqrt(1 + x^2))))/(2*sqrt(2)*sqrt(1 + x^2)*sqrt(1 + (2*x)/(1 + x^2)))]

    @test_int [sqrt(1 + 2*x/(1 + x^2))/(1 + x^2), x, 3, -(((1 - x)*sqrt(1 + (2*x)/(1 + x^2)))/(1 + x))]

    #= Piecewise*constant*extraction*and*simplification*caused*infinite*recursion*prior*to*version*4.89.0. =#
    @test_int [F(x)*sqrt(x - x^2), x, 0, CannotIntegrate(sqrt(x - x^2)*F(x), x)]
    @test_int [F(x)/sqrt(x - x^2), x, 0, CannotIntegrate(F(x)/sqrt(x - x^2), x)]

    @test_int [F(x)*(sqrt(1 - x)*sqrt(x)), x, 1, CannotIntegrate(sqrt(x - x^2)*F(x), x)]
    @test_int [F(x)/(sqrt(1 - x)*sqrt(x)), x, 1, CannotIntegrate(F(x)/sqrt(x - x^2), x)]

    #= ::Section::Closed:: =#
    #=Integrands*involving*roots*of*improper*binomials=#

    #= Integrands*of*the*form*F(x^m*(a+b*x^n)^p)*where*m==-n*p =#
    @test_int [F((a + b*x)/x), x, 1, CannotIntegrate(F(b + a/x), x)]
    @test_int [F((a + b*x^2)/x^2), x, 1, CannotIntegrate(F(b + a/x^2), x)]
    @test_int [F(x/(a + b*x)), x, 0, CannotIntegrate(F(x/(a + b*x)), x)]
    @test_int [F(x^2/(a + b*x^2)), x, 0, CannotIntegrate(F(x^2/(a + b*x^2)), x)]
    @test_int [F(x^2/(a + b*x)^2), x, 0, CannotIntegrate(F(x^2/(a + b*x)^2), x)]
    @test_int [F(x^4/(a + b*x^2)^2), x, 0, CannotIntegrate(F(x^4/(a + b*x^2)^2), x)]

    #= ::Section::Closed:: =#
    #=Integrands*involving*nested*radicals=#

    @test_int [sqrt(b*x^2 + sqrt(a + b^2*x^4))/sqrt(a + b^2*x^4), x, 2, atanh((sqrt(2)*sqrt(b)*x)/sqrt(b*x^2 + sqrt(a + b^2*x^4)))/(sqrt(2)*sqrt(b))]
    @test_int [sqrt(-b*x^2 + sqrt(a + b^2*x^4))/sqrt(a + b^2*x^4), x, 2, atan((sqrt(2)*sqrt(b)*x)/sqrt((-b)*x^2 + sqrt(a + b^2*x^4)))/(sqrt(2)*sqrt(b))]

    @test_int [sqrt(2*x^2 + sqrt(3 + 4*x^4))/((c + d*x)*sqrt(3 + 4*x^4)), x, 5, ((1/2 - im/2)*atan((sqrt(3)*d + 2*im*c*x)/(sqrt(2*im*c^2 - sqrt(3)*d^2)*sqrt(sqrt(3) - 2*im*x^2))))/sqrt(2*im*c^2 - sqrt(3)*d^2) - ((1/2 + im/2)*atanh((sqrt(3)*d - 2*im*c*x)/(sqrt(2*im*c^2 + sqrt(3)*d^2)*sqrt(sqrt(3) + 2*im*x^2))))/sqrt(2*im*c^2 + sqrt(3)*d^2)]
    @test_int [sqrt(2*x^2 + sqrt(3 + 4*x^4))/((c + d*x)^2*sqrt(3 + 4*x^4)), x, 7, ((1/2 - im/2)*d*sqrt(sqrt(3) - 2*im*x^2))/((2*im*c^2 - sqrt(3)*d^2)*(c + d*x)) - ((1/2 + im/2)*d*sqrt(sqrt(3) + 2*im*x^2))/((2*im*c^2 + sqrt(3)*d^2)*(c + d*x)) + ((1 + im)*c*atan((sqrt(3)*d + 2*im*c*x)/(sqrt(2*im*c^2 - sqrt(3)*d^2)*sqrt(sqrt(3) - 2*im*x^2))))/(2*im*c^2 - sqrt(3)*d^2)^(3/2) + ((1 - im)*c*atanh((sqrt(3)*d - 2*im*c*x)/(sqrt(2*im*c^2 + sqrt(3)*d^2)*sqrt(sqrt(3) + 2*im*x^2))))/(2*im*c^2 + sqrt(3)*d^2)^(3/2)]

    #= ::Section::Closed:: =#
    #=Miscellaneous*algebraic*function*integrands=#

    @test_int [(-4 + x)/((1 + x^(1/3))*sqrt(x)), x, 6, -30*x^(1/6) + 2*sqrt(x) - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 30*atan(x^(1/6))]
    @test_int [(1 + sqrt(x))/(x^(5/6) + x^(7/6)), x, 7, 3*x^(1/3) + 6*atan(x^(1/6)) - 3*log(1 + x^(1/3))]
    @test_int [(1 + sqrt(x))/((1 + x^(1/3))*sqrt(x)), x, 8, 6*x^(1/6) - 3*x^(1/3) + (3*x^(2/3))/2 - 6*atan(x^(1/6)) + 3*log(1 + x^(1/3))]

    @test_int [sqrt(2 + b/x^2)/(b + 2*x^2), x, 3, -(acsch((sqrt(2)*x)/sqrt(b))/sqrt(b))]
    @test_int [sqrt(2 - b/x^2)/(-b + 2*x^2), x, 3, -(acsc((sqrt(2)*x)/sqrt(b))/sqrt(b))]

    @test_int [sqrt(a + c/x^2)/(d + e*x), x, 11, (sqrt(a)*atanh(sqrt(a + c/x^2)/sqrt(a)))/e - (sqrt(a*d^2 + c*e^2)*atanh((a*d - (c*e)/x)/(sqrt(a*d^2 + c*e^2)*sqrt(a + c/x^2))))/(d*e) - (sqrt(c)*atanh(sqrt(c)/(sqrt(a + c/x^2)*x)))/d]
    @test_int [sqrt(a + b/x + c/x^2)/(d + e*x), x, 10, (sqrt(a)*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + c/x^2 + b/x))))/e - (sqrt(c)*atanh((b + (2*c)/x)/(2*sqrt(c)*sqrt(a + c/x^2 + b/x))))/d - (sqrt(a*d^2 - e*(b*d - c*e))*atanh((2*a*d - b*e + (b*d - 2*c*e)/x)/(2*sqrt(a*d^2 - e*(b*d - c*e))*sqrt(a + c/x^2 + b/x))))/(d*e)]

    @test_int [(x^(1/6) + (x^3)^(1/5))/sqrt(x), x, 4, (3*x^(2/3))/2 + (10*sqrt(x)*(x^3)^(1/5))/11]

    @test_int [(2 + x)/sqrt(4*x - x^2), x, 3, -sqrt(4*x - x^2) - 4*asin(1 - x/2)]
    @test_int [(3 + x)/(6*x + x^2)^(1/3), x, 1, (3/4)*(6*x + x^2)^(2/3)]
    @test_int [(4 + x)/(6*x - x^2)^(3/2), x, 1, -((12 - 7*x)/(9*sqrt(6*x - x^2)))]
    @test_int [1/((1 + x)*sqrt(2*x + x^2)), x, 2, atan(sqrt(2*x + x^2))]
    @test_int [1/((1 + 2*x)*sqrt(x + x^2)), x, 2, atan(2*sqrt(x + x^2))]
    @test_int [(-1 + x)/sqrt(2*x - x^2), x, 1, -sqrt(2*x - x^2)]
    @test_int [sqrt(x - x^2)/(1 + x), x, 6, sqrt(x - x^2) - (3/2)*asin(1 - 2*x) + sqrt(2)*atan((1 - 3*x)/(2*sqrt(2)*sqrt(x - x^2)))]

    @test_int [sqrt(x^(1/4) + x), x, 5, (1/3)*x^(1/4)*sqrt(x^(1/4) + x) + (2/3)*x*sqrt(x^(1/4) + x) - (1/3)*atanh(sqrt(x)/sqrt(x^(1/4) + x))]
    @test_int [sqrt(x + x^(3/2)), x, 3, (32*(x + x^(3/2))^(3/2))/(105*x^(3/2)) - (16*(x + x^(3/2))^(3/2))/(35*x) + (4*(x + x^(3/2))^(3/2))/(7*sqrt(x))]
    @test_int [x*sqrt(x + x^(3/2)), x, 5, (-(32/99))*(x + x^(3/2))^(3/2) + (512*(x + x^(3/2))^(3/2))/(3465*x^(3/2)) - (256*(x + x^(3/2))^(3/2))/(1155*x) + (64*(x + x^(3/2))^(3/2))/(231*sqrt(x)) + (4/11)*sqrt(x)*(x + x^(3/2))^(3/2)]

    @test_int [(1 - x^2)*sqrt(1/(2 - x^2)), x, 2, x/(2*sqrt(1/(2 - x^2)))]

    @test_int [sqrt(x^2 + x^3 - x^4), x, 5, -(((1 - 2*x)*sqrt(x^2 + x^3 - x^4))/(8*x)) - ((1 + x - x^2)*sqrt(x^2 + x^3 - x^4))/(3*x) - (5*sqrt(x^2 + x^3 - x^4)*asin((1 - 2*x)/sqrt(5)))/(16*x*sqrt(1 + x - x^2))]

    @test_int [1/sqrt((a^2 + x^2)^3), x, 2, (x*(a^2 + x^2))/(a^2*sqrt((a^2 + x^2)^3))]

    @test_int [sqrt(x)/(1 + sqrt(x) + x), x, 6, 2*sqrt(x) - (2*atan((1 + 2*sqrt(x))/sqrt(3)))/sqrt(3) - log(1 + sqrt(x) + x)]
    @test_int [x/(1 + sqrt(x) + x), x, 5, -2*sqrt(x) + x + (4*atan((1 + 2*sqrt(x))/sqrt(3)))/sqrt(3)]
    @test_int [1/(sqrt(x)*(1 + sqrt(x) + x)^(7/2)), x, 4, (4*(1 + 2*sqrt(x)))/(15*(1 + sqrt(x) + x)^(5/2)) + (64*(1 + 2*sqrt(x)))/(135*(1 + sqrt(x) + x)^(3/2)) + (512*(1 + 2*sqrt(x)))/(405*sqrt(1 + sqrt(x) + x))]

    #= [sqrt(1+x^2)/(1-x^3), x, 0] =#

    @test_int [(-1 + x)/(1 + sqrt(1 + x^2)), x, 10, -(1/x) + sqrt(1 + x^2) + sqrt(1 + x^2)/x - asinh(x) - log(1 + sqrt(1 + x^2))]

    @test_int [1/((1 + x)^(2/3)*(-1 + x^2)^(2/3)), x, 1, (3*(-1 + x^2)^(1/3))/(2*(1 + x)^(2/3))]

    @test_int [(1 - x^6)^(2/3) + (1 - x^6)^(2/3)/x^6, x, -3, -((1 - x^6)^(2/3)/(5*x^5)) + (1/5)*x*(1 - x^6)^(2/3)]

    @test_int [(x^(-1 + m)*(2*a*m + b*(2*m - n)*x^n))/(2*(a + b*x^n)^(3/2)), x, 2, x^m/sqrt(a + b*x^n)]

    @test_int [(x - 2*x^3)/sqrt(2 + 3*x), x, 3, (-(4/81))*sqrt(2 + 3*x) - (10/81)*(2 + 3*x)^(3/2) + (8/135)*(2 + 3*x)^(5/2) - (4/567)*(2 + 3*x)^(7/2)]

    @test_int [1/((1 + x)^(1/4) + sqrt(1 + x)), x, 5, -4*(1 + x)^(1/4) + 2*sqrt(1 + x) + 4*log(1 + (1 + x)^(1/4))]
    @test_int [(1 + 2*x)/sqrt(x + x^2), x, 1, 2*sqrt(x + x^2)]
    @test_int [1/(2*sqrt(x)*(1 + x)), x, 3, atan(sqrt(x))]
    @test_int [1/(x*sqrt(6*x - x^2)), x, 1, -(sqrt(6*x - x^2)/(3*x))]
    @test_int [(1 + sqrt(x))*sqrt(x), x, 2, (2*x^(3/2))/3 + x^2/2]
    @test_int [(1 - sqrt(x))/x^(1/3), x, 2, (3*x^(2/3))/2 - (6*x^(7/6))/7]
    @test_int [sqrt(x)/(1 + x^(1/3)), x, 7, -6*x^(1/6) + 2*sqrt(x) - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*atan(x^(1/6))]

    @test_int [(1 + sqrt(x))^(1/3)/x, x, 6, 6*(1 + sqrt(x))^(1/3) - 2*sqrt(3)*atan((1 + 2*(1 + sqrt(x))^(1/3))/sqrt(3)) + 3*log(1 - (1 + sqrt(x))^(1/3)) - log(x)/2]

    @test_int [1 - sqrt(x), x, 1, x - (2*x^(3/2))/3]
    @test_int [1 - x^(1/4), x, 1, x - (4*x^(5/4))/5]
    @test_int [(1 - sqrt(x))/(1 + x^(1/4)), x, 2, x - (4*x^(5/4))/5]

    @test_int [1/sqrt((a + b*x)*(c + d*x)), x, 3, atanh((b*c + a*d + 2*b*d*x)/(2*sqrt(b)*sqrt(d)*sqrt(a*c + (b*c + a*d)*x + b*d*x^2)))/(sqrt(b)*sqrt(d))]
    @test_int [1/sqrt((a + b*x)*(c - d*x)), x, 3, -(atan((b*c - a*d - 2*b*d*x)/(2*sqrt(b)*sqrt(d)*sqrt(a*c + (b*c - a*d)*x - b*d*x^2)))/(sqrt(b)*sqrt(d)))]

    @test_int [1/(sqrt(x)*(1 - x^2)), x, 4, atan(sqrt(x)) + atanh(sqrt(x))]
    @test_int [sqrt(x)/(x - x^3), x, 5, atan(sqrt(x)) + atanh(sqrt(x))]

    @test_int [x/(2 - sqrt(3) + (1 + sqrt(3))*x + x^2), x, 4, sqrt((1/23)*(13 + 8*sqrt(3)))*atanh((1 + sqrt(3) + 2*x)/sqrt(2*(-2 + 3*sqrt(3)))) + (1/2)*log(2 - sqrt(3) + (1 + sqrt(3))*x + x^2)]
    @test_int [sqrt(x^2 + x^3), x, 2, -((4*(x^2 + x^3)^(3/2))/(15*x^3)) + (2*(x^2 + x^3)^(3/2))/(5*x^2)]
    @test_int [1/((1 + x)*sqrt(2*x + x^2)), x, 2, atan(sqrt(2*x + x^2))]
    @test_int [sqrt(1 - sqrt(x) - x)*sqrt(x), x, 6, (9/32)*(1 + 2*sqrt(x))*sqrt(1 - sqrt(x) - x) + (5/12)*(1 - sqrt(x) - x)^(3/2) - (1/2)*(1 - sqrt(x) - x)^(3/2)*sqrt(x) + (45/64)*asin((1 + 2*sqrt(x))/sqrt(5))]

    @test_int [(1 + sqrt(-3 + x))^(1/3), x, 4, (-(3/2))*(1 + sqrt(-3 + x))^(4/3) + (6/7)*(1 + sqrt(-3 + x))^(7/3)]
    @test_int [1/sqrt(3 + sqrt(-1 + 2*x)), x, 4, -6*sqrt(3 + sqrt(-1 + 2*x)) + (2/3)*(3 + sqrt(-1 + 2*x))^(3/2)]

    #= [(sqrt(x)+x)^(2/3), x, 0] =#
    #= [(-3*x+x^2)^(-1/3), x, 0] =#

    @test_int [sqrt(1 - x)/(1 + sqrt(x)), x, 4, -((2 - sqrt(x))*sqrt(1 - x)) - asin(sqrt(x))]
    @test_int [sqrt(1 - x)/(1 - sqrt(x)), x, 4, -((2 + sqrt(x))*sqrt(1 - x)) + asin(sqrt(x))]

    @test_int [x/(x - sqrt(1 + x^2)), x, 3, -(x^3/3) - (1/3)*(1 + x^2)^(3/2)]
    @test_int [x/(x - sqrt(1 - x^2)), x, 7, x/2 + sqrt(1 - x^2)/2 - atanh(sqrt(2)*x)/(2*sqrt(2)) - atanh(sqrt(2)*sqrt(1 - x^2))/(2*sqrt(2))]
    @test_int [x/(x - sqrt(1 + 2*x^2)), x, 7, -x - sqrt(1 + 2*x^2) + atan(x) + atan(sqrt(1 + 2*x^2))]

    @test_int [sqrt(x)*sqrt(sqrt(x) + x), x, 6, (5/32)*(1 + 2*sqrt(x))*sqrt(sqrt(x) + x) - (5/12)*(sqrt(x) + x)^(3/2) + (1/2)*sqrt(x)*(sqrt(x) + x)^(3/2) - (5/32)*atanh(sqrt(x)/sqrt(sqrt(x) + x))]

    @test_int [(1 + x^(1/3))/(1 + sqrt(x)), x, 10, -3*x^(1/3) + 2*sqrt(x) + (6*x^(5/6))/5 - 2*sqrt(3)*atan((1 - 2*x^(1/6))/sqrt(3)) - 4*log(1 + x^(1/6)) - log(1 - x^(1/6) + x^(1/3))]
    @test_int [(1 + x^(1/3))/(1 + x^(1/4)), x, 11, 12*x^(1/12) + 4*x^(1/4) - 3*x^(1/3) - 2*sqrt(x) + (12*x^(7/12))/7 + (4*x^(3/4))/3 - (6*x^(5/6))/5 + (12*x^(13/12))/13 + 4*sqrt(3)*atan((1 - 2*x^(1/12))/sqrt(3)) - 8*log(1 + x^(1/12)) - 2*log(1 - x^(1/12) + x^(1/6))]

    #= [1/sqrt(a*x+b*x^3), x, 0] =#

    @test_int [x^2/(-1 + x^2 + sqrt(1 - x^2)), x, 3, x + asin(x)]

    @test_int [sqrt((1 + x)/x), x, 5, sqrt(1 + 1/x)*x + atanh(sqrt(1 + 1/x))]
    @test_int [sqrt((1 - x)/x), x, 5, sqrt(-1 + 1/x)*x - atan(sqrt(-1 + 1/x))]
    @test_int [sqrt((-1 + x)/x), x, 5, sqrt(-1 + x)*sqrt(x) - asinh(sqrt(-1 + x)), sqrt((-1 + x)/x)*x - atanh(sqrt((-1 + x)/x))]
    @test_int [sqrt((1 + x)/x)/x, x, 5, -2*sqrt(1 + 1/x) + 2*atanh(sqrt(1 + 1/x))]

    @test_int [sqrt(x/(1 + x)), x, 4, sqrt(x)*sqrt(1 + x) - asinh(sqrt(x))]

    @test_int [1/sqrt((-1 - x)/x), x, 5, (-x)*sqrt(-((1 + x)/x)) + atan(sqrt(-((1 + x)/x)))]

    @test_int [sqrt((4 - x)*x), x, 4, (-(1/2))*(2 - x)*sqrt(4*x - x^2) - 2*asin(1 - x/2)]
    @test_int [1/sqrt((1 - x)*x), x, 3, -asin(1 - 2*x)]

    @test_int [x/(x*(2 + x))^(3/2), x, 2, x/sqrt(2*x + x^2)]

    @test_int [sqrt(1 + 1/x)/(1 - x^2), x, 5, sqrt(2)*atanh(sqrt(1 + 1/x)/sqrt(2))]

    @test_int [1/(1 + sqrt(5) - x^2 + sqrt(5)*x^2), x, 2, (1/2)*atan(sqrt((1/2)*(3 - sqrt(5)))*x)]

    #= Integrands*equivalent*to*expressions*of*the*form*1/sqrt(a*x + b*x^2) =#
    @test_int [1/sqrt(a*x + b*x^2), x, 2, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]
    @test_int [1/sqrt(x*(a + b*x)), x, 3, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]
    @test_int [1/sqrt(x^2*(b + a/x)), x, 3, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]
    @test_int [1/sqrt(x^3*(b/x + a/x^2)), x, 3, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]
    @test_int [1/sqrt((a*x^2 + b*x^3)/x), x, 3, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]
    @test_int [1/sqrt((a*x^3 + b*x^4)/x^2), x, 3, (2*atanh((sqrt(b)*x)/sqrt(a*x + b*x^2)))/sqrt(b)]

    #= Integrands*equivalent*to*expressions*of*the*form*1/sqrt(a*c*x + b*c*x^2) =#
    @test_int [1/sqrt(a*c*x + b*c*x^2), x, 2, (2*atanh((sqrt(b)*sqrt(c)*x)/sqrt(a*c*x + b*c*x^2)))/(sqrt(b)*sqrt(c))]
    @test_int [1/sqrt(c*(a*x + b*x^2)), x, 3, (2*atanh((sqrt(b)*sqrt(c)*x)/sqrt(a*c*x + b*c*x^2)))/(sqrt(b)*sqrt(c))]
    @test_int [1/sqrt(c*x*(a + b*x)), x, 3, (2*atanh((sqrt(b)*sqrt(c)*x)/sqrt(a*c*x + b*c*x^2)))/(sqrt(b)*sqrt(c))]
    @test_int [1/sqrt(c*x^2*(b + a/x)), x, 3, (2*atanh((sqrt(b)*sqrt(c)*x)/sqrt(a*c*x + b*c*x^2)))/(sqrt(b)*sqrt(c))]

    #= Subproblems*of*Charlwood*Fifty*problems =#
    @test_int [sqrt(1 - x^2 + x*sqrt(-1 + x^2)), x, -1, (1/4)*(3*x + sqrt(-1 + x^2))*sqrt(1 - x^2 + x*sqrt(-1 + x^2)) + (3*asin(x - sqrt(-1 + x^2)))/(4*sqrt(2))]
    @test_int [sqrt(-x + sqrt(x)*sqrt(1 + x))/sqrt(1 + x), x, -1, (1/2)*(sqrt(x) + 3*sqrt(1 + x))*sqrt(-x + sqrt(x)*sqrt(1 + x)) - (3*asin(sqrt(x) - sqrt(1 + x)))/(2*sqrt(2))]
    @test_int [-((x + 2*sqrt(1 + x^2))/(x + x^3 + sqrt(1 + x^2))), x, -25, (-sqrt(2*(1 + sqrt(5))))*atan(sqrt(-2 + sqrt(5))*(x + sqrt(1 + x^2))) + sqrt(2*(-1 + sqrt(5)))*atanh(sqrt(2 + sqrt(5))*(x + sqrt(1 + x^2)))]
    @test_int [(1 + 2*x)/((1 + x^2)*sqrt(2 + 2*x + x^2)), x, 5, (-sqrt((1/2)*(1 + sqrt(5))))*atan((2*sqrt(5) - (5 + sqrt(5))*x)/(sqrt(10*(1 + sqrt(5)))*sqrt(2 + 2*x + x^2))) - sqrt((1/2)*(-1 + sqrt(5)))*atanh((2*sqrt(5) + (5 - sqrt(5))*x)/(sqrt(10*(-1 + sqrt(5)))*sqrt(2 + 2*x + x^2)))]

    @test_int [1/((1 + x^4)*sqrt(-x^2 + sqrt(1 + x^4))), x, 2, atan(x/sqrt(-x^2 + sqrt(1 + x^4)))]

    @test_int [1/((a + b*x^4)*sqrt(c*x^2 + d*sqrt(a + b*x^4))), x, 2, atanh((sqrt(c)*x)/sqrt(c*x^2 + d*sqrt(a + b*x^4)))/(a*sqrt(c))]
    @test_int [1/((a + b*x^4)*sqrt((-c)*x^2 + d*sqrt(a + b*x^4))), x, 2, atan((sqrt(c)*x)/sqrt((-c)*x^2 + d*sqrt(a + b*x^4)))/(a*sqrt(c))]

    @test_int [x/sqrt(a + b*c^4 + 4*b*c^3*d*x + 6*b*c^2*d^2*x^2 + 4*b*c*d^3*x^3 + b*d^4*x^4), x, 7, atanh((sqrt(b)*d^2*(c/d + x)^2)/sqrt(a + b*d^4*(c/d + x)^4))/(2*sqrt(b)*d^2) - (c*(sqrt(a) + sqrt(b)*d^2*(c/d + x)^2)*sqrt((a + b*d^4*(c/d + x)^4)/(sqrt(a) + sqrt(b)*d^2*(c/d + x)^2)^2)*Elliptic.F(2*atan((b^(1/4)*(c + d*x))/a^(1/4)), 1/2))/(2*a^(1/4)*b^(1/4)*d^2*sqrt(a + b*d^4*(c/d + x)^4))]
    @test_int [1/sqrt(a + b*c^4 + 4*b*c^3*d*x + 6*b*c^2*d^2*x^2 + 4*b*c*d^3*x^3 + b*d^4*x^4), x, 2, ((sqrt(a) + sqrt(b)*d^2*(c/d + x)^2)*sqrt((a + b*d^4*(c/d + x)^4)/(sqrt(a) + sqrt(b)*d^2*(c/d + x)^2)^2)*Elliptic.F(2*atan((b^(1/4)*(c + d*x))/a^(1/4)), 1/2))/(2*a^(1/4)*b^(1/4)*d*sqrt(a + b*d^4*(c/d + x)^4))]

    @test_int [(a - c*x^4)/(sqrt(a + b*x^2 + c*x^4)*(a*d + a*e*x^2 + c*d*x^4)), x, 2, atanh((sqrt(b*d - a*e)*x)/(sqrt(d)*sqrt(a + b*x^2 + c*x^4)))/(sqrt(d)*sqrt(b*d - a*e))]
    @test_int [(a - c*x^4)/(sqrt(a - b*x^2 + c*x^4)*(a*d + a*e*x^2 + c*d*x^4)), x, 2, atan((sqrt(b*d + a*e)*x)/(sqrt(d)*sqrt(a - b*x^2 + c*x^4)))/(sqrt(d)*sqrt(b*d + a*e))]

    @test_int [1/(sqrt(5 - 2*x + x^2)*(8 + x^3)), x, 9, -(atan((1 - x)/(sqrt(3)*sqrt(5 - 2*x + x^2)))/(4*sqrt(3))) - atanh((7 - 3*x)/(sqrt(13)*sqrt(5 - 2*x + x^2)))/(12*sqrt(13)) + (1/12)*atanh(sqrt(5 - 2*x + x^2))]

    @test_int [sqrt(x^2/(1 + x^2)), x, 3, (sqrt(x^2)*sqrt(1 + x^2))/x]
    @test_int [sqrt(x^n/(1 + x^n)), x, 3, (2*x*sqrt(x^n)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 + 2/n), (1/2)*(3 + 2/n), -x^n))/(2 + n)]

    @test_int [(e*f - e*f*x^2)/((a*d + b*d*x + a*d*x^2)*sqrt(a + b*x + c*x^2 + b*x^3 + a*x^4)), x, 1, (e*f*atan((a*b + (4*a^2 + b^2 - 2*a*c)*x + a*b*x^2)/(2*a*sqrt(2*a - c)*sqrt(a + b*x + c*x^2 + b*x^3 + a*x^4))))/(a*sqrt(2*a - c)*d)]
    @test_int [(e*f - e*f*x^2)/(((-a)*d + b*d*x - a*d*x^2)*sqrt(-a + b*x + c*x^2 + b*x^3 - a*x^4)), x, 1, (e*f*atanh((a*b - (4*a^2 + b^2 + 2*a*c)*x + a*b*x^2)/(2*a*sqrt(2*a + c)*sqrt(-a + b*x + c*x^2 + b*x^3 - a*x^4))))/(a*sqrt(2*a + c)*d)]

    @test_int [sqrt(a*x^2 + b*x*sqrt(-(a/b^2) + (a^2*x^2)/b^2))/(x*sqrt(-(a/b^2) + (a^2*x^2)/b^2)), x, 2, (sqrt(2)*b*asinh((a*x + b*sqrt(-(a/b^2) + (a^2*x^2)/b^2))/sqrt(a)))/sqrt(a)]
    @test_int [sqrt((-a)*x^2 + b*x*sqrt(a/b^2 + (a^2*x^2)/b^2))/(x*sqrt(a/b^2 + (a^2*x^2)/b^2)), x, 2, (sqrt(2)*b*asin((a*x - b*sqrt(a/b^2 + (a^2*x^2)/b^2))/sqrt(a)))/sqrt(a)]

    @test_int [sqrt(x*(a*x + b*sqrt(-(a/b^2) + (a^2*x^2)/b^2)))/(x*sqrt(-(a/b^2) + (a^2*x^2)/b^2)), x, 3, (sqrt(2)*b*asinh((a*x + b*sqrt(-(a/b^2) + (a^2*x^2)/b^2))/sqrt(a)))/sqrt(a)]
    @test_int [sqrt(x*((-a)*x + b*sqrt(a/b^2 + (a^2*x^2)/b^2)))/(x*sqrt(a/b^2 + (a^2*x^2)/b^2)), x, 3, (sqrt(2)*b*asin((a*x - b*sqrt(a/b^2 + (a^2*x^2)/b^2))/sqrt(a)))/sqrt(a)]

    @test_int [(sqrt(x - 1)*x - 4*sqrt(x - 1) + sqrt(x - 4)*x - sqrt(x - 4))/((1 + sqrt(x - 4) + sqrt(x - 1))*(x^2 - 5*x + 4)), x, 3, 2*log(1 + sqrt(-4 + x) + sqrt(-1 + x))]

    @test_int [1/(x*(3 + 3*x + x^2)*(3 + 3*x + 3*x^2 + x^3)^(1/3)), x, 3, -(atan((1 + (2*3^(1/3)*(1 + x))/(2 + (1 + x)^3)^(1/3))/sqrt(3))/3^(5/6)) - log(1 - (1 + x)^3)/(6*3^(1/3)) + log(3^(1/3)*(1 + x) - (2 + (1 + x)^3)^(1/3))/(2*3^(1/3))]

    @test_int [(1 - x^2)/((1 - x + x^2)*(1 - x^3)^(2/3)), x, -42, (sqrt(3)*atan((1 - (2*2^(1/3)*(1 - x))/(1 - x^3)^(1/3))/sqrt(3)))/2^(2/3) - log(1 + 2*(1 - x)^3 - x^3)/(2*2^(2/3)) + (3*log(2^(1/3)*(1 - x) + (1 - x^3)^(1/3)))/(2*2^(2/3))]

    @test_int [x^2/(sqrt(-1 + x^4)*(1 + x^4)), x, -9, (-(1/4))*atan((1 + x^2)/(x*sqrt(-1 + x^4))) - (1/4)*atanh((1 - x^2)/(x*sqrt(-1 + x^4)))]

    @test_int [(a - c*x^4)/((a*e + c*d*x^2)*(d + e*x^2)*sqrt(a + b*x^2 + c*x^4)), x, 2, atan((sqrt(c*d^2 - b*d*e + a*e^2)*x)/(sqrt(d)*sqrt(e)*sqrt(a + b*x^2 + c*x^4)))/(sqrt(d)*sqrt(e)*sqrt(c*d^2 - b*d*e + a*e^2))]

    #= [(1 - sqrt(3) + x)/((1 + sqrt(3) + x)*sqrt(-4 + 2*sqrt(3) + x^2)*sqrt(2 - (-2 + sqrt(3))*x^2)), x, -10, (1/3^(3/4))*atanh((1 - sqrt(3) + x)^2/(3^(3/4)*sqrt(-4 + 2*sqrt(3) + x^2)*sqrt(2 - (-2 + sqrt(3))*x^2)))]
    @test_int [(1 - sqrt(3) + x)/((1 + sqrt(3) + x)*sqrt((-4 + 2*sqrt(3) + x^2)*(2 - (sqrt(3) - 2)*x^2))), x, -11, (1/3^(3/4))*atanh((1 - sqrt(3) + x)^2/(3^(3/4)*sqrt((-4 + 2*sqrt(3) + x^2)*(2 - (sqrt(3) - 2)*x^2))))] =#

    #= Lack*of*gcd*cancellation*used*to*cause*a*zero-divide*error*in*IntSum. =#
    @test_int [x + (1 - x^2)/(1 + x), x, 1, x]
    @test_int [1/(1/x + sqrt(1 - x^2)), x, 12, asin(x) - atan((1 - 2*x^2)/sqrt(3))/sqrt(3) - atan(x/(sqrt(-((im - sqrt(3))/(im + sqrt(3))))*sqrt(1 - x^2)))/sqrt(3) - atan((sqrt(-((im - sqrt(3))/(im + sqrt(3))))*x)/sqrt(1 - x^2))/sqrt(3)]
    @test_int [x*(sqrt(1 - x^2)/(x - x^3 + sqrt(1 - x^2))), x, 13, asin(x) - atan((1 - 2*x^2)/sqrt(3))/sqrt(3) - atan(x/(sqrt(-((im - sqrt(3))/(im + sqrt(3))))*sqrt(1 - x^2)))/sqrt(3) - atan((sqrt(-((im - sqrt(3))/(im + sqrt(3))))*x)/sqrt(1 - x^2))/sqrt(3), (1/4)*(1 - x)^2 - x^2/2 + (1/4)*(1 + x)^2 + asin(x) - atan((1 - 2*x^2)/sqrt(3))/sqrt(3) - atan(x/(sqrt(-((im - sqrt(3))/(im + sqrt(3))))*sqrt(1 - x^2)))/sqrt(3) - atan((sqrt(-((im - sqrt(3))/(im + sqrt(3))))*x)/sqrt(1 - x^2))/sqrt(3)]

    @test_int [(1 - x^4)^n/(1 + x + x^2 + x^3)^n, x, -1, -(((1 - x)*(1 - x^4)^n)/((1 + x + x^2 + x^3)^n*(1 + n)))]

    #= Manuel*Bronstein*pseudo-elliptic*integrals: =#
    @test_int [x/sqrt(-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4), x, 1, (1/(18432*c^2))*log(20738073600000000*b^8*c^4 + 597005697024000000*b^6*c^6*x^2 + 2583100705996800000*b^5*c^7*x^3 + 951050714480640000*b^4*c^8*x^4 + 21641687369515008000*b^3*c^9*x^5 + 32462531054272512000*b^2*c^10*x^6 + 149587343098087735296*c^12*x^8 + 5308416*sqrt(-44375*b^4 + 576000*b^3*c*x + 576000*b^2*c^2*x^2 + 5308416*c^4*x^4)*(12203125*b^6*c^4 + 79200000*b^5*c^5*x + 38880000*b^4*c^6*x^2 + 1105920000*b^3*c^7*x^3 + 1990656000*b^2*c^8*x^4 + 12230590464*c^10*x^6))]

    @test_int [(1 + 4*x)/sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4), x, 2, (1/16)*log(921 + 2864*x + 9280*x^2 + 13440*x^3 + 17024*x^4 + 19456*x^5 + 12288*x^6 + 8192*x^7 + 4096*x^8 + sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4)*(179 + 444*x + 744*x^2 + 1280*x^3 + 960*x^4 + 768*x^5 + 512*x^6)), (1/16)*log(921 + 2864*x + 9280*x^2 + 13440*x^3 + 17024*x^4 + 19456*x^5 + 12288*x^6 + 8192*x^7 + 4096*x^8 + 179*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 444*x*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 744*x^2*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 1280*x^3*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 960*x^4*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 768*x^5*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4) + 512*x^6*sqrt(9 + 120*x + 64*x^2 + 64*x^3 + 64*x^4))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.3 Miscellaneous/1.3.2 Algebraic functions.jl