text:
@testset "1.1.3.3 (a+b x^n)^p (c+d x^n)^q" begin
    (a, b, c, d, e, m, n, p, q, x, ) = @variables a b c d e m n p q x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^p*(c+d*x^n)^q*with*integer*n>2=#

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

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

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

    @test_int [(a + b*x^3)*(c + d*x^3)^4, x, 2, a*c^4*x + (1/4)*c^3*(b*c + 4*a*d)*x^4 + (2/7)*c^2*d*(2*b*c + 3*a*d)*x^7 + (1/5)*c*d^2*(3*b*c + 2*a*d)*x^10 + (1/13)*d^3*(4*b*c + a*d)*x^13 + (1/16)*b*d^4*x^16]
    @test_int [(a + b*x^3)*(c + d*x^3)^3, x, 2, a*c^3*x + (1/4)*c^2*(b*c + 3*a*d)*x^4 + (3/7)*c*d*(b*c + a*d)*x^7 + (1/10)*d^2*(3*b*c + a*d)*x^10 + (1/13)*b*d^3*x^13]
    @test_int [(a + b*x^3)*(c + d*x^3)^2, x, 2, a*c^2*x + (1/4)*c*(b*c + 2*a*d)*x^4 + (1/7)*d*(2*b*c + a*d)*x^7 + (1/10)*b*d^2*x^10]
    @test_int [(a + b*x^3)*(c + d*x^3)^1, x, 2, a*c*x + (1/4)*(b*c + a*d)*x^4 + (1/7)*b*d*x^7]
    @test_int [(a + b*x^3)/(c + d*x^3)^1, x, 7, (b*x)/d + ((b*c - a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(4/3)) - ((b*c - a*d)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(4/3)) + ((b*c - a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(4/3))]
    @test_int [(a + b*x^3)/(c + d*x^3)^2, x, 7, -(((b*c - a*d)*x)/(3*c*d*(c + d*x^3))) - ((b*c + 2*a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(3*sqrt(3)*c^(5/3)*d^(4/3)) + ((b*c + 2*a*d)*log(c^(1/3) + d^(1/3)*x))/(9*c^(5/3)*d^(4/3)) - ((b*c + 2*a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(18*c^(5/3)*d^(4/3))]
    @test_int [(a + b*x^3)/(c + d*x^3)^3, x, 8, -(((b*c - a*d)*x)/(6*c*d*(c + d*x^3)^2)) + ((b*c + 5*a*d)*x)/(18*c^2*d*(c + d*x^3)) - ((b*c + 5*a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(9*sqrt(3)*c^(8/3)*d^(4/3)) + ((b*c + 5*a*d)*log(c^(1/3) + d^(1/3)*x))/(27*c^(8/3)*d^(4/3)) - ((b*c + 5*a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(54*c^(8/3)*d^(4/3))]

    @test_int [(a + b*x^3)^2*(c + d*x^3)^3, x, 2, a^2*c^3*x + (1/4)*a*c^2*(2*b*c + 3*a*d)*x^4 + (1/7)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7 + (1/10)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^10 + (1/13)*b*d^2*(3*b*c + 2*a*d)*x^13 + (1/16)*b^2*d^3*x^16]
    @test_int [(a + b*x^3)^2*(c + d*x^3)^2, x, 2, a^2*c^2*x + (1/2)*a*c*(b*c + a*d)*x^4 + (1/7)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7 + (1/5)*b*d*(b*c + a*d)*x^10 + (1/13)*b^2*d^2*x^13]
    @test_int [(a + b*x^3)^2*(c + d*x^3)^1, x, 2, a^2*c*x + (1/4)*a*(2*b*c + a*d)*x^4 + (1/7)*b*(b*c + 2*a*d)*x^7 + (1/10)*b^2*d*x^10]
    @test_int [(a + b*x^3)^2/(c + d*x^3)^1, x, 8, -((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^4)/(4*d) - ((b*c - a*d)^2*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*d^(7/3)) + ((b*c - a*d)^2*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*d^(7/3)) - ((b*c - a*d)^2*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*d^(7/3))]
    @test_int [(a + b*x^3)^2/(c + d*x^3)^2, x, 9, (b^2*x)/d^2 + ((b*c - a*d)^2*x)/(3*c*d^2*(c + d*x^3)) + (2*(b*c - a*d)*(2*b*c + a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(3*sqrt(3)*c^(5/3)*d^(7/3)) - (2*(b*c - a*d)*(2*b*c + a*d)*log(c^(1/3) + d^(1/3)*x))/(9*c^(5/3)*d^(7/3)) + ((b*c - a*d)*(2*b*c + a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(9*c^(5/3)*d^(7/3))]
    @test_int [(a + b*x^3)^2/(c + d*x^3)^3, x, 8, -(((b*c - a*d)*x*(a + b*x^3))/(6*c*d*(c + d*x^3)^2)) - ((b*c - a*d)*(4*b*c + 5*a*d)*x)/(18*c^2*d^2*(c + d*x^3)) - ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(9*sqrt(3)*c^(8/3)*d^(7/3)) + ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*log(c^(1/3) + d^(1/3)*x))/(27*c^(8/3)*d^(7/3)) - ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(54*c^(8/3)*d^(7/3))]

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

    @test_int [1/(a + b*x^3)*(c + d*x^3)^4, x, 8, (d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^3*(4*b*c - a*d)*x^7)/(7*b^2) + (d^4*x^10)/(10*b) - ((b*c - a*d)^4*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(13/3)) + ((b*c - a*d)^4*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*b^(13/3)) - ((b*c - a*d)^4*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*b^(13/3))]
    @test_int [1/(a + b*x^3)*(c + d*x^3)^3, x, 8, (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^4)/(4*b^2) + (d^3*x^7)/(7*b) - ((b*c - a*d)^3*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(10/3)) + ((b*c - a*d)^3*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*b^(10/3)) - ((b*c - a*d)^3*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*b^(10/3))]
    @test_int [1/(a + b*x^3)*(c + d*x^3)^2, x, 8, (d*(2*b*c - a*d)*x)/b^2 + (d^2*x^4)/(4*b) - ((b*c - a*d)^2*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(7/3)) + ((b*c - a*d)^2*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*b^(7/3)) - ((b*c - a*d)^2*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*b^(7/3))]
    @test_int [1/(a + b*x^3)*(c + d*x^3)^1, x, 7, (d*x)/b - ((b*c - a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*b^(4/3)) + ((b*c - a*d)*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*b^(4/3)) - ((b*c - a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*b^(4/3))]
    @test_int [1/(a + b*x^3)/(c + d*x^3)^1, x, 13, -((b^(2/3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*(b*c - a*d))) + (d^(2/3)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*(b*c - a*d)) + (b^(2/3)*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*(b*c - a*d)) - (d^(2/3)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*(b*c - a*d)) - (b^(2/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*(b*c - a*d)) + (d^(2/3)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*(b*c - a*d))]
    @test_int [1/(a + b*x^3)/(c + d*x^3)^2, x, 14, -((d*x)/(3*c*(b*c - a*d)*(c + d*x^3))) - (b^(5/3)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(sqrt(3)*a^(2/3)*(b*c - a*d)^2) + (d^(2/3)*(5*b*c - 2*a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^2) + (b^(5/3)*log(a^(1/3) + b^(1/3)*x))/(3*a^(2/3)*(b*c - a*d)^2) - (d^(2/3)*(5*b*c - 2*a*d)*log(c^(1/3) + d^(1/3)*x))/(9*c^(5/3)*(b*c - a*d)^2) - (b^(5/3)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(6*a^(2/3)*(b*c - a*d)^2) + (d^(2/3)*(5*b*c - 2*a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(18*c^(5/3)*(b*c - a*d)^2)]

    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^5, x, 9, (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^4)/(4*b^4) + (d^4*(5*b*c - 2*a*d)*x^7)/(7*b^3) + (d^5*x^10)/(10*b^2) + ((b*c - a*d)^5*x)/(3*a*b^5*(a + b*x^3)) - ((b*c - a*d)^4*(2*b*c + 13*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(16/3)) + ((b*c - a*d)^4*(2*b*c + 13*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(16/3)) - ((b*c - a*d)^4*(2*b*c + 13*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(18*a^(5/3)*b^(16/3))]
    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^4, x, 9, (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (d^3*(2*b*c - a*d)*x^4)/(2*b^3) + (d^4*x^7)/(7*b^2) + ((b*c - a*d)^4*x)/(3*a*b^4*(a + b*x^3)) - (2*(b*c - a*d)^3*(b*c + 5*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(13/3)) + (2*(b*c - a*d)^3*(b*c + 5*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(13/3)) - ((b*c - a*d)^3*(b*c + 5*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(9*a^(5/3)*b^(13/3))]
    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^3, x, 9, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^4)/(4*b^2) + ((b*c - a*d)^3*x)/(3*a*b^3*(a + b*x^3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(10/3)) + ((b*c - a*d)^2*(2*b*c + 7*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(10/3)) - ((b*c - a*d)^2*(2*b*c + 7*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(18*a^(5/3)*b^(10/3))]
    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^2, x, 9, (d^2*x)/b^2 + ((b*c - a*d)^2*x)/(3*a*b^2*(a + b*x^3)) - (2*(b*c - a*d)*(b*c + 2*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(7/3)) + (2*(b*c - a*d)*(b*c + 2*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(7/3)) - ((b*c - a*d)*(b*c + 2*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(9*a^(5/3)*b^(7/3))]
    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^1, x, 7, ((b*c - a*d)*x)/(3*a*b*(a + b*x^3)) - ((2*b*c + a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*b^(4/3)) + ((2*b*c + a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*b^(4/3)) - ((2*b*c + a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(18*a^(5/3)*b^(4/3))]
    @test_int [1/(a + b*x^3)^2/(c + d*x^3)^1, x, 14, (b*x)/(3*a*(b*c - a*d)*(a + b*x^3)) - (b^(2/3)*(2*b*c - 5*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*(b*c - a*d)^2) - (d^(5/3)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(sqrt(3)*c^(2/3)*(b*c - a*d)^2) + (b^(2/3)*(2*b*c - 5*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*(b*c - a*d)^2) + (d^(5/3)*log(c^(1/3) + d^(1/3)*x))/(3*c^(2/3)*(b*c - a*d)^2) - (b^(2/3)*(2*b*c - 5*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(18*a^(5/3)*(b*c - a*d)^2) - (d^(5/3)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(6*c^(2/3)*(b*c - a*d)^2)]
    @test_int [1/(a + b*x^3)^2/(c + d*x^3)^2, x, 15, (d*(b*c + a*d)*x)/(3*a*c*(b*c - a*d)^2*(c + d*x^3)) + (b*x)/(3*a*(b*c - a*d)*(a + b*x^3)*(c + d*x^3)) - (2*b^(5/3)*(b*c - 4*a*d)*atan((a^(1/3) - 2*b^(1/3)*x)/(sqrt(3)*a^(1/3))))/(3*sqrt(3)*a^(5/3)*(b*c - a*d)^3) - (2*d^(5/3)*(4*b*c - a*d)*atan((c^(1/3) - 2*d^(1/3)*x)/(sqrt(3)*c^(1/3))))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^3) + (2*b^(5/3)*(b*c - 4*a*d)*log(a^(1/3) + b^(1/3)*x))/(9*a^(5/3)*(b*c - a*d)^3) + (2*d^(5/3)*(4*b*c - a*d)*log(c^(1/3) + d^(1/3)*x))/(9*c^(5/3)*(b*c - a*d)^3) - (b^(5/3)*(b*c - 4*a*d)*log(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2))/(9*a^(5/3)*(b*c - a*d)^3) - (d^(5/3)*(4*b*c - a*d)*log(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2))/(9*c^(5/3)*(b*c - a*d)^3)]

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

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

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

    @test_int [(a + b*x^3)^(2/3)*(a - b*x^3), x, 3, (7/18)*a*x*(a + b*x^3)^(2/3) - (1/6)*x*(a + b*x^3)^(5/3) + (7*a^2*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*b^(1/3)) - (7*a^2*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*b^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(1/3), x, 2, (-(1/3))*x*(a + b*x^3)^(2/3) + (4*a*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(1/3)) - (2*a*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(3*b^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(4/3), x, 2, (2*x)/(a + b*x^3)^(1/3) - atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3))/(sqrt(3)*b^(1/3)) + log((-b^(1/3))*x + (a + b*x^3)^(1/3))/(2*b^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(7/3), x, 2, (x*(a - b*x^3))/(4*a*(a + b*x^3)^(4/3)) + (3*x)/(4*a*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(10/3), x, 3, (2*x)/(7*(a + b*x^3)^(7/3)) + (5*x)/(28*a*(a + b*x^3)^(4/3)) + (15*x)/(28*a^2*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(13/3), x, 4, x/(5*(a + b*x^3)^(10/3)) + (4*x)/(35*a*(a + b*x^3)^(7/3)) + (6*x)/(35*a^2*(a + b*x^3)^(4/3)) + (18*x)/(35*a^3*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)/(a + b*x^3)^(16/3), x, 5, (2*x)/(13*(a + b*x^3)^(13/3)) + (11*x)/(130*a*(a + b*x^3)^(10/3)) + (99*x)/(910*a^2*(a + b*x^3)^(7/3)) + (297*x)/(1820*a^3*(a + b*x^3)^(4/3)) + (891*x)/(1820*a^4*(a + b*x^3)^(1/3))]

    @test_int [(a + b*x^3)^(7/3)/(a - b*x^3), x, 22, (-(7/5))*a*x*(a + b*x^3)^(1/3) - (1/5)*x*(a + b*x^3)^(4/3) - (4*2^(1/3)*a^(5/3)*atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (2*2^(1/3)*a^(5/3)*atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (7*a^2*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(5*(a + b*x^3)^(2/3)) - (2*2^(1/3)*a^(5/3)*log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3)))/(3*b^(1/3)) + (2*2^(1/3)*a^(5/3)*log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*b^(1/3)) - (4*2^(1/3)*a^(5/3)*log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*b^(1/3)) + (2^(1/3)*a^(5/3)*log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*b^(1/3))]
    @test_int [(a + b*x^3)^(4/3)/(a - b*x^3), x, 21, (-(1/2))*x*(a + b*x^3)^(1/3) - (2*2^(1/3)*a^(2/3)*atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (2^(1/3)*a^(2/3)*atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(1/3)) - (a*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)) - (2^(1/3)*a^(2/3)*log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3)))/(3*b^(1/3)) + (2^(1/3)*a^(2/3)*log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*b^(1/3)) - (2*2^(1/3)*a^(2/3)*log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*b^(1/3)) + (a^(2/3)*log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*2^(2/3)*b^(1/3))]
    @test_int [(a + b*x^3)^(1/3)/(a - b*x^3), x, 14, -((2^(1/3)*atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*a^(1/3)*b^(1/3))) - atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3)*a^(1/3)*b^(1/3)) - log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3))/(3*2^(2/3)*a^(1/3)*b^(1/3)) + log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(3*2^(2/3)*a^(1/3)*b^(1/3)) - (2^(1/3)*log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3)))/(3*a^(1/3)*b^(1/3)) + log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(6*2^(2/3)*a^(1/3)*b^(1/3))]
    @test_int [1/((a + b*x^3)^(2/3)*(a - b*x^3)), x, 17, -(atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(2^(2/3)*sqrt(3)*a^(4/3)*b^(1/3))) - atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(2*2^(2/3)*sqrt(3)*a^(4/3)*b^(1/3)) + (x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*a*(a + b*x^3)^(2/3)) - log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3))/(6*2^(2/3)*a^(4/3)*b^(1/3)) + log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(6*2^(2/3)*a^(4/3)*b^(1/3)) - log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(3*2^(2/3)*a^(4/3)*b^(1/3)) + log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(12*2^(2/3)*a^(4/3)*b^(1/3))]
    @test_int [1/((a + b*x^3)^(5/3)*(a - b*x^3)), x, 21, x/(4*a^2*(a + b*x^3)^(2/3)) - atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(2*2^(2/3)*sqrt(3)*a^(7/3)*b^(1/3)) - atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(4*2^(2/3)*sqrt(3)*a^(7/3)*b^(1/3)) + (x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*a^2*(a + b*x^3)^(2/3)) - log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3))/(12*2^(2/3)*a^(7/3)*b^(1/3)) + log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(12*2^(2/3)*a^(7/3)*b^(1/3)) - log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(6*2^(2/3)*a^(7/3)*b^(1/3)) + log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(24*2^(2/3)*a^(7/3)*b^(1/3))]
    @test_int [1/((a + b*x^3)^(8/3)*(a - b*x^3)), x, 22, x/(10*a^2*(a + b*x^3)^(5/3)) + (13*x)/(40*a^3*(a + b*x^3)^(2/3)) - atan((1 - (2*2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(4*2^(2/3)*sqrt(3)*a^(10/3)*b^(1/3)) - atan((1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/sqrt(3))/(8*2^(2/3)*sqrt(3)*a^(10/3)*b^(1/3)) + (9*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(20*a^3*(a + b*x^3)^(2/3)) - log(2^(2/3) - (a^(1/3) + b^(1/3)*x)/(a + b*x^3)^(1/3))/(24*2^(2/3)*a^(10/3)*b^(1/3)) + log(1 + (2^(2/3)*(a^(1/3) + b^(1/3)*x)^2)/(a + b*x^3)^(2/3) - (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(24*2^(2/3)*a^(10/3)*b^(1/3)) - log(1 + (2^(1/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(12*2^(2/3)*a^(10/3)*b^(1/3)) + log(2*2^(1/3) + (a^(1/3) + b^(1/3)*x)^2/(a + b*x^3)^(2/3) + (2^(2/3)*(a^(1/3) + b^(1/3)*x))/(a + b*x^3)^(1/3))/(48*2^(2/3)*a^(10/3)*b^(1/3))]

    @test_int [(a + b*x^3)^(2/3)*(a - b*x^3)^2, x, 4, (38/81)*a^2*x*(a + b*x^3)^(2/3) - (8/27)*a*x*(a + b*x^3)^(5/3) - (1/9)*x*(a - b*x^3)*(a + b*x^3)^(5/3) + (76*a^3*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(81*sqrt(3)*b^(1/3)) - (38*a^3*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(81*b^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(1/3), x, 3, (-(13/18))*a*x*(a + b*x^3)^(2/3) - (1/6)*x*(a - b*x^3)*(a + b*x^3)^(2/3) + (17*a^2*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*b^(1/3)) - (17*a^2*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*b^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(4/3), x, 3, (2*x*(a - b*x^3))/(a + b*x^3)^(1/3) + (7/3)*x*(a + b*x^3)^(2/3) - (10*a*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(1/3)) + (5*a*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(3*b^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(7/3), x, 3, (x*(a - b*x^3))/(2*(a + b*x^3)^(4/3)) - x/(2*(a + b*x^3)^(1/3)) + atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3))/(sqrt(3)*b^(1/3)) - log((-b^(1/3))*x + (a + b*x^3)^(1/3))/(2*b^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(10/3), x, 3, (x*(a - b*x^3)^2)/(7*a*(a + b*x^3)^(7/3)) + (3*x*(a - b*x^3))/(14*a*(a + b*x^3)^(4/3)) + (9*x)/(14*a*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(13/3), x, 4, (x*(a - b*x^3)^3)/(20*a^2*(a + b*x^3)^(10/3)) + (19*x*(a - b*x^3)^2)/(140*a^2*(a + b*x^3)^(7/3)) + (57*x*(a - b*x^3))/(280*a^2*(a + b*x^3)^(4/3)) + (171*x)/(280*a^2*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(16/3), x, 5, (2*x*(a - b*x^3))/(13*(a + b*x^3)^(13/3)) + (8*x)/(65*(a + b*x^3)^(10/3)) + (47*x)/(455*a*(a + b*x^3)^(7/3)) + (141*x)/(910*a^2*(a + b*x^3)^(4/3)) + (423*x)/(910*a^3*(a + b*x^3)^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(19/3), x, 6, (x*(a - b*x^3))/(8*(a + b*x^3)^(16/3)) + (11*x)/(104*(a + b*x^3)^(13/3)) + x/(13*a*(a + b*x^3)^(10/3)) + (9*x)/(91*a^2*(a + b*x^3)^(7/3)) + (27*x)/(182*a^3*(a + b*x^3)^(4/3)) + (81*x)/(182*a^4*(a + b*x^3)^(1/3))]

    @test_int [(a + b*x^3)^(4/3)*(a - b*x^3)^2, x, 4, (-(9/44))*a*x*(a + b*x^3)^(7/3) - (1/11)*x*(a - b*x^3)*(a + b*x^3)^(7/3) + (57*a^3*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), 1/3, 4/3, -((b*x^3)/a)))/(44*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)*(a - b*x^3)^2, x, 4, (-(3/8))*a*x*(a + b*x^3)^(4/3) - (1/8)*x*(a - b*x^3)*(a + b*x^3)^(4/3) + (3*a^2*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), 1/3, 4/3, -((b*x^3)/a)))/(2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(2/3), x, 4, (-(6/5))*a*x*(a + b*x^3)^(1/3) - (1/5)*x*(a - b*x^3)*(a + b*x^3)^(1/3) + (12*a^2*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(5*(a + b*x^3)^(2/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(5/3), x, 4, (x*(a - b*x^3))/(a + b*x^3)^(2/3) + (3*b*x^4*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(2/3, 4/3, 7/3, -((b*x^3)/a)))/(4*(a + b*x^3)^(2/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(8/3), x, 4, (2*x*(a - b*x^3))/(5*(a + b*x^3)^(5/3)) + (3*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(5*(a + b*x^3)^(2/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(11/3), x, 4, (x*(a - b*x^3))/(4*(a + b*x^3)^(8/3)) + (3*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 8/3, 4/3, -((b*x^3)/a)))/(4*a*(a + b*x^3)^(2/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(14/3), x, 4, (2*x*(a - b*x^3))/(11*(a + b*x^3)^(11/3)) + (3*x)/(22*(a + b*x^3)^(8/3)) + (15*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 8/3, 4/3, -((b*x^3)/a)))/(22*a^2*(a + b*x^3)^(2/3))]
    @test_int [(a - b*x^3)^2/(a + b*x^3)^(17/3), x, 4, (x*(a - b*x^3))/(7*(a + b*x^3)^(14/3)) + (9*x)/(77*(a + b*x^3)^(11/3)) + (57*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 11/3, 4/3, -((b*x^3)/a)))/(77*a^3*(a + b*x^3)^(2/3))]

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

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

    @test_int [(a + b*x^3)^(5/3)*(c + d*x^3), x, 4, (5*a*(9*b*c - a*d)*x*(a + b*x^3)^(2/3))/(162*b) + ((9*b*c - a*d)*x*(a + b*x^3)^(5/3))/(54*b) + (d*x*(a + b*x^3)^(8/3))/(9*b) + (5*a^2*(9*b*c - a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(81*sqrt(3)*b^(4/3)) - (5*a^2*(9*b*c - a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(162*b^(4/3))]
    @test_int [(a + b*x^3)^(2/3)*(c + d*x^3), x, 3, ((6*b*c - a*d)*x*(a + b*x^3)^(2/3))/(18*b) + (d*x*(a + b*x^3)^(5/3))/(6*b) + (a*(6*b*c - a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*b^(4/3)) - (a*(6*b*c - a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*b^(4/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(1/3), x, 2, (d*x*(a + b*x^3)^(2/3))/(3*b) + ((3*b*c - a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(4/3)) - ((3*b*c - a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(6*b^(4/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(4/3), x, 2, ((b*c - a*d)*x)/(a*b*(a + b*x^3)^(1/3)) + (d*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(4/3)) - (d*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(4/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(7/3), x, 2, (3*c*x)/(4*a^2*(a + b*x^3)^(1/3)) + (x*(c + d*x^3))/(4*a*(a + b*x^3)^(4/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(10/3), x, 3, ((b*c - a*d)*x)/(7*a*b*(a + b*x^3)^(7/3)) + ((6*b*c + a*d)*x)/(28*a^2*b*(a + b*x^3)^(4/3)) + (3*(6*b*c + a*d)*x)/(28*a^3*b*(a + b*x^3)^(1/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(13/3), x, 4, ((b*c - a*d)*x)/(10*a*b*(a + b*x^3)^(10/3)) + ((9*b*c + a*d)*x)/(70*a^2*b*(a + b*x^3)^(7/3)) + (3*(9*b*c + a*d)*x)/(140*a^3*b*(a + b*x^3)^(4/3)) + (9*(9*b*c + a*d)*x)/(140*a^4*b*(a + b*x^3)^(1/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(16/3), x, 5, ((b*c - a*d)*x)/(13*a*b*(a + b*x^3)^(13/3)) + ((12*b*c + a*d)*x)/(130*a^2*b*(a + b*x^3)^(10/3)) + (9*(12*b*c + a*d)*x)/(910*a^3*b*(a + b*x^3)^(7/3)) + (27*(12*b*c + a*d)*x)/(1820*a^4*b*(a + b*x^3)^(4/3)) + (81*(12*b*c + a*d)*x)/(1820*a^5*b*(a + b*x^3)^(1/3))]

    @test_int [(a + b*x^3)^(7/3)*(c + d*x^3), x, 3, (d*x*(a + b*x^3)^(10/3))/(11*b) + (a^2*(11*b*c - a*d)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(7/3), 1/3, 4/3, -((b*x^3)/a)))/(11*b*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(4/3)*(c + d*x^3), x, 3, (d*x*(a + b*x^3)^(7/3))/(8*b) + (a*(8*b*c - a*d)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), 1/3, 4/3, -((b*x^3)/a)))/(8*b*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)*(c + d*x^3), x, 3, (d*x*(a + b*x^3)^(4/3))/(5*b) + ((5*b*c - a*d)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), 1/3, 4/3, -((b*x^3)/a)))/(5*b*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(2/3), x, 3, (d*x*(a + b*x^3)^(1/3))/(2*b) + ((2*b*c - a*d)*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))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(5/3), x, 3, ((b*c - a*d)*x)/(2*a*b*(a + b*x^3)^(2/3)) + ((b*c + a*d)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*a*b*(a + b*x^3)^(2/3))]
    @test_int [(c + d*x^3)/(a + b*x^3)^(8/3), x, 3, ((b*c - a*d)*x)/(5*a*b*(a + b*x^3)^(5/3)) + ((4*b*c + a*d)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 5/3, 4/3, -((b*x^3)/a)))/(5*a^2*b*(a + b*x^3)^(2/3))]

    @test_int [(a + b*x^3)^(5/3)*(c + d*x^3)^2, x, 5, (5*a*(27*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*(a + b*x^3)^(2/3))/(486*b^2) + ((27*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*(a + b*x^3)^(5/3))/(162*b^2) + (d*(15*b*c - 4*a*d)*x*(a + b*x^3)^(8/3))/(108*b^2) + (d*x*(a + b*x^3)^(8/3)*(c + d*x^3))/(12*b) + (5*a^2*(27*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(243*sqrt(3)*b^(7/3)) - (5*a^2*(27*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(486*b^(7/3))]
    @test_int [(a + b*x^3)^(2/3)*(c + d*x^3)^2, x, 4, ((27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*x*(a + b*x^3)^(2/3))/(81*b^2) + (2*d*(3*b*c - a*d)*x*(a + b*x^3)^(5/3))/(27*b^2) + (d*x*(a + b*x^3)^(5/3)*(c + d*x^3))/(9*b) + (2*a*(27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(81*sqrt(3)*b^(7/3)) - (a*(27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(81*b^(7/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(1/3), x, 3, (d*(9*b*c - 4*a*d)*x*(a + b*x^3)^(2/3))/(18*b^2) + (d*x*(a + b*x^3)^(2/3)*(c + d*x^3))/(6*b) + ((9*b^2*c^2 - 6*a*b*c*d + 2*a^2*d^2)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*b^(7/3)) - ((9*b^2*c^2 - 6*a*b*c*d + 2*a^2*d^2)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*b^(7/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(4/3), x, 3, -((d*(3*b*c - 4*a*d)*x*(a + b*x^3)^(2/3))/(3*a*b^2)) + ((b*c - a*d)*x*(c + d*x^3))/(a*b*(a + b*x^3)^(1/3)) + (2*d*(3*b*c - 2*a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*b^(7/3)) - (d*(3*b*c - 2*a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(3*b^(7/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(7/3), x, 3, ((b*c - a*d)*(3*b*c + 4*a*d)*x)/(4*a^2*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(4*a*b*(a + b*x^3)^(4/3)) + (d^2*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*b^(7/3)) - (d^2*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*b^(7/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(10/3), x, 3, (9*c^2*x)/(14*a^3*(a + b*x^3)^(1/3)) + (3*c*x*(c + d*x^3))/(14*a^2*(a + b*x^3)^(4/3)) + (x*(c + d*x^3)^2)/(7*a*(a + b*x^3)^(7/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(13/3), x, 4, (9*c^2*(9*b*c - 10*a*d)*x)/(140*a^4*(b*c - a*d)*(a + b*x^3)^(1/3)) + (3*c*(9*b*c - 10*a*d)*x*(c + d*x^3))/(140*a^3*(b*c - a*d)*(a + b*x^3)^(4/3)) + ((9*b*c - 10*a*d)*x*(c + d*x^3)^2)/(70*a^2*(b*c - a*d)*(a + b*x^3)^(7/3)) + (b*x*(c + d*x^3)^3)/(10*a*(b*c - a*d)*(a + b*x^3)^(10/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(16/3), x, 5, (2*(b*c - a*d)*(3*b*c + a*d)*x)/(65*a^2*b^2*(a + b*x^3)^(10/3)) + ((54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(455*a^3*b^2*(a + b*x^3)^(7/3)) + (3*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^4*b^2*(a + b*x^3)^(4/3)) + (9*(54*b^2*c^2 + 9*a*b*c*d + 2*a^2*d^2)*x)/(910*a^5*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(13*a*b*(a + b*x^3)^(13/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(19/3), x, 6, ((b*c - a*d)*(15*b*c + 4*a*d)*x)/(208*a^2*b^2*(a + b*x^3)^(13/3)) + ((45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(520*a^3*b^2*(a + b*x^3)^(10/3)) + (9*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(3640*a^4*b^2*(a + b*x^3)^(7/3)) + (27*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(7280*a^5*b^2*(a + b*x^3)^(4/3)) + (81*(45*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x)/(7280*a^6*b^2*(a + b*x^3)^(1/3)) + ((b*c - a*d)*x*(c + d*x^3))/(16*a*b*(a + b*x^3)^(16/3))]

    @test_int [(a + b*x^3)^(7/3)*(c + d*x^3)^2, x, 4, (d*(17*b*c - 4*a*d)*x*(a + b*x^3)^(10/3))/(154*b^2) + (d*x*(a + b*x^3)^(10/3)*(c + d*x^3))/(14*b) + (a^2*(77*b^2*c^2 - 14*a*b*c*d + 2*a^2*d^2)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(7/3), 1/3, 4/3, -((b*x^3)/a)))/(77*b^2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(4/3)*(c + d*x^3)^2, x, 4, (d*(7*b*c - 2*a*d)*x*(a + b*x^3)^(7/3))/(44*b^2) + (d*x*(a + b*x^3)^(7/3)*(c + d*x^3))/(11*b) + (a*(44*b^2*c^2 - 11*a*b*c*d + 2*a^2*d^2)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(4/3), 1/3, 4/3, -((b*x^3)/a)))/(44*b^2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)*(c + d*x^3)^2, x, 4, (d*(11*b*c - 4*a*d)*x*(a + b*x^3)^(4/3))/(40*b^2) + (d*x*(a + b*x^3)^(4/3)*(c + d*x^3))/(8*b) + ((10*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x*(a + b*x^3)^(1/3)*HypergeometricFunctions._₂F₁(-(1/3), 1/3, 4/3, -((b*x^3)/a)))/(10*b^2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(2/3), x, 4, (2*d*(2*b*c - a*d)*x*(a + b*x^3)^(1/3))/(5*b^2) + (d*x*(a + b*x^3)^(1/3)*(c + d*x^3))/(5*b) + ((5*b^2*c^2 - 5*a*b*c*d + 2*a^2*d^2)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(5*b^2*(a + b*x^3)^(2/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(5/3), x, 4, -((d*(b*c - 2*a*d)*x*(a + b*x^3)^(1/3))/(2*a*b^2)) + ((b*c - a*d)*x*(c + d*x^3))/(2*a*b*(a + b*x^3)^(2/3)) + ((b^2*c^2 + 2*a*b*c*d - 2*a^2*d^2)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(2*a*b^2*(a + b*x^3)^(2/3))]
    @test_int [(c + d*x^3)^2/(a + b*x^3)^(8/3), x, 4, (2*(c^2/a^2 - d^2/b^2)*x)/(5*(a + b*x^3)^(2/3)) + ((b*c - a*d)*x*(c + d*x^3))/(5*a*b*(a + b*x^3)^(5/3)) + ((2*b^2*c^2 + a*b*c*d + 2*a^2*d^2)*x*(1 + (b*x^3)/a)^(2/3)*HypergeometricFunctions._₂F₁(1/3, 2/3, 4/3, -((b*x^3)/a)))/(5*a^2*b^2*(a + b*x^3)^(2/3))]

    @test_int [(a + b*x^3)^3/(c + d*x^3)^(13/3), x, 4, (x*(a + b*x^3)^3)/(10*c*(c + d*x^3)^(10/3)) + (9*a*x*(a + b*x^3)^2)/(70*c^2*(c + d*x^3)^(7/3)) + (27*a^2*x*(a + b*x^3))/(140*c^3*(c + d*x^3)^(4/3)) + (81*a^3*x)/(140*c^4*(c + d*x^3)^(1/3))]

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

    @test_int [(a + b*x^3)^(8/3)/(c + d*x^3), x, 5, -((b*(6*b*c - 11*a*d)*x*(a + b*x^3)^(2/3))/(18*d^2)) + (b*x*(a + b*x^3)^(5/3))/(6*d) + (b^(2/3)*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*d^3) - ((b*c - a*d)^(8/3)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*d^3) - ((b*c - a*d)^(8/3)*log(c + d*x^3))/(6*c^(2/3)*d^3) + ((b*c - a*d)^(8/3)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*d^3) - (b^(2/3)*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*d^3)]
    @test_int [(a + b*x^3)^(5/3)/(c + d*x^3), x, 4, (b*x*(a + b*x^3)^(2/3))/(3*d) - (b^(2/3)*(3*b*c - 5*a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*d^2) + ((b*c - a*d)^(5/3)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*d^2) + ((b*c - a*d)^(5/3)*log(c + d*x^3))/(6*c^(2/3)*d^2) - ((b*c - a*d)^(5/3)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*d^2) + (b^(2/3)*(3*b*c - 5*a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(6*d^2)]
    @test_int [(a + b*x^3)^(2/3)/(c + d*x^3), x, 3, (b^(2/3)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d) - ((b*c - a*d)^(2/3)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*d) - ((b*c - a*d)^(2/3)*log(c + d*x^3))/(6*c^(2/3)*d) + ((b*c - a*d)^(2/3)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*d) - (b^(2/3)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*d)]
    @test_int [1/((a + b*x^3)^(1/3)*(c + d*x^3)), x, 1, atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3))/(sqrt(3)*c^(2/3)*(b*c - a*d)^(1/3)) + log(c + d*x^3)/(6*c^(2/3)*(b*c - a*d)^(1/3)) - log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3))/(2*c^(2/3)*(b*c - a*d)^(1/3))]
    @test_int [1/((a + b*x^3)^(4/3)*(c + d*x^3)), x, 2, (b*x)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)) - (d*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*(b*c - a*d)^(4/3)) - (d*log(c + d*x^3))/(6*c^(2/3)*(b*c - a*d)^(4/3)) + (d*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*(b*c - a*d)^(4/3))]
    @test_int [1/((a + b*x^3)^(7/3)*(c + d*x^3)), x, 4, (b*x)/(4*a*(b*c - a*d)*(a + b*x^3)^(4/3)) + (b*(3*b*c - 7*a*d)*x)/(4*a^2*(b*c - a*d)^2*(a + b*x^3)^(1/3)) + (d^2*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*(b*c - a*d)^(7/3)) + (d^2*log(c + d*x^3))/(6*c^(2/3)*(b*c - a*d)^(7/3)) - (d^2*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*(b*c - a*d)^(7/3))]
    @test_int [1/((a + b*x^3)^(10/3)*(c + d*x^3)), x, 5, (b*x)/(7*a*(b*c - a*d)*(a + b*x^3)^(7/3)) + (b*(6*b*c - 13*a*d)*x)/(28*a^2*(b*c - a*d)^2*(a + b*x^3)^(4/3)) + (b*(18*b^2*c^2 - 57*a*b*c*d + 67*a^2*d^2)*x)/(28*a^3*(b*c - a*d)^3*(a + b*x^3)^(1/3)) - (d^3*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(sqrt(3)*c^(2/3)*(b*c - a*d)^(10/3)) - (d^3*log(c + d*x^3))/(6*c^(2/3)*(b*c - a*d)^(10/3)) + (d^3*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(2*c^(2/3)*(b*c - a*d)^(10/3))]

    @test_int [(a + b*x^3)^(4/3)/(c + d*x^3), x, 2, (a*x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(4/3), 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)/(c + d*x^3), x, 2, (x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(1/3), 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c*(1 + (b*x^3)/a)^(1/3))]
    @test_int [1/((a + b*x^3)^(2/3)*(c + d*x^3)), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(5/3)*(c + d*x^3)), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 5/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a*c*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(8/3)*(c + d*x^3)), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 8/3, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a^2*c*(a + b*x^3)^(2/3))]

    @test_int [(a + b*x^3)^(8/3)/(c + d*x^3)^2, x, 5, (b*(2*b*c - a*d)*x*(a + b*x^3)^(2/3))/(3*c*d^2) - ((b*c - a*d)*x*(a + b*x^3)^(5/3))/(3*c*d*(c + d*x^3)) - (2*b^(5/3)*(3*b*c - 4*a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*d^3) + (2*(b*c - a*d)^(5/3)*(3*b*c + a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*d^3) + ((b*c - a*d)^(5/3)*(3*b*c + a*d)*log(c + d*x^3))/(9*c^(5/3)*d^3) - ((b*c - a*d)^(5/3)*(3*b*c + a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(3*c^(5/3)*d^3) + (b^(5/3)*(3*b*c - 4*a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(3*d^3)]
    @test_int [(a + b*x^3)^(5/3)/(c + d*x^3)^2, x, 4, -(((b*c - a*d)*x*(a + b*x^3)^(2/3))/(3*c*d*(c + d*x^3))) + (b^(5/3)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^2) - ((b*c - a*d)^(2/3)*(3*b*c + 2*a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*d^2) - ((b*c - a*d)^(2/3)*(3*b*c + 2*a*d)*log(c + d*x^3))/(18*c^(5/3)*d^2) + ((b*c - a*d)^(2/3)*(3*b*c + 2*a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(6*c^(5/3)*d^2) - (b^(5/3)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*d^2)]
    @test_int [(a + b*x^3)^(2/3)/(c + d*x^3)^2, x, 2, (x*(a + b*x^3)^(2/3))/(3*c*(c + d*x^3)) + (2*a*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^(1/3)) + (a*log(c + d*x^3))/(9*c^(5/3)*(b*c - a*d)^(1/3)) - (a*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(3*c^(5/3)*(b*c - a*d)^(1/3))]
    @test_int [1/((a + b*x^3)^(1/3)*(c + d*x^3)^2), x, 2, -((d*x*(a + b*x^3)^(2/3))/(3*c*(b*c - a*d)*(c + d*x^3))) + ((3*b*c - 2*a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^(4/3)) + ((3*b*c - 2*a*d)*log(c + d*x^3))/(18*c^(5/3)*(b*c - a*d)^(4/3)) - ((3*b*c - 2*a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(6*c^(5/3)*(b*c - a*d)^(4/3))]
    @test_int [1/((a + b*x^3)^(4/3)*(c + d*x^3)^2), x, 4, (b*(3*b*c + a*d)*x)/(3*a*c*(b*c - a*d)^2*(a + b*x^3)^(1/3)) - (d*x)/(3*c*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3)) - (2*d*(3*b*c - a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^(7/3)) - (d*(3*b*c - a*d)*log(c + d*x^3))/(9*c^(5/3)*(b*c - a*d)^(7/3)) + (d*(3*b*c - a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(3*c^(5/3)*(b*c - a*d)^(7/3))]
    @test_int [1/((a + b*x^3)^(7/3)*(c + d*x^3)^2), x, 5, (b*(3*b*c + 4*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^3)^(4/3)) + (b*(9*b^2*c^2 - 33*a*b*c*d - 4*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*(a + b*x^3)^(1/3)) - (d*x)/(3*c*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)) + (d^2*(9*b*c - 2*a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(3*sqrt(3)*c^(5/3)*(b*c - a*d)^(10/3)) + (d^2*(9*b*c - 2*a*d)*log(c + d*x^3))/(18*c^(5/3)*(b*c - a*d)^(10/3)) - (d^2*(9*b*c - 2*a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(6*c^(5/3)*(b*c - a*d)^(10/3))]

    @test_int [(a + b*x^3)^(4/3)/(c + d*x^3)^2, x, 2, (a*x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(4/3), 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)/(c + d*x^3)^2, x, 2, (x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(1/3), 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^2*(1 + (b*x^3)/a)^(1/3))]
    @test_int [1/((a + b*x^3)^(2/3)*(c + d*x^3)^2), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^2*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(5/3)*(c + d*x^3)^2), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 5/3, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a*c^2*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(8/3)*(c + d*x^3)^2), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 8/3, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a^2*c^2*(a + b*x^3)^(2/3))]

    @test_int [(a + b*x^3)^(14/3)/(c + d*x^3)^3, x, 7, -((b*(2*b*c - a*d)*(18*b^2*c^2 - 18*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/(18*c^2*d^4)) + (b*(18*b^2*c^2 - 10*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(5/3))/(18*c^2*d^3) - ((b*c - a*d)*x*(a + b*x^3)^(11/3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(12*b*c + 5*a*d)*x*(a + b*x^3)^(8/3))/(18*c^2*d^2*(c + d*x^3)) + (b^(8/3)*(54*b^2*c^2 - 126*a*b*c*d + 77*a^2*d^2)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(9*sqrt(3)*d^5) - ((b*c - a*d)^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*d^5) - ((b*c - a*d)^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*d^5) + ((b*c - a*d)^(8/3)*(54*b^2*c^2 + 18*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*d^5) - (b^(8/3)*(54*b^2*c^2 - 126*a*b*c*d + 77*a^2*d^2)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(18*d^5)]
    @test_int [(a + b*x^3)^(11/3)/(c + d*x^3)^3, x, 6, (b*(18*b^2*c^2 - 7*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/(18*c^2*d^3) - ((b*c - a*d)*x*(a + b*x^3)^(8/3))/(6*c*d*(c + d*x^3)^2) - ((b*c - a*d)*(9*b*c + 5*a*d)*x*(a + b*x^3)^(5/3))/(18*c^2*d^2*(c + d*x^3)) - (b^(8/3)*(9*b*c - 11*a*d)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(3*sqrt(3)*d^4) + ((b*c - a*d)^(5/3)*(27*b^2*c^2 + 12*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*d^4) + ((b*c - a*d)^(5/3)*(27*b^2*c^2 + 12*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*d^4) - ((b*c - a*d)^(5/3)*(27*b^2*c^2 + 12*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*d^4) + (b^(8/3)*(9*b*c - 11*a*d)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(6*d^4)]
    @test_int [(a + b*x^3)^(8/3)/(c + d*x^3)^3, x, 5, -(((b*c - a*d)*x*(a + b*x^3)^(5/3))/(6*c*d*(c + d*x^3)^2)) - ((b*c - a*d)*(6*b*c + 5*a*d)*x*(a + b*x^3)^(2/3))/(18*c^2*d^2*(c + d*x^3)) + (b^(8/3)*atan((1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/sqrt(3)))/(sqrt(3)*d^3) - ((b*c - a*d)^(2/3)*(9*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*d^3) - ((b*c - a*d)^(2/3)*(9*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*d^3) + ((b*c - a*d)^(2/3)*(9*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*d^3) - (b^(8/3)*log((-b^(1/3))*x + (a + b*x^3)^(1/3)))/(2*d^3)]
    @test_int [(a + b*x^3)^(5/3)/(c + d*x^3)^3, x, 3, (x*(a + b*x^3)^(5/3))/(6*c*(c + d*x^3)^2) + (5*a*x*(a + b*x^3)^(2/3))/(18*c^2*(c + d*x^3)) + (5*a^2*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*(b*c - a*d)^(1/3)) + (5*a^2*log(c + d*x^3))/(54*c^(8/3)*(b*c - a*d)^(1/3)) - (5*a^2*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*(b*c - a*d)^(1/3))]
    @test_int [(a + b*x^3)^(2/3)/(c + d*x^3)^3, x, 3, -((d*x*(a + b*x^3)^(5/3))/(6*c*(b*c - a*d)*(c + d*x^3)^2)) + ((6*b*c - 5*a*d)*x*(a + b*x^3)^(2/3))/(18*c^2*(b*c - a*d)*(c + d*x^3)) + (a*(6*b*c - 5*a*d)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*(b*c - a*d)^(4/3)) + (a*(6*b*c - 5*a*d)*log(c + d*x^3))/(54*c^(8/3)*(b*c - a*d)^(4/3)) - (a*(6*b*c - 5*a*d)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*(b*c - a*d)^(4/3))]
    @test_int [1/((a + b*x^3)^(1/3)*(c + d*x^3)^3), x, 4, -((d*x*(a + b*x^3)^(2/3))/(6*c*(b*c - a*d)*(c + d*x^3)^2)) - (d*(9*b*c - 5*a*d)*x*(a + b*x^3)^(2/3))/(18*c^2*(b*c - a*d)^2*(c + d*x^3)) + ((9*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*(b*c - a*d)^(7/3)) + ((9*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*(b*c - a*d)^(7/3)) - ((9*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*(b*c - a*d)^(7/3))]
    @test_int [1/((a + b*x^3)^(4/3)*(c + d*x^3)^3), x, 5, -((d*x)/(6*c*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3)^2)) + (b*(6*b*c + a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^3)^(1/3)*(c + d*x^3)) + (d*(18*b^2*c^2 + 15*a*b*c*d - 5*a^2*d^2)*x*(a + b*x^3)^(2/3))/(18*a*c^2*(b*c - a*d)^3*(c + d*x^3)) - (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*(b*c - a*d)^(10/3)) - (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*(b*c - a*d)^(10/3)) + (d*(27*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*(b*c - a*d)^(10/3))]
    @test_int [1/((a + b*x^3)^(7/3)*(c + d*x^3)^3), x, 6, -((d*x)/(6*c*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)^2)) + (b*(3*b*c + 2*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^3)^(4/3)*(c + d*x^3)) + (b*(9*b^2*c^2 - 42*a*b*c*d - 2*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*(a + b*x^3)^(1/3)*(c + d*x^3)) + (d*(27*b^3*c^3 - 135*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 10*a^3*d^3)*x*(a + b*x^3)^(2/3))/(36*a^2*c^2*(b*c - a*d)^4*(c + d*x^3)) + (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^2)*atan((1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/sqrt(3)))/(9*sqrt(3)*c^(8/3)*(b*c - a*d)^(13/3)) + (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^2)*log(c + d*x^3))/(54*c^(8/3)*(b*c - a*d)^(13/3)) - (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^2)*log(((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)))/(18*c^(8/3)*(b*c - a*d)^(13/3))]

    @test_int [(a + b*x^3)^(4/3)/(c + d*x^3)^3, x, 2, (a*x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(4/3), 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^3*(1 + (b*x^3)/a)^(1/3))]
    @test_int [(a + b*x^3)^(1/3)/(c + d*x^3)^3, x, 2, (x*(a + b*x^3)^(1/3)*AppellF1(1/3, -(1/3), 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^3*(1 + (b*x^3)/a)^(1/3))]
    @test_int [1/((a + b*x^3)^(2/3)*(c + d*x^3)^3), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 2/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(c^3*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(5/3)*(c + d*x^3)^3), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 5/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a*c^3*(a + b*x^3)^(2/3))]
    @test_int [1/((a + b*x^3)^(8/3)*(c + d*x^3)^3), x, 2, (x*(1 + (b*x^3)/a)^(2/3)*AppellF1(1/3, 8/3, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/(a^2*c^3*(a + b*x^3)^(2/3))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^3)^(p/4)*(c+d*x^3)^(q/12)*with*3*(p+q+1)+1=0 =#

    @test_int [(a + b*x^3)^(7/4)/(c + d*x^3)^(37/12), x, 3, (4*x*(a + b*x^3)^(7/4))/(25*c*(c + d*x^3)^(25/12)) + (84*a*x*(a + b*x^3)^(3/4))/(325*c^2*(c + d*x^3)^(13/12)) + (189*a^2*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/4)*HypergeometricFunctions._₂F₁(1/4, 1/3, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(325*c^3*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12))]
    @test_int [(a + b*x^3)^(5/4)/(c + d*x^3)^(31/12), x, 3, (4*x*(a + b*x^3)^(5/4))/(19*c*(c + d*x^3)^(19/12)) + (60*a*x*(a + b*x^3)^(1/4))/(133*c^2*(c + d*x^3)^(7/12)) + (45*a^2*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*HypergeometricFunctions._₂F₁(1/3, 3/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(133*c^3*(a + b*x^3)^(3/4))]
    @test_int [(a + b*x^3)^(3/4)/(c + d*x^3)^(25/12), x, 2, (4*x*(a + b*x^3)^(3/4))/(13*c*(c + d*x^3)^(13/12)) + (9*a*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/4)*HypergeometricFunctions._₂F₁(1/4, 1/3, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(13*c^2*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12))]
    @test_int [(a + b*x^3)^(1/4)/(c + d*x^3)^(19/12), x, 2, (4*x*(a + b*x^3)^(1/4))/(7*c*(c + d*x^3)^(7/12)) + (3*a*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*HypergeometricFunctions._₂F₁(1/3, 3/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(7*c^2*(a + b*x^3)^(3/4))]
    @test_int [1/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12)), x, 1, (x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/4)*HypergeometricFunctions._₂F₁(1/4, 1/3, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(c*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12))]
    @test_int [1/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12)), x, 1, (x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*HypergeometricFunctions._₂F₁(1/3, 3/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(c*(a + b*x^3)^(3/4))]
    @test_int [1/((a + b*x^3)^(5/4)*(c + d*x^3)^(1/12)), x, 1, (x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*HypergeometricFunctions._₂F₁(1/3, 5/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(c*(a + b*x^3)^(5/4))]
    @test_int [(c + d*x^3)^(5/12)/(a + b*x^3)^(7/4), x, 2, (4*x*(c + d*x^3)^(5/12))/(9*a*(a + b*x^3)^(3/4)) + (5*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*HypergeometricFunctions._₂F₁(1/3, 3/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(9*a*(a + b*x^3)^(3/4))]
    @test_int [(c + d*x^3)^(11/12)/(a + b*x^3)^(9/4), x, 2, (4*x*(c + d*x^3)^(11/12))/(15*a*(a + b*x^3)^(5/4)) + (11*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*HypergeometricFunctions._₂F₁(1/3, 5/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(15*a*(a + b*x^3)^(5/4))]
    @test_int [(c + d*x^3)^(17/12)/(a + b*x^3)^(11/4), x, 3, (68*c*x*(c + d*x^3)^(5/12))/(189*a^2*(a + b*x^3)^(3/4)) + (4*x*(c + d*x^3)^(17/12))/(21*a*(a + b*x^3)^(7/4)) + (85*c*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(3/4)*(c + d*x^3)^(5/12)*HypergeometricFunctions._₂F₁(1/3, 3/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(189*a^2*(a + b*x^3)^(3/4))]
    @test_int [(c + d*x^3)^(23/12)/(a + b*x^3)^(13/4), x, 3, (92*c*x*(c + d*x^3)^(11/12))/(405*a^2*(a + b*x^3)^(5/4)) + (4*x*(c + d*x^3)^(23/12))/(27*a*(a + b*x^3)^(9/4)) + (253*c*x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*HypergeometricFunctions._₂F₁(1/3, 5/4, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)))))/(405*a^2*(a + b*x^3)^(5/4))]

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

    @test_int [(a + b*x^3)^m*(c + d*x^3)^p, x, 3, (x*(a + b*x^3)^m*(c + d*x^3)^p*AppellF1(1/3, -m, -p, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (b*x^3)/a)^m*(1 + (d*x^3)/c)^p)]

    @test_int [(a + b*x^3)^2*(c + d*x^3)^q, x, 4, If(13>=8, -((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(4 + 3*q)*(7 + 3*q))) + (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^(1 + q)*HypergeometricFunctions._₂F₁(1, 4/3 + q, 4/3, -((d*x^3)/c)))/(c*d^2*(4 + 3*q)*(7 + 3*q)), -((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(28 + 33*q + 9*q^2))) + (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^(1 + q)*HypergeometricFunctions._₂F₁(1, 4/3 + q, 4/3, -((d*x^3)/c)))/(c*d^2*(28 + 33*q + 9*q^2))), If(13>=8, -((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(4 + 3*q)*(7 + 3*q))) + (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^q*HypergeometricFunctions._₂F₁(1/3, -q, 4/3, -((d*x^3)/c)))/((1 + (d*x^3)/c)^q*(d^2*(4 + 3*q)*(7 + 3*q))), -((b*(4*b*c - a*d*(10 + 3*q))*x*(c + d*x^3)^(1 + q))/(d^2*(28 + 33*q + 9*q^2))) + (b*x*(a + b*x^3)*(c + d*x^3)^(1 + q))/(d*(7 + 3*q)) + ((4*b^2*c^2 - 2*a*b*c*d*(7 + 3*q) + a^2*d^2*(28 + 33*q + 9*q^2))*x*(c + d*x^3)^q*HypergeometricFunctions._₂F₁(1/3, -q, 4/3, -((d*x^3)/c)))/((1 + (d*x^3)/c)^q*(d^2*(28 + 33*q + 9*q^2))))]
    @test_int [(a + b*x^3)^1*(c + d*x^3)^q, x, 3, (b*x*(c + d*x^3)^(1 + q))/(d*(4 + 3*q)) - ((b*c - a*d*(4 + 3*q))*x*(c + d*x^3)^(1 + q)*HypergeometricFunctions._₂F₁(1, 4/3 + q, 4/3, -((d*x^3)/c)))/(c*d*(4 + 3*q)), (b*x*(c + d*x^3)^(1 + q))/(d*(4 + 3*q)) - ((b*c - a*d*(4 + 3*q))*x*(c + d*x^3)^q*HypergeometricFunctions._₂F₁(1/3, -q, 4/3, -((d*x^3)/c)))/((1 + (d*x^3)/c)^q*(d*(4 + 3*q)))]
    @test_int [1/(a + b*x^3)^1*(c + d*x^3)^q, x, 2, (x*(c + d*x^3)^q*AppellF1(1/3, 1, -q, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (d*x^3)/c)^q*a)]
    @test_int [1/(a + b*x^3)^2*(c + d*x^3)^q, x, 2, (x*(c + d*x^3)^q*AppellF1(1/3, 2, -q, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (d*x^3)/c)^q*a^2)]

    @test_int [(a + b*x^3)^m*(c + d*x^3)^3, x, 5, If(13>=8, (d*(28*a^2*d^2 - a*b*c*d*(92 + 15*m) + b^2*c^2*(118 + 60*m + 9*m^2))*x*(a + b*x^3)^(1 + m))/(b^3*(4 + 3*m)*(7 + 3*m)*(10 + 3*m)) - (d*(7*a*d - b*c*(16 + 3*m))*x*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b^2*(7 + 3*m)*(10 + 3*m)) + (d*x*(a + b*x^3)^(1 + m)*(c + d*x^3)^2)/(b*(10 + 3*m)) - ((28*a^3*d^3 - 12*a^2*b*c*d^2*(10 + 3*m) + 3*a*b^2*c^2*d*(70 + 51*m + 9*m^2) - b^3*c^3*(280 + 414*m + 189*m^2 + 27*m^3))*x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^m*(b^3*(4 + 3*m)*(7 + 3*m)*(10 + 3*m))), (d*(28*a^2*d^2 - a*b*c*d*(92 + 15*m) + b^2*c^2*(118 + 60*m + 9*m^2))*x*(a + b*x^3)^(1 + m))/(b^3*(10 + 3*m)*(28 + 33*m + 9*m^2)) - (d*(7*a*d - b*c*(16 + 3*m))*x*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b^2*(70 + 51*m + 9*m^2)) + (d*x*(a + b*x^3)^(1 + m)*(c + d*x^3)^2)/(b*(10 + 3*m)) - ((28*a^3*d^3 - 12*a^2*b*c*d^2*(10 + 3*m) + 3*a*b^2*c^2*d*(70 + 51*m + 9*m^2) - b^3*c^3*(280 + 414*m + 189*m^2 + 27*m^3))*x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^m*(b^3*(10 + 3*m)*(28 + 33*m + 9*m^2))))]
    @test_int [(a + b*x^3)^m*(c + d*x^3)^2, x, 4, If(13>=8, -((d*(4*a*d - b*c*(10 + 3*m))*x*(a + b*x^3)^(1 + m))/(b^2*(4 + 3*m)*(7 + 3*m))) + (d*x*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b*(7 + 3*m)) + ((4*a^2*d^2 - 2*a*b*c*d*(7 + 3*m) + b^2*c^2*(28 + 33*m + 9*m^2))*x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^m*(b^2*(4 + 3*m)*(7 + 3*m))), -((d*(4*a*d - b*c*(10 + 3*m))*x*(a + b*x^3)^(1 + m))/(b^2*(28 + 33*m + 9*m^2))) + (d*x*(a + b*x^3)^(1 + m)*(c + d*x^3))/(b*(7 + 3*m)) + ((4*a^2*d^2 - 2*a*b*c*d*(7 + 3*m) + b^2*c^2*(28 + 33*m + 9*m^2))*x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^m*(b^2*(28 + 33*m + 9*m^2))))]
    @test_int [(a + b*x^3)^m*(c + d*x^3)^1, x, 3, (d*x*(a + b*x^3)^(1 + m))/(b*(4 + 3*m)) - ((a*d - b*c*(4 + 3*m))*x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/((1 + (b*x^3)/a)^m*(b*(4 + 3*m)))]
    @test_int [(a + b*x^3)^m*(c + d*x^3)^0, x, 2, (x*(a + b*x^3)^m*HypergeometricFunctions._₂F₁(1/3, -m, 4/3, -((b*x^3)/a)))/(1 + (b*x^3)/a)^m]
    @test_int [(a + b*x^3)^m/(c + d*x^3)^1, x, 2, (x*(a + b*x^3)^m*AppellF1(1/3, -m, 1, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (b*x^3)/a)^m*c)]
    @test_int [(a + b*x^3)^m/(c + d*x^3)^2, x, 2, (x*(a + b*x^3)^m*AppellF1(1/3, -m, 2, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (b*x^3)/a)^m*c^2)]
    @test_int [(a + b*x^3)^m/(c + d*x^3)^3, x, 2, (x*(a + b*x^3)^m*AppellF1(1/3, -m, 3, 4/3, -((b*x^3)/a), -((d*x^3)/c)))/((1 + (b*x^3)/a)^m*c^3)]

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

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

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

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

    @test_int [(a + b*x^4)*(c + d*x^4)^4, x, 2, a*c^4*x + (1/5)*c^3*(b*c + 4*a*d)*x^5 + (2/9)*c^2*d*(2*b*c + 3*a*d)*x^9 + (2/13)*c*d^2*(3*b*c + 2*a*d)*x^13 + (1/17)*d^3*(4*b*c + a*d)*x^17 + (1/21)*b*d^4*x^21]
    @test_int [(a + b*x^4)*(c + d*x^4)^3, x, 2, a*c^3*x + (1/5)*c^2*(b*c + 3*a*d)*x^5 + (1/3)*c*d*(b*c + a*d)*x^9 + (1/13)*d^2*(3*b*c + a*d)*x^13 + (1/17)*b*d^3*x^17]
    @test_int [(a + b*x^4)*(c + d*x^4)^2, x, 2, a*c^2*x + (1/5)*c*(b*c + 2*a*d)*x^5 + (1/9)*d*(2*b*c + a*d)*x^9 + (1/13)*b*d^2*x^13]
    @test_int [(a + b*x^4)*(c + d*x^4)^1, x, 2, a*c*x + (1/5)*(b*c + a*d)*x^5 + (1/9)*b*d*x^9]
    @test_int [(a + b*x^4)/(c + d*x^4)^1, x, 10, (b*x)/d + ((b*c - a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*d^(5/4)) - ((b*c - a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*d^(5/4)) + ((b*c - a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*d^(5/4)) - ((b*c - a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*d^(5/4))]
    @test_int [(a + b*x^4)/(c + d*x^4)^2, x, 10, -(((b*c - a*d)*x)/(4*c*d*(c + d*x^4))) - ((b*c + 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*d^(5/4)) + ((b*c + 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*d^(5/4)) - ((b*c + 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*d^(5/4)) + ((b*c + 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*d^(5/4))]
    @test_int [(a + b*x^4)/(c + d*x^4)^3, x, 11, -(((b*c - a*d)*x)/(8*c*d*(c + d*x^4)^2)) + ((b*c + 7*a*d)*x)/(32*c^2*d*(c + d*x^4)) - (3*(b*c + 7*a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(64*sqrt(2)*c^(11/4)*d^(5/4)) + (3*(b*c + 7*a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(64*sqrt(2)*c^(11/4)*d^(5/4)) - (3*(b*c + 7*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(128*sqrt(2)*c^(11/4)*d^(5/4)) + (3*(b*c + 7*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(128*sqrt(2)*c^(11/4)*d^(5/4))]

    @test_int [(a + b*x^4)^2*(c + d*x^4)^4, x, 2, a^2*c^4*x + (2/5)*a*c^3*(b*c + 2*a*d)*x^5 + (1/9)*c^2*(b^2*c^2 + 8*a*b*c*d + 6*a^2*d^2)*x^9 + (4/13)*c*d*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^13 + (1/17)*d^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^2)*x^17 + (2/21)*b*d^3*(2*b*c + a*d)*x^21 + (1/25)*b^2*d^4*x^25]
    @test_int [(a + b*x^4)^2*(c + d*x^4)^3, x, 2, a^2*c^3*x + (1/5)*a*c^2*(2*b*c + 3*a*d)*x^5 + (1/9)*c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9 + (1/13)*d*(3*b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^13 + (1/17)*b*d^2*(3*b*c + 2*a*d)*x^17 + (1/21)*b^2*d^3*x^21]
    @test_int [(a + b*x^4)^2*(c + d*x^4)^2, x, 2, a^2*c^2*x + (2/5)*a*c*(b*c + a*d)*x^5 + (1/9)*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^9 + (2/13)*b*d*(b*c + a*d)*x^13 + (1/17)*b^2*d^2*x^17]
    @test_int [(a + b*x^4)^2*(c + d*x^4)^1, x, 2, a^2*c*x + (1/5)*a*(2*b*c + a*d)*x^5 + (1/9)*b*(b*c + 2*a*d)*x^9 + (1/13)*b^2*d*x^13]
    @test_int [(a + b*x^4)^2/(c + d*x^4)^1, x, 11, -((b*(b*c - 2*a*d)*x)/d^2) + (b^2*x^5)/(5*d) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*d^(9/4)) - ((b*c - a*d)^2*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*d^(9/4)) + ((b*c - a*d)^2*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*d^(9/4))]
    @test_int [(a + b*x^4)^2/(c + d*x^4)^2, x, 12, (b^2*x)/d^2 + ((b*c - a*d)^2*x)/(4*c*d^2*(c + d*x^4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*d^(9/4)) + ((b*c - a*d)*(5*b*c + 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*d^(9/4)) - ((b*c - a*d)*(5*b*c + 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*d^(9/4))]
    @test_int [(a + b*x^4)^2/(c + d*x^4)^3, x, 11, -(((b*c - a*d)*x*(a + b*x^4))/(8*c*d*(c + d*x^4)^2)) - ((b*c - a*d)*(5*b*c + 7*a*d)*x)/(32*c^2*d^2*(c + d*x^4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(64*sqrt(2)*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(64*sqrt(2)*c^(11/4)*d^(9/4)) - ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(128*sqrt(2)*c^(11/4)*d^(9/4)) + ((5*b^2*c^2 + 6*a*b*c*d + 21*a^2*d^2)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(128*sqrt(2)*c^(11/4)*d^(9/4))]

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

    @test_int [1/(a + b*x^4)*(c + d*x^4)^4, x, 11, (d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x)/b^4 + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x^5)/(5*b^3) + (d^3*(4*b*c - a*d)*x^9)/(9*b^2) + (d^4*x^13)/(13*b) - ((b*c - a*d)^4*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(17/4)) - ((b*c - a*d)^4*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(17/4)) + ((b*c - a*d)^4*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(17/4))]
    @test_int [1/(a + b*x^4)*(c + d*x^4)^3, x, 11, (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x)/b^3 + (d^2*(3*b*c - a*d)*x^5)/(5*b^2) + (d^3*x^9)/(9*b) - ((b*c - a*d)^3*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(13/4))]
    @test_int [1/(a + b*x^4)*(c + d*x^4)^2, x, 11, (d*(2*b*c - a*d)*x)/b^2 + (d^2*x^5)/(5*b) - ((b*c - a*d)^2*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(9/4)) + ((b*c - a*d)^2*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(9/4)) - ((b*c - a*d)^2*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(9/4)) + ((b*c - a*d)^2*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(9/4))]
    @test_int [1/(a + b*x^4)*(c + d*x^4)^1, x, 10, (d*x)/b - ((b*c - a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(5/4)) + ((b*c - a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*b^(5/4)) - ((b*c - a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(5/4)) + ((b*c - a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*b^(5/4))]
    @test_int [1/(a + b*x^4)/(c + d*x^4)^1, x, 19, -((b^(3/4)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(b*c - a*d))) + (b^(3/4)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)) + (d^(3/4)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*(b*c - a*d)) - (d^(3/4)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*(b*c - a*d)) - (b^(3/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)) + (b^(3/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)) + (d^(3/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*(b*c - a*d)) - (d^(3/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*(b*c - a*d))]
    @test_int [1/(a + b*x^4)/(c + d*x^4)^2, x, 20, -((d*x)/(4*c*(b*c - a*d)*(c + d*x^4))) - (b^(7/4)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(2*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*(7*b*c - 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (d^(3/4)*(7*b*c - 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (b^(7/4)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (b^(7/4)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(4*sqrt(2)*a^(3/4)*(b*c - a*d)^2) + (d^(3/4)*(7*b*c - 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*(b*c - a*d)^2) - (d^(3/4)*(7*b*c - 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*(b*c - a*d)^2)]

    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^5, x, 12, (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^5)/(5*b^4) + (d^4*(5*b*c - 2*a*d)*x^9)/(9*b^3) + (d^5*x^13)/(13*b^2) + ((b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(21/4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(21/4))]
    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^4, x, 12, (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^5)/(5*b^3) + (d^4*x^9)/(9*b^2) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(17/4)) - ((b*c - a*d)^3*(3*b*c + 13*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(17/4)) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(17/4))]
    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^3, x, 12, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^5)/(5*b^2) + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(13/4)) - (3*(b*c - a*d)^2*(b*c + 3*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(13/4)) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(13/4))]
    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^2, x, 12, (d^2*x)/b^2 + ((b*c - a*d)^2*x)/(4*a*b^2*(a + b*x^4)) - ((b*c - a*d)*(3*b*c + 5*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(9/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(9/4)) - ((b*c - a*d)*(3*b*c + 5*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(9/4)) + ((b*c - a*d)*(3*b*c + 5*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(9/4))]
    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^1, x, 10, ((b*c - a*d)*x)/(4*a*b*(a + b*x^4)) - ((3*b*c + a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(5/4)) + ((3*b*c + a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*b^(5/4)) - ((3*b*c + a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(5/4)) + ((3*b*c + a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*b^(5/4))]
    @test_int [1/(a + b*x^4)^2/(c + d*x^4)^1, x, 20, (b*x)/(4*a*(b*c - a*d)*(a + b*x^4)) - (b^(3/4)*(3*b*c - 7*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^2) + (b^(3/4)*(3*b*c - 7*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*(b*c - a*d)^2) + (d^(7/4)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(2*sqrt(2)*c^(3/4)*(b*c - a*d)^2) - (b^(3/4)*(3*b*c - 7*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*(b*c - a*d)^2) + (b^(3/4)*(3*b*c - 7*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*(b*c - a*d)^2) - (d^(7/4)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*(b*c - a*d)^2) + (d^(7/4)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(4*sqrt(2)*c^(3/4)*(b*c - a*d)^2)]
    @test_int [1/(a + b*x^4)^2/(c + d*x^4)^2, x, 21, (d*(b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^4)) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^4)*(c + d*x^4)) - (b^(7/4)*(3*b*c - 11*a*d)*atan(1 - (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*atan(1 + (sqrt(2)*b^(1/4)*x)/a^(1/4)))/(8*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*atan(1 - (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*atan(1 + (sqrt(2)*d^(1/4)*x)/c^(1/4)))/(8*sqrt(2)*c^(7/4)*(b*c - a*d)^3) - (b^(7/4)*(3*b*c - 11*a*d)*log(sqrt(a) - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*(b*c - a*d)^3) + (b^(7/4)*(3*b*c - 11*a*d)*log(sqrt(a) + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b)*x^2))/(16*sqrt(2)*a^(7/4)*(b*c - a*d)^3) - (d^(7/4)*(11*b*c - 3*a*d)*log(sqrt(c) - sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*(b*c - a*d)^3) + (d^(7/4)*(11*b*c - 3*a*d)*log(sqrt(c) + sqrt(2)*c^(1/4)*d^(1/4)*x + sqrt(d)*x^2))/(16*sqrt(2)*c^(7/4)*(b*c - a*d)^3)]

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

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

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

    @test_int [(a - b*x^4)^(5/2)/(c - d*x^4), x, 10, -((b*(7*b*c - 13*a*d)*x*sqrt(a - b*x^4))/(21*d^2)) + (b*x*(a - b*x^4)^(3/2))/(7*d) + (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 56*a*b*c*d + 47*a^2*d^2)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(21*d^3*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - a*d)^3*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d^3*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - a*d)^3*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d^3*sqrt(a - b*x^4))]
    @test_int [(a - b*x^4)^(3/2)/(c - d*x^4), x, 9, (b*x*sqrt(a - b*x^4))/(3*d) - (a^(1/4)*b^(3/4)*(3*b*c - 5*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(3*d^2*sqrt(a - b*x^4)) + (a^(1/4)*(b*c - a*d)^2*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d^2*sqrt(a - b*x^4)) + (a^(1/4)*(b*c - a*d)^2*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d^2*sqrt(a - b*x^4))]
    @test_int [(a - b*x^4)^(1/2)/(c - d*x^4), x, 8, (a^(1/4)*b^(3/4)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(d*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*d*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(1/2)*(c - d*x^4)), x, 5, (a^(1/4)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*sqrt(a - b*x^4)) + (a^(1/4)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(3/2)*(c - d*x^4)), x, 9, (b*x)/(2*a*(b*c - a*d)*sqrt(a - b*x^4)) + (b^(3/4)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(2*a^(3/4)*(b*c - a*d)*sqrt(a - b*x^4)) - (a^(1/4)*d*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)*sqrt(a - b*x^4)) - (a^(1/4)*d*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(5/2)*(c - d*x^4)), x, 10, (b*x)/(6*a*(b*c - a*d)*(a - b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c - a*d)^2*sqrt(a - b*x^4)) + (b^(3/4)*(5*b*c - 11*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(12*a^(7/4)*(b*c - a*d)^2*sqrt(a - b*x^4)) + (a^(1/4)*d^2*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^2*sqrt(a - b*x^4)) + (a^(1/4)*d^2*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^2*sqrt(a - b*x^4))]

    @test_int [(a + b*x^4)^(3/2)/(c + d*x^4), x, 10, (b*x*sqrt(a + b*x^4))/(3*d) - ((b*c - a*d)^(3/2)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*d^(7/4)) - (((-b)*c + a*d)^(3/2)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*d^(7/4)) - (b^(3/4)*(3*b*c - 5*a*d)*(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))/(6*a^(1/4)*d^2*sqrt(a + b*x^4)) + (b^(1/4)*(sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))*(b*c - a*d)^2*(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))/(4*a^(1/4)*sqrt(-c)*d^2*(b*c + a*d)*sqrt(a + b*x^4)) + (b^(1/4)*(sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))*(b*c - a*d)^2*(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))/(4*a^(1/4)*sqrt(-c)*d^2*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*(b*c - a*d)^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*d^2*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*(b*c - a*d)^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*d^2*(b*c + a*d)*sqrt(a + b*x^4))]
    @test_int [(a + b*x^4)^(1/2)/(c + d*x^4), x, 9, (sqrt(b*c - a*d)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*d^(3/4)) - (sqrt((-b)*c + a*d)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*d^(3/4)) + (b^(3/4)*(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)*d*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))*(b*c - a*d)*(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))/(4*a^(1/4)*sqrt(-c)*d*(b*c + a*d)*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b) + (sqrt(a)*sqrt(d))/sqrt(-c))*(b*c - a*d)*(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))/(4*a^(1/4)*d*(b*c + a*d)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*(b*c - a*d)*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*d*(b*c + a*d)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*(b*c - a*d)*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*d*(b*c + a*d)*sqrt(a + b*x^4))]
    @test_int [1/((a + b*x^4)^(1/2)*(c + d*x^4)), x, 7, -((d^(1/4)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*sqrt(b*c - a*d))) - (d^(1/4)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*sqrt((-b)*c + a*d)) + (b^(1/4)*(sqrt(b) + (sqrt(a)*sqrt(d))/sqrt(-c))*(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))/(4*a^(1/4)*(b*c + a*d)*sqrt(a + b*x^4)) + (b^(1/4)*(sqrt(b)*c + sqrt(a)*sqrt(-c)*sqrt(d))*(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))/(4*a^(1/4)*c*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c + a*d)*sqrt(a + b*x^4))]
    @test_int [1/((a + b*x^4)^(3/2)*(c + d*x^4)), x, 10, If(13>=8, (b*x)/(2*a*(b*c - a*d)*sqrt(a + b*x^4)) + (d^(5/4)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*(b*c - a*d)^(3/2)) - (d^(5/4)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*((-b)*c + a*d)^(3/2)) + (b^(3/4)*(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))/(4*a^(5/4)*(b*c - a*d)*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b) + (sqrt(a)*sqrt(d))/sqrt(-c))*d*(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))/(4*a^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b)*c + sqrt(a)*sqrt(-c)*sqrt(d))*d*(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))/(4*a^(1/4)*c*(b^2*c^2 - a^2*d^2)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*d*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c - a*d)*(b*c + a*d)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*d*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c - a*d)*(b*c + a*d)*sqrt(a + b*x^4)), (b*x)/(2*a*(b*c - a*d)*sqrt(a + b*x^4)) + (d^(5/4)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*(b*c - a*d)^(3/2)) - (d^(5/4)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*((-b)*c + a*d)^(3/2)) + (b^(3/4)*(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))/(4*a^(5/4)*(b*c - a*d)*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b) + (sqrt(a)*sqrt(d))/sqrt(-c))*d*(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))/(4*a^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(a + b*x^4)) - (b^(1/4)*(sqrt(b)*c + sqrt(a)*sqrt(-c)*sqrt(d))*d*(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))/(4*a^(1/4)*c*(b^2*c^2 - a^2*d^2)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*d*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b^2*c^2 - a^2*d^2)*sqrt(a + b*x^4)) - ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*d*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b^2*c^2 - a^2*d^2)*sqrt(a + b*x^4)))]
    @test_int [1/((a + b*x^4)^(5/2)*(c + d*x^4)), x, 11, (b*x)/(6*a*(b*c - a*d)*(a + b*x^4)^(3/2)) + (b*(5*b*c - 11*a*d)*x)/(12*a^2*(b*c - a*d)^2*sqrt(a + b*x^4)) - (d^(9/4)*atan((sqrt(b*c - a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*(b*c - a*d)^(5/2)) - (d^(9/4)*atan((sqrt((-b)*c + a*d)*x)/((-c)^(1/4)*d^(1/4)*sqrt(a + b*x^4))))/(4*(-c)^(3/4)*((-b)*c + a*d)^(5/2)) + (b^(3/4)*(5*b*c - 11*a*d)*(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))/(24*a^(9/4)*(b*c - a*d)^2*sqrt(a + b*x^4)) + (b^(1/4)*(sqrt(b)*c - sqrt(a)*sqrt(-c)*sqrt(d))*d^2*(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))/(4*a^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*sqrt(a + b*x^4)) + (b^(1/4)*(sqrt(b)*c + sqrt(a)*sqrt(-c)*sqrt(d))*d^2*(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))/(4*a^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2*d^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi(-((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d))), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*sqrt(a + b*x^4)) + ((sqrt(b)*sqrt(-c) - sqrt(a)*sqrt(d))^2*d^2*(sqrt(a) + sqrt(b)*x^2)*sqrt((a + b*x^4)/(sqrt(a) + sqrt(b)*x^2)^2)*Elliptic.Pi((sqrt(b)*sqrt(-c) + sqrt(a)*sqrt(d))^2/(4*sqrt(a)*sqrt(b)*sqrt(-c)*sqrt(d)), 2*atan((b^(1/4)*x)/a^(1/4)), 1/2))/(8*a^(1/4)*b^(1/4)*c*(b*c - a*d)^2*(b*c + a*d)*sqrt(a + b*x^4))]

    @test_int [(a - b*x^4)^(7/2)/(c - d*x^4)^2, x, 11, -((b*(77*b^2*c^2 - 122*a*b*c*d + 21*a^2*d^2)*x*sqrt(a - b*x^4))/(84*c*d^3)) + (b*(11*b*c - 7*a*d)*x*(a - b*x^4)^(3/2))/(28*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(5/2))/(4*c*d*(c - d*x^4)) + (1/(84*c*d^4*sqrt(a - b*x^4)))*(a^(1/4)*b^(3/4)*(231*b^3*c^3 - 553*a*b^2*c^2*d + 349*a^2*b*c*d^2 + 21*a^3*d^3)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1)) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^4*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - a*d)^3*(11*b*c + 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^4*sqrt(a - b*x^4))]
    @test_int [(a - b*x^4)^(5/2)/(c - d*x^4)^2, x, 10, (b*(7*b*c - 3*a*d)*x*sqrt(a - b*x^4))/(12*c*d^2) - ((b*c - a*d)*x*(a - b*x^4)^(3/2))/(4*c*d*(c - d*x^4)) - (a^(1/4)*b^(3/4)*(21*b^2*c^2 - 26*a*b*c*d - 3*a^2*d^2)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(12*c*d^3*sqrt(a - b*x^4)) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^3*sqrt(a - b*x^4)) + (a^(1/4)*(b*c - a*d)^2*(7*b*c + 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^3*sqrt(a - b*x^4))]
    @test_int [(a - b*x^4)^(3/2)/(c - d*x^4)^2, x, 9, -(((b*c - a*d)*x*sqrt(a - b*x^4))/(4*c*d*(c - d*x^4))) + (a^(1/4)*b^(3/4)*(3*b*c + a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(4*c*d^2*sqrt(a - b*x^4)) - (3*a^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^2*sqrt(a - b*x^4)) - (3*a^(1/4)*(b*c - a*d)*(b*c + a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d^2*sqrt(a - b*x^4))]
    @test_int [(a - b*x^4)^(1/2)/(c - d*x^4)^2, x, 9, (x*sqrt(a - b*x^4))/(4*c*(c - d*x^4)) + (a^(1/4)*b^(3/4)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(4*c*d*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d*sqrt(a - b*x^4)) - (a^(1/4)*(b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*d*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(1/2)*(c - d*x^4)^2), x, 9, -((d*x*sqrt(a - b*x^4))/(4*c*(b*c - a*d)*(c - d*x^4))) - (a^(1/4)*b^(3/4)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(4*c*(b*c - a*d)*sqrt(a - b*x^4)) + (a^(1/4)*(5*b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)*sqrt(a - b*x^4)) + (a^(1/4)*(5*b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(3/2)*(c - d*x^4)^2), x, 10, (b*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*sqrt(a - b*x^4)) - (d*x)/(4*c*(b*c - a*d)*sqrt(a - b*x^4)*(c - d*x^4)) + (b^(3/4)*(2*b*c + a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(4*a^(3/4)*c*(b*c - a*d)^2*sqrt(a - b*x^4)) - (3*a^(1/4)*d*(3*b*c - a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^2*sqrt(a - b*x^4)) - (3*a^(1/4)*d*(3*b*c - a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^2*sqrt(a - b*x^4))]
    @test_int [1/((a - b*x^4)^(5/2)*(c - d*x^4)^2), x, 11, (b*(2*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a - b*x^4)^(3/2)) + (b*(5*b^2*c^2 - 17*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*sqrt(a - b*x^4)) - (d*x)/(4*c*(b*c - a*d)*(a - b*x^4)^(3/2)*(c - d*x^4)) + (b^(3/4)*(5*b^2*c^2 - 17*a*b*c*d - 3*a^2*d^2)*sqrt(1 - (b*x^4)/a)*Elliptic.F(asin((b^(1/4)*x)/a^(1/4)), -1))/(12*a^(7/4)*c*(b*c - a*d)^3*sqrt(a - b*x^4)) + (a^(1/4)*d^2*(13*b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi(-((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^3*sqrt(a - b*x^4)) + (a^(1/4)*d^2*(13*b*c - 3*a*d)*sqrt(1 - (b*x^4)/a)*Elliptic.Pi((sqrt(a)*sqrt(d))/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/a^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^3*sqrt(a - b*x^4))]

    #= [(a + b*x^4)^(7/2)/(c + d*x^4)^2, x, 11, 0]
    @test_int [(a + b*x^4)^(5/2)/(c + d*x^4)^2, x, 10, 0]
    @test_int [(a + b*x^4)^(3/2)/(c + d*x^4)^2, x, 9, 0]
    @test_int [(a + b*x^4)^(1/2)/(c + d*x^4)^2, x, 9, 0]
    @test_int [1/((a + b*x^4)^(1/2)*(c + d*x^4)^2), x, 9, 0]
    @test_int [1/((a + b*x^4)^(3/2)*(c + d*x^4)^2), x, 10, 0]
    @test_int [1/((a + b*x^4)^(5/2)*(c + d*x^4)^2), x, 11, 0] =#

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

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

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

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

    #= Good*examples*of*elementary*integrands*and*antiderivatives =#
    @test_int [(a + b*x^4)^(7/4)/(c + d*x^4), x, 10, (b*x*(a + b*x^4)^(3/4))/(4*d) - (b^(3/4)*(4*b*c - 7*a*d)*atan((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(8*d^2) + ((b*c - a*d)^(7/4)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*d^2) - (b^(3/4)*(4*b*c - 7*a*d)*atanh((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(8*d^2) + ((b*c - a*d)^(7/4)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*d^2)]
    @test_int [(a + b*x^4)^(3/4)/(c + d*x^4), x, 9, (b^(3/4)*atan((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(2*d) - ((b*c - a*d)^(3/4)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*d) + (b^(3/4)*atanh((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(2*d) - ((b*c - a*d)^(3/4)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*d)]
    @test_int [1/((a + b*x^4)^(1/4)*(c + d*x^4)), x, 4, atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4)))/(2*c^(3/4)*(b*c - a*d)^(1/4)) + atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4)))/(2*c^(3/4)*(b*c - a*d)^(1/4))]
    @test_int [1/((a + b*x^4)^(5/4)*(c + d*x^4)), x, 5, (b*x)/(a*(b*c - a*d)*(a + b*x^4)^(1/4)) - (d*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(5/4)) - (d*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(5/4))]
    @test_int [1/((a + b*x^4)^(9/4)*(c + d*x^4)), x, 7, (b*x)/(5*a*(b*c - a*d)*(a + b*x^4)^(5/4)) + (b*(4*b*c - 9*a*d)*x)/(5*a^2*(b*c - a*d)^2*(a + b*x^4)^(1/4)) + (d^2*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(9/4)) + (d^2*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(9/4))]
    @test_int [1/((a + b*x^4)^(13/4)*(c + d*x^4)), x, 8, (b*x)/(9*a*(b*c - a*d)*(a + b*x^4)^(9/4)) + (b*(8*b*c - 17*a*d)*x)/(45*a^2*(b*c - a*d)^2*(a + b*x^4)^(5/4)) + (b*(32*b^2*c^2 - 100*a*b*c*d + 113*a^2*d^2)*x)/(45*a^3*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d^3*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(13/4)) - (d^3*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(2*c^(3/4)*(b*c - a*d)^(13/4))]

    @test_int [(a + b*x^4)^(9/4)/(c + d*x^4), x, 11, -((b*(6*b*c - 11*a*d)*x*(a + b*x^4)^(1/4))/(12*d^2)) + (b*x*(a + b*x^4)^(5/4))/(6*d) + (sqrt(a)*b^(3/2)*(6*b*c - 11*a*d)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(12*d^2*(a + b*x^4)^(3/4)) + ((b*c - a*d)^2*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*d^2) + ((b*c - a*d)^2*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*d^2)]
    @test_int [(a + b*x^4)^(5/4)/(c + d*x^4), x, 10, (b*x*(a + b*x^4)^(1/4))/(2*d) - (sqrt(a)*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(2*d*(a + b*x^4)^(3/4)) - ((b*c - a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*d) - ((b*c - a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*d)]
    @test_int [(a + b*x^4)^(1/4)/(c + d*x^4), x, 4, (sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c) + (sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c)]
    @test_int [1/((a + b*x^4)^(3/4)*(c + d*x^4)), x, 9, -((b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(sqrt(a)*(b*c - a*d)*(a + b*x^4)^(3/4))) - (d*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)) - (d*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d))]
    @test_int [1/((a + b*x^4)^(7/4)*(c + d*x^4)), x, 10, (b*x)/(3*a*(b*c - a*d)*(a + b*x^4)^(3/4)) - (b^(3/2)*(2*b*c - 5*a*d)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(3*a^(3/2)*(b*c - a*d)^2*(a + b*x^4)^(3/4)) + (d^2*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^2) + (d^2*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^2)]
    @test_int [1/((a + b*x^4)^(11/4)*(c + d*x^4)), x, 11, (b*x)/(7*a*(b*c - a*d)*(a + b*x^4)^(7/4)) + (b*(6*b*c - 13*a*d)*x)/(21*a^2*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (b^(3/2)*(12*b^2*c^2 - 38*a*b*c*d + 47*a^2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(21*a^(5/2)*(b*c - a*d)^3*(a + b*x^4)^(3/4)) - (d^3*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^3) - (d^3*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(2*b^(1/4)*c*(b*c - a*d)^3)]

    #= Good*examples*of*elementary*integrands*and*antiderivatives =#
    @test_int [(a + b*x^4)^(11/4)/(c + d*x^4)^2, x, 11, (b*(2*b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d^2) - ((b*c - a*d)*x*(a + b*x^4)^(7/4))/(4*c*d*(c + d*x^4)) - (b^(7/4)*(8*b*c - 11*a*d)*atan((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a*d)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*d^3) - (b^(7/4)*(8*b*c - 11*a*d)*atanh((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(8*d^3) + ((b*c - a*d)^(7/4)*(8*b*c + 3*a*d)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*d^3)]
    @test_int [(a + b*x^4)^(7/4)/(c + d*x^4)^2, x, 10, -(((b*c - a*d)*x*(a + b*x^4)^(3/4))/(4*c*d*(c + d*x^4))) + (b^(7/4)*atan((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*d^2) + (b^(7/4)*atanh((b^(1/4)*x)/(a + b*x^4)^(1/4)))/(2*d^2) - ((b*c - a*d)^(3/4)*(4*b*c + 3*a*d)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*d^2)]
    @test_int [(a + b*x^4)^(3/4)/(c + d*x^4)^2, x, 5, (x*(a + b*x^4)^(3/4))/(4*c*(c + d*x^4)) + (3*a*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(1/4)) + (3*a*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(1/4))]
    @test_int [1/((a + b*x^4)^(1/4)*(c + d*x^4)^2), x, 5, -((d*x*(a + b*x^4)^(3/4))/(4*c*(b*c - a*d)*(c + d*x^4))) + ((4*b*c - 3*a*d)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(5/4)) + ((4*b*c - 3*a*d)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(5/4))]
    @test_int [1/((a + b*x^4)^(5/4)*(c + d*x^4)^2), x, 7, (b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(1/4)*(c + d*x^4)) - (d*(8*b*c - 3*a*d)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(9/4)) - (d*(8*b*c - 3*a*d)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(9/4))]
    @test_int [1/((a + b*x^4)^(9/4)*(c + d*x^4)^2), x, 8, (b*(4*b*c + 5*a*d)*x)/(20*a*c*(b*c - a*d)^2*(a + b*x^4)^(5/4)) + (b*(16*b^2*c^2 - 56*a*b*c*d - 5*a^2*d^2)*x)/(20*a^2*c*(b*c - a*d)^3*(a + b*x^4)^(1/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(5/4)*(c + d*x^4)) + (3*d^2*(4*b*c - a*d)*atan(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(13/4)) + (3*d^2*(4*b*c - a*d)*atanh(((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))))/(8*c^(7/4)*(b*c - a*d)^(13/4))]

    @test_int [(a + b*x^4)^(9/4)/(c + d*x^4)^2, x, 11, (b*(3*b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d^2) - ((b*c - a*d)*x*(a + b*x^4)^(5/4))/(4*c*d*(c + d*x^4)) - (sqrt(a)*b^(3/2)*(3*b*c - a*d)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(4*c*d^2*(a + b*x^4)^(3/4)) - (3*(b*c - a*d)*(2*b*c + a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*d^2) - (3*(b*c - a*d)*(2*b*c + a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*d^2)]
    @test_int [(a + b*x^4)^(5/4)/(c + d*x^4)^2, x, 10, -(((b*c - a*d)*x*(a + b*x^4)^(1/4))/(4*c*d*(c + d*x^4))) + (sqrt(a)*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(4*c*d*(a + b*x^4)^(3/4)) + ((2*b*c + 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*d) + ((2*b*c + 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*d)]
    @test_int [(a + b*x^4)^(1/4)/(c + d*x^4)^2, x, 10, (x*(a + b*x^4)^(1/4))/(4*c*(c + d*x^4)) - (sqrt(a)*b^(3/2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(4*c*(b*c - a*d)*(a + b*x^4)^(3/4)) + ((2*b*c - 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)) + ((2*b*c - 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d))]
    @test_int [1/((a + b*x^4)^(3/4)*(c + d*x^4)^2), x, 10, -((d*x*(a + b*x^4)^(1/4))/(4*c*(b*c - a*d)*(c + d*x^4))) - (b^(3/2)*(4*b*c - a*d)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(4*sqrt(a)*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (3*d*(2*b*c - a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^2) - (3*d*(2*b*c - a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^2)]
    @test_int [1/((a + b*x^4)^(7/4)*(c + d*x^4)^2), x, 11, (b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^4)^(3/4)) - (d*x)/(4*c*(b*c - a*d)*(a + b*x^4)^(3/4)*(c + d*x^4)) - (b^(3/2)*(8*b^2*c^2 - 32*a*b*c*d + 3*a^2*d^2)*(1 + a/(b*x^4))^(3/4)*x^3*Elliptic.F((1/2)*acot((sqrt(b)*x^2)/sqrt(a)), 2))/(12*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^4)^(3/4)) + (d^2*(10*b*c - 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(-(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c))), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^3) + (d^2*(10*b*c - 3*a*d)*sqrt(a/(a + b*x^4))*sqrt(a + b*x^4)*Elliptic.Pi(sqrt(b*c - a*d)/(sqrt(b)*sqrt(c)), asin((b^(1/4)*x)/(a + b*x^4)^(1/4)), -1))/(8*b^(1/4)*c^2*(b*c - a*d)^3)]

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

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

    @test_int [(a + b*x^4)^p*(c + d*x^4)^q, x, 3, (x*(a + b*x^4)^p*(c + d*x^4)^q*AppellF1(1/4, -p, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)))/((1 + (b*x^4)/a)^p*(1 + (d*x^4)/c)^q)]

    @test_int [(a + b*x^4)^2*(c + d*x^4)^q, x, 4, If(13>=8, -((b*(5*b*c - a*d*(13 + 4*q))*x*(c + d*x^4)^(1 + q))/(d^2*(5 + 4*q)*(9 + 4*q))) + (b*x*(a + b*x^4)*(c + d*x^4)^(1 + q))/(d*(9 + 4*q)) + ((5*b^2*c^2 - 2*a*b*c*d*(9 + 4*q) + a^2*d^2*(45 + 56*q + 16*q^2))*x*(c + d*x^4)^q*HypergeometricFunctions._₂F₁(1/4, -q, 5/4, -((d*x^4)/c)))/((1 + (d*x^4)/c)^q*(d^2*(5 + 4*q)*(9 + 4*q))), -((b*(5*b*c - a*d*(13 + 4*q))*x*(c + d*x^4)^(1 + q))/(d^2*(45 + 56*q + 16*q^2))) + (b*x*(a + b*x^4)*(c + d*x^4)^(1 + q))/(d*(9 + 4*q)) + ((5*b^2*c^2 - 2*a*b*c*d*(9 + 4*q) + a^2*d^2*(45 + 56*q + 16*q^2))*x*(c + d*x^4)^q*HypergeometricFunctions._₂F₁(1/4, -q, 5/4, -((d*x^4)/c)))/((1 + (d*x^4)/c)^q*(d^2*(45 + 56*q + 16*q^2))))]
    @test_int [(a + b*x^4)^1*(c + d*x^4)^q, x, 3, (b*x*(c + d*x^4)^(1 + q))/(d*(5 + 4*q)) - ((b*c - a*d*(5 + 4*q))*x*(c + d*x^4)^q*HypergeometricFunctions._₂F₁(1/4, -q, 5/4, -((d*x^4)/c)))/((1 + (d*x^4)/c)^q*(d*(5 + 4*q)))]
    @test_int [1/(a + b*x^4)^1*(c + d*x^4)^q, x, 2, (x*(c + d*x^4)^q*AppellF1(1/4, 1, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)))/((1 + (d*x^4)/c)^q*a)]
    @test_int [1/(a + b*x^4)^2*(c + d*x^4)^q, x, 2, (x*(c + d*x^4)^q*AppellF1(1/4, 2, -q, 5/4, -((b*x^4)/a), -((d*x^4)/c)))/((1 + (d*x^4)/c)^q*a^2)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^5)^p*(c+d*x^5)^q=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^5)^(p/5)*(c+d*x^4)^q=#

    @test_int [1/((a + b*x^5)^(1/5)*(c + d*x^5)), x, 7, -((sqrt((1/2)*(5 + sqrt(5)))*atan(sqrt((1/5)*(5 - 2*sqrt(5))) - (2*sqrt(2/(5 + sqrt(5)))*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))))/(5*c^(4/5)*(b*c - a*d)^(1/5))) + (sqrt((1/2)*(5 - sqrt(5)))*atan(sqrt((1/5)*(5 + 2*sqrt(5))) + (sqrt((2/5)*(5 + sqrt(5)))*(b*c - a*d)^(1/5)*x)/(c^(1/5)*(a + b*x^5)^(1/5))))/(5*c^(4/5)*(b*c - a*d)^(1/5)) - log(c^(1/5) - ((b*c - a*d)^(1/5)*x)/(a + b*x^5)^(1/5))/(5*c^(4/5)*(b*c - a*d)^(1/5)) + ((1 - sqrt(5))*log((2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) - sqrt(5)*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) + 2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)))/(20*c^(4/5)*(b*c - a*d)^(1/5)) + ((1 + sqrt(5))*log((2*(b*c - a*d)^(2/5)*x^2 + c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) + sqrt(5)*c^(1/5)*(b*c - a*d)^(1/5)*x*(a + b*x^5)^(1/5) + 2*c^(2/5)*(a + b*x^5)^(2/5))/(a + b*x^5)^(2/5)))/(20*c^(4/5)*(b*c - a*d)^(1/5))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^p*(c+d*x^n)^q*with*integer*n<0=#

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

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

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

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

    @test_int [sqrt(a + b/x)*(c + d/x)^3, x, 6, (-7*d*sqrt(a + b/x)*(c + d/x)^2)/5 - (d*sqrt(a + b/x)*(2*(57*b^2*c^2 + 15*a*b*c*d - 2*a^2*d^2) + (b*d*(33*b*c + 2*a*d))/x))/(15*b^2) + sqrt(a + b/x)*(c + d/x)^3*x + (c^2*(b*c + 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/sqrt(a)]
    @test_int [sqrt(a + b/x)*(c + d/x)^2, x, 6, -((c*(b*c + 4*a*d)*sqrt(a + b/x))/a) - (2*d^2*(a + b/x)^(3/2))/(3*b) + (c^2*(a + b/x)^(3/2)*x)/a + (c*(b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/sqrt(a)]
    @test_int [sqrt(a + b/x)*(c + d/x), x, 5, -(((b*c + 2*a*d)*sqrt(a + b/x))/a) + (c*(a + b/x)^(3/2)*x)/a + ((b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/sqrt(a)]
    @test_int [sqrt(a + b/x), x, 4, sqrt(a + b/x)*x + (b*atanh(sqrt(a + b/x)/sqrt(a)))/sqrt(a)]
    @test_int [sqrt(a + b/x)/(c + d/x), x, 7, (sqrt(a + b/x)*x)/c + (2*sqrt(d)*sqrt(b*c - a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/c^2 + ((b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(sqrt(a)*c^2)]
    @test_int [sqrt(a + b/x)/(c + d/x)^2, x, 8, (2*d*sqrt(a + b/x))/(c^2*(c + d/x)) + (sqrt(a + b/x)*x)/(c*(c + d/x)) + (sqrt(d)*(3*b*c - 4*a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*sqrt(b*c - a*d)) + ((b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(sqrt(a)*c^3)]
    @test_int [sqrt(a + b/x)/(c + d/x)^3, x, 9, (3*d*sqrt(a + b/x))/(2*c^2*(c + d/x)^2) + (d*(11*b*c - 12*a*d)*sqrt(a + b/x))/(4*c^3*(b*c - a*d)*(c + d/x)) + (sqrt(a + b/x)*x)/(c*(c + d/x)^2) + (sqrt(d)*(15*b^2*c^2 - 40*a*b*c*d + 24*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*(b*c - a*d)^(3/2)) + ((b*c - 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(sqrt(a)*c^4)]

    @test_int [(a + b/x)^(3/2)*(c + d/x)^3, x, 7, -3*c^2*(b*c + 2*a*d)*sqrt(a + b/x) - (9/7)*d*(a + b/x)^(3/2)*(c + d/x)^2 - (d*(a + b/x)^(3/2)*(2*(13*b*c - a*d)*(5*b*c + 2*a*d) + (3*b*d*(19*b*c + 2*a*d))/x))/(35*b^2) + (a + b/x)^(3/2)*(c + d/x)^3*x + 3*sqrt(a)*c^2*(b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(3/2)*(c + d/x)^2, x, 7, -(c*(3*b*c + 4*a*d)*sqrt(a + b/x)) - (c*(3*b*c + 4*a*d)*(a + b/x)^(3/2))/(3*a) - (2*d^2*(a + b/x)^(5/2))/(5*b) + (c^2*(a + b/x)^(5/2)*x)/a + sqrt(a)*c*(3*b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(3/2)*(c + d/x), x, 6, -((3*b*c + 2*a*d)*sqrt(a + b/x)) - ((3*b*c + 2*a*d)*(a + b/x)^(3/2))/(3*a) + (c*(a + b/x)^(5/2)*x)/a + sqrt(a)*(3*b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(3/2), x, 5, -3*b*sqrt(a + b/x) + (a + b/x)^(3/2)*x + 3*sqrt(a)*b*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(3/2)/(c + d/x), x, 7, (a*sqrt(a + b/x)*x)/c - (2*(b*c - a*d)^(3/2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^2*sqrt(d)) + (sqrt(a)*(3*b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^2]
    @test_int [(a + b/x)^(3/2)/(c + d/x)^2, x, 8, -(((b*c - 2*a*d)*sqrt(a + b/x))/(c^2*(c + d/x))) + (a*sqrt(a + b/x)*x)/(c*(c + d/x)) - ((b*c - 4*a*d)*sqrt(b*c - a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*sqrt(d)) + (sqrt(a)*(3*b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^3]
    @test_int [(a + b/x)^(3/2)/(c + d/x)^3, x, 9, -((b*c - 3*a*d)*sqrt(a + b/x))/(2*c^2*(c + d/x)^2) - (3*(b*c - 4*a*d)*sqrt(a + b/x))/(4*c^3*(c + d/x)) + (a*sqrt(a + b/x)*x)/(c*(c + d/x)^2) - (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*sqrt(d)*sqrt(b*c - a*d)) + (3*sqrt(a)*(b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^4]

    @test_int [(a + b/x)^(5/2)*(c + d/x)^3, x, 8, (-a)*c^2*(5*b*c + 6*a*d)*sqrt(a + b/x) - (1/3)*c^2*(5*b*c + 6*a*d)*(a + b/x)^(3/2) - (11/9)*d*(a + b/x)^(5/2)*(c + d/x)^2 - (d*(a + b/x)^(5/2)*(2*(469*b^2*c^2 + 135*a*b*c*d - 10*a^2*d^2) + (5*b*d*(89*b*c + 10*a*d))/x))/(315*b^2) + (a + b/x)^(5/2)*(c + d/x)^3*x + a^(3/2)*c^2*(5*b*c + 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(5/2)*(c + d/x)^2, x, 8, -(a*c*(5*b*c + 4*a*d)*sqrt(a + b/x)) - (c*(5*b*c + 4*a*d)*(a + b/x)^(3/2))/3 - (c*(5*b*c + 4*a*d)*(a + b/x)^(5/2))/(5*a) - (2*d^2*(a + b/x)^(7/2))/(7*b) + (c^2*(a + b/x)^(7/2)*x)/a + a^(3/2)*c*(5*b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(5/2)*(c + d/x), x, 7, (-a)*(5*b*c + 2*a*d)*sqrt(a + b/x) - (1/3)*(5*b*c + 2*a*d)*(a + b/x)^(3/2) - ((5*b*c + 2*a*d)*(a + b/x)^(5/2))/(5*a) + (c*(a + b/x)^(7/2)*x)/a + a^(3/2)*(5*b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(5/2), x, 6, -5*a*b*sqrt(a + b/x) - (5*b*(a + b/x)^(3/2))/3 + (a + b/x)^(5/2)*x + 5*a^(3/2)*b*atanh(sqrt(a + b/x)/sqrt(a))]
    @test_int [(a + b/x)^(5/2)/(c + d/x), x, 8, -((b*(2*b*c + a*d)*sqrt(a + b/x))/(c*d)) + (a*(a + b/x)^(3/2)*x)/c + (2*(b*c - a*d)^(5/2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^2*d^(3/2)) + (a^(3/2)*(5*b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^2]
    @test_int [(a + b/x)^(5/2)/(c + d/x)^2, x, 8, ((b*c - 2*a*d)*(b*c - a*d)*sqrt(a + b/x))/(c^2*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)/(c*(c + d/x)) - ((b*c - a*d)^(3/2)*(b*c + 4*a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*d^(3/2)) + (a^(3/2)*(5*b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^3]
    @test_int [(a + b/x)^(5/2)/(c + d/x)^3, x, 9, ((b*c - 3*a*d)*(b*c - a*d)*sqrt(a + b/x))/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*a*b*c*d - 12*a^2*d^2)*sqrt(a + b/x))/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)/(c*(c + d/x)^2) - (sqrt(b*c - a*d)*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/c^4]

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

    @test_int [(c + d/x)^3/sqrt(a + b/x), x, 5, -(d*sqrt(a + b/x)*(2*(3*b^2*c^2 + 9*a*b*c*d - 2*a^2*d^2) + (b*d*(3*b*c + 2*a*d))/x))/(3*a*b^2) + (c*sqrt(a + b/x)*(c + d/x)^2*x)/a - (c^2*(b*c - 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(3/2)]
    @test_int [(c + d/x)^2/sqrt(a + b/x), x, 5, (-2*d^2*sqrt(a + b/x))/b + (c^2*sqrt(a + b/x)*x)/a - (c*(b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(3/2)]
    @test_int [(c + d/x)/sqrt(a + b/x), x, 4, (c*sqrt(a + b/x)*x)/a - ((b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(3/2)]
    @test_int [1/sqrt(a + b/x), x, 4, (sqrt(a + b/x)*x)/a - (b*atanh(sqrt(a + b/x)/sqrt(a)))/a^(3/2)]
    @test_int [1/(sqrt(a + b/x)*(c + d/x)), x, 7, (sqrt(a + b/x)*x)/(a*c) - (2*d^(3/2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^2*sqrt(b*c - a*d)) - ((b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(3/2)*c^2)]
    @test_int [1/(sqrt(a + b/x)*(c + d/x)^2), x, 8, (d*(b*c - 2*a*d)*sqrt(a + b/x))/(a*c^2*(b*c - a*d)*(c + d/x)) + (sqrt(a + b/x)*x)/(a*c*(c + d/x)) - (d^(3/2)*(5*b*c - 4*a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(3/2)*c^3)]
    @test_int [1/(sqrt(a + b/x)*(c + d/x)^3), x, 9, (d*(2*b*c - 3*a*d)*sqrt(a + b/x))/(2*a*c^2*(b*c - a*d)*(c + d/x)^2) + (d*(b*c - 4*a*d)*(4*b*c - 3*a*d)*sqrt(a + b/x))/(4*a*c^3*(b*c - a*d)^2*(c + d/x)) + (sqrt(a + b/x)*x)/(a*c*(c + d/x)^2) - (d^(3/2)*(35*b^2*c^2 - 56*a*b*c*d + 24*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*(b*c - a*d)^(5/2)) - ((b*c + 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(3/2)*c^4)]

    @test_int [(c + d/x)^3/(a + b/x)^(3/2), x, 5, ((b*c - 2*a*d)*(3*b^2*c^2 - 2*a*b*c*d + 2*a^2*d^2) - (a*b*d^2*(b*c + 2*a*d))/x)/(a^2*b^2*sqrt(a + b/x)) + (c*(c + d/x)^2*x)/(a*sqrt(a + b/x)) - (3*c^2*(b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(5/2)]
    @test_int [(c + d/x)^2/(a + b/x)^(3/2), x, 5, (2*a^2*d^2 + b*c*(3*b*c - 4*a*d))/(a^2*b*sqrt(a + b/x)) + (c^2*x)/(a*sqrt(a + b/x)) - (c*(3*b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(5/2)]
    @test_int [(c + d/x)/(a + b/x)^(3/2), x, 5, (3*b*c - 2*a*d)/(a^2*sqrt(a + b/x)) + (c*x)/(a*sqrt(a + b/x)) - ((3*b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(5/2)]
    @test_int [1/(a + b/x)^(3/2), x, 5, (3*b)/(a^2*sqrt(a + b/x)) + x/(a*sqrt(a + b/x)) - (3*b*atanh(sqrt(a + b/x)/sqrt(a)))/a^(5/2)]
    @test_int [1/((a + b/x)^(3/2)*(c + d/x)), x, 8, (b*(3*b*c - a*d))/(a^2*c*(b*c - a*d)*sqrt(a + b/x)) + x/(a*c*sqrt(a + b/x)) + (2*d^(5/2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^2*(b*c - a*d)^(3/2)) - ((3*b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(5/2)*c^2)]
    @test_int [1/((a + b/x)^(3/2)*(c + d/x)^2), x, 9, (b*(3*b^2*c^2 - 2*a*b*c*d + 2*a^2*d^2))/(a^2*c^2*(b*c - a*d)^2*sqrt(a + b/x)) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*sqrt(a + b/x)*(c + d/x)) + x/(a*c*sqrt(a + b/x)*(c + d/x)) + (d^(5/2)*(7*b*c - 4*a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*(b*c - a*d)^(5/2)) - ((3*b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(5/2)*c^3)]
    @test_int [1/((a + b/x)^(3/2)*(c + d/x)^3), x, 10, (3*b*(2*b*c - a*d)*(2*b^2*c^2 - a*b*c*d + 4*a^2*d^2))/(4*a^2*c^3*(b*c - a*d)^3*sqrt(a + b/x)) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*sqrt(a + b/x)*(c + d/x)^2) + (d*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*sqrt(a + b/x)*(c + d/x)) + x/(a*c*sqrt(a + b/x)*(c + d/x)^2) + (3*d^(5/2)*(21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*(b*c - a*d)^(7/2)) - (3*(b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(5/2)*c^4)]

    @test_int [(c + d/x)^3/(a + b/x)^(5/2), x, 5, (c*(c + d/x)^2*x)/(a*(a + b/x)^(3/2)) + ((b*c - a*d)*(15*b^3*c^2 - 4*a^3*d^2*x - 2*a^2*b*d*(3*d + 5*c*x) + a*b^2*c*(-3*d + 20*c*x)))/(3*a^3*b^2*(a + b/x)^(3/2)*x) - (c^2*(5*b*c - 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(7/2)]
    @test_int [(c + d/x)^2/(a + b/x)^(5/2), x, 6, (2*a^2*d^2 + b*c*(5*b*c - 4*a*d))/(3*a^2*b*(a + b/x)^(3/2)) + (c*(5*b*c - 4*a*d))/(a^3*sqrt(a + b/x)) + (c^2*x)/(a*(a + b/x)^(3/2)) - (c*(5*b*c - 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(7/2)]
    @test_int [(c + d/x)/(a + b/x)^(5/2), x, 6, (5*b*c - 2*a*d)/(3*a^2*(a + b/x)^(3/2)) + (5*b*c - 2*a*d)/(a^3*sqrt(a + b/x)) + (c*x)/(a*(a + b/x)^(3/2)) - ((5*b*c - 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/a^(7/2)]
    @test_int [1/(a + b/x)^(5/2), x, 6, (5*b)/(3*a^2*(a + b/x)^(3/2)) + (5*b)/(a^3*sqrt(a + b/x)) + x/(a*(a + b/x)^(3/2)) - (5*b*atanh(sqrt(a + b/x)/sqrt(a)))/a^(7/2)]
    @test_int [1/((a + b/x)^(5/2)*(c + d/x)), x, 9, (b*(5*b*c - 3*a*d))/(3*a^2*c*(b*c - a*d)*(a + b/x)^(3/2)) + (b*(5*b^2*c^2 - 8*a*b*c*d + a^2*d^2))/(a^3*c*(b*c - a*d)^2*sqrt(a + b/x)) + x/(a*c*(a + b/x)^(3/2)) - (2*d^(7/2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^2*(b*c - a*d)^(5/2)) - ((5*b*c + 2*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(7/2)*c^2)]
    @test_int [1/((a + b/x)^(5/2)*(c + d/x)^2), x, 10, (b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)) + (b*(b*c - 2*a*d)*(5*b^2*c^2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*sqrt(a + b/x)) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(7/2)*c^3)]
    @test_int [1/((a + b/x)^(5/2)*(c + d/x)^3), x, 11, (b*(20*b^3*c^3 - 36*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(12*a^2*c^3*(b*c - a*d)^3*(a + b/x)^(3/2)) + (b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 - 35*a^3*b*c*d^3 + 12*a^4*d^4))/(4*a^3*c^3*(b*c - a*d)^4*sqrt(a + b/x)) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)^2) + (d*(4*b^2*c^2 - 23*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*(a + b/x)^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) - (d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*atan((sqrt(d)*sqrt(a + b/x))/sqrt(b*c - a*d)))/(4*c^4*(b*c - a*d)^(9/2)) - ((5*b*c + 6*a*d)*atanh(sqrt(a + b/x)/sqrt(a)))/(a^(7/2)*c^4)]

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

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

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b/x)^p*(c+d/x)^q*with*p*and*q*symbolic=#

    @test_int [(a + b/x)^p*(c + d/x)^q, x, 3, -((b*(a + b/x)^(1 + p)*(c + d/x)^q*AppellF1(1 + p, -q, 2, 2 + p, -((d*(a + b/x))/(b*c - a*d)), (a + b/x)/a))/(((b*(c + d/x))/(b*c - a*d))^q*(a^2*(1 + p))))]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    #= ::Subsection:: =#
    #=Integrands*of*the*form*(a+b/x^3)^p*(c+d/x^3)^q*with*p*and*q*symbolic=#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^p*(c+d*x^n)^q*with*fractional*n=#

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

    #= ::Subsubsection::Closed:: =#
    #=n>0=#

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

    @test_int [(1 + x^(2/3))/(-1 + x^(2/3)), x, 4, 6*x^(1/3) + x - 6*atanh(x^(1/3))]
    @test_int [(-16 + x^(3/4))/(16 + x^(3/4)), x, 9, -128*x^(1/4) + x - (256*2^(1/3)*atan((2^(1/3) - x^(1/4))/(2^(1/3)*sqrt(3))))/sqrt(3) + (256/3)*2^(1/3)*log(2*2^(1/3) + x^(1/4)) - (128/3)*2^(1/3)*log(4*2^(2/3) - 2*2^(1/3)*x^(1/4) + sqrt(x))]

    #= ::Subsubsection::Closed:: =#
    #=n<0=#

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

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^p*(c+d*x^n)^q*with*b*c+a*d=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^(p/2)*(c+d*x^n)^(p/2)*with*b*c+a*d=0=#

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^n)^p*(c+d*x^n)^p*with*b*c+a*d=0*and*p*symbolic=#

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

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

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

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

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

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

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

    @test_int [1/(a + b*x^n)*(c + d*x^n)^4, x, 5, If(13>=8, -((d*(a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - b^3*c^3*(1 + 7*n + 18*n^2 + 24*n^3) - a^2*b*c*d^2*(3 + 19*n + 38*n^2 + 24*n^3) + a*b^2*c^2*d*(3 + 20*n + 45*n^2 + 36*n^3))*x)/(b^4*(1 + n)*(1 + 2*n)*(1 + 3*n))) - (d*(2*a*b*c*d*(1 + 3*n)^2 - a^2*d^2*(1 + 5*n + 6*n^2) - b^2*c^2*(1 + 7*n + 18*n^2))*x*(c + d*x^n))/(b^3*(1 + n)*(1 + 2*n)*(1 + 3*n)) - (d*(a*d*(1 + 3*n) - b*(c + 6*c*n))*x*(c + d*x^n)^2)/(b^2*(1 + 5*n + 6*n^2)) + (d*x*(c + d*x^n)^3)/(b*(1 + 3*n)) + ((b*c - a*d)^4*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b^4), -((d*(a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - b^3*c^3*(1 + 7*n + 18*n^2 + 24*n^3) - a^2*b*c*d^2*(3 + 19*n + 38*n^2 + 24*n^3) + a*b^2*c^2*d*(3 + 20*n + 45*n^2 + 36*n^3))*x)/(b^4*(1 + 6*n + 11*n^2 + 6*n^3))) - (d*(2*a*b*c*d*(1 + 3*n)^2 - a^2*d^2*(1 + 5*n + 6*n^2) - b^2*c^2*(1 + 7*n + 18*n^2))*x*(c + d*x^n))/(b^3*(1 + 6*n + 11*n^2 + 6*n^3)) - (d*(a*d*(1 + 3*n) - b*(c + 6*c*n))*x*(c + d*x^n)^2)/(b^2*(1 + 5*n + 6*n^2)) + (d*x*(c + d*x^n)^3)/(b*(1 + 3*n)) + ((b*c - a*d)^4*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b^4))]
    @test_int [1/(a + b*x^n)*(c + d*x^n)^3, x, 4, If(13>=8, (d*(a^2*d^2*(1 + 3*n + 2*n^2) + b^2*c^2*(1 + 4*n + 6*n^2) - a*b*c*d*(2 + 7*n + 6*n^2))*x)/(b^3*(1 + n)*(1 + 2*n)) - (d*(a*d*(1 + 2*n) - b*(c + 4*c*n))*x*(c + d*x^n))/(b^2*(1 + n)*(1 + 2*n)) + (d*x*(c + d*x^n)^2)/(b*(1 + 2*n)) + ((b*c - a*d)^3*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b^3), (d*(a^2*d^2*(1 + 3*n + 2*n^2) + b^2*c^2*(1 + 4*n + 6*n^2) - a*b*c*d*(2 + 7*n + 6*n^2))*x)/(b^3*(1 + 3*n + 2*n^2)) - (d*(a*d*(1 + 2*n) - b*(c + 4*c*n))*x*(c + d*x^n))/(b^2*(1 + 3*n + 2*n^2)) + (d*x*(c + d*x^n)^2)/(b*(1 + 2*n)) + ((b*c - a*d)^3*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b^3))]
    @test_int [1/(a + b*x^n)*(c + d*x^n)^2, x, 3, -((d*(a*d*(1 + n) - b*(c + 2*c*n))*x)/(b^2*(1 + n))) + (d*x*(c + d*x^n))/(b*(1 + n)) + ((b*c - a*d)^2*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b^2)]
    @test_int [1/(a + b*x^n)*(c + d*x^n)^1, x, 2, (d*x)/b + ((b*c - a*d)*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*b)]
    @test_int [1/(a + b*x^n)/(c + d*x^n)^1, x, 3, (b*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*(b*c - a*d)) - (d*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(c*(b*c - a*d))]
    @test_int [1/(a + b*x^n)/(c + d*x^n)^2, x, 4, -((d*x)/(c*(b*c - a*d)*n*(c + d*x^n))) + (b^2*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*(b*c - a*d)^2) + (d*(b*c*(1 - 2*n) - a*d*(1 - n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(c^2*(b*c - a*d)^2*n)]
    @test_int [1/(a + b*x^n)/(c + d*x^n)^3, x, 5, -((d*x)/(2*c*(b*c - a*d)*n*(c + d*x^n)^2)) - (d*(a*d*(1 - 2*n) - b*(c - 4*c*n))*x)/(2*c^2*(b*c - a*d)^2*n^2*(c + d*x^n)) + (b^3*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a*(b*c - a*d)^3) - (d*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*b*c*d*(1 - 4*n + 3*n^2) + b^2*c^2*(1 - 5*n + 6*n^2))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(2*c^3*(b*c - a*d)^3*n^2)]

    @test_int [1/(a + b*x^n)^2*(c + d*x^n)^4, x, 5, If(13>=8, -((d*(b^3*c^3*(1 + 3*n + 2*n^2) - a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - a*b^2*c^2*d*(3 + 12*n + 17*n^2 + 12*n^3) + a^2*b*c*d^2*(3 + 15*n + 26*n^2 + 16*n^3))*x)/(a*b^4*n*(1 + n)*(1 + 2*n))) - (d*(b^2*c^2*(1 + 3*n + 2*n^2) - 2*a*b*c*d*(1 + 4*n + 5*n^2) + a^2*d^2*(1 + 5*n + 6*n^2))*x*(c + d*x^n))/(a*b^3*n*(1 + n)*(1 + 2*n)) + (d*(a*d*(1 + 3*n) - b*(c + 2*c*n))*x*(c + d*x^n)^2)/(a*b^2*n*(1 + 2*n)) + ((b*c - a*d)*x*(c + d*x^n)^3)/(a*b*n*(a + b*x^n)) - ((b*c - a*d)^3*(b*c*(1 - n) - a*d*(1 + 3*n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*b^4*n), -((1/(a*b^4*n*(1 + 3*n + 2*n^2)))*(d*(b^3*c^3*(1 + 3*n + 2*n^2) - a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - a*b^2*c^2*d*(3 + 12*n + 17*n^2 + 12*n^3) + a^2*b*c*d^2*(3 + 15*n + 26*n^2 + 16*n^3))*x)) - (d*(b^2*c^2*(1 + 3*n + 2*n^2) - 2*a*b*c*d*(1 + 4*n + 5*n^2) + a^2*d^2*(1 + 5*n + 6*n^2))*x*(c + d*x^n))/(a*b^3*n*(1 + 3*n + 2*n^2)) - (d*(b*c - a*d + 2*b*c*n - 3*a*d*n)*x*(c + d*x^n)^2)/(a*b^2*n*(1 + 2*n)) + ((b*c - a*d)*x*(c + d*x^n)^3)/(a*b*n*(a + b*x^n)) - ((b*c - a*d)^3*(b*c*(1 - n) - a*d*(1 + 3*n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*b^4*n))]
    @test_int [1/(a + b*x^n)^2*(c + d*x^n)^3, x, 4, -((d*(b^2*c^2*(1 + n) + a^2*d^2*(1 + 3*n + 2*n^2) - a*b*c*d*(2 + 4*n + 3*n^2))*x)/(a*b^3*n*(1 + n))) - (d*(b*c*(1 + n) - a*d*(1 + 2*n))*x*(c + d*x^n))/(a*b^2*n*(1 + n)) + ((b*c - a*d)*x*(c + d*x^n)^2)/(a*b*n*(a + b*x^n)) - ((b*c - a*d)^2*(b*c*(1 - n) - a*d*(1 + 2*n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*b^3*n)]
    @test_int [1/(a + b*x^n)^2*(c + d*x^n)^2, x, 3, -((d*(b*c - a*d*(1 + n))*x)/(a*b^2*n)) + ((b*c - a*d)*x*(c + d*x^n))/(a*b*n*(a + b*x^n)) - ((b*c - a*d)*(b*c*(1 - n) - a*d*(1 + n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*b^2*n)]
    @test_int [1/(a + b*x^n)^2*(c + d*x^n)^1, x, 2, ((b*c - a*d)*x)/(a*b*n*(a + b*x^n)) + ((a*d - b*c*(1 - n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*b*n)]
    @test_int [1/(a + b*x^n)^2/(c + d*x^n)^1, x, 4, (b*x)/(a*(b*c - a*d)*n*(a + b*x^n)) + (b*(a*d*(1 - 2*n) - b*c*(1 - n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*(b*c - a*d)^2*n) + (d^2*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(c*(b*c - a*d)^2)]
    @test_int [1/(a + b*x^n)^2/(c + d*x^n)^2, x, 5, (d*(b*c + a*d)*x)/(a*c*(b*c - a*d)^2*n*(c + d*x^n)) + (b*x)/(a*(b*c - a*d)*n*(a + b*x^n)*(c + d*x^n)) + (b^2*(a*d*(1 - 3*n) - b*(c - c*n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*(b*c - a*d)^3*n) - (d^2*(b*c*(1 - 3*n) - a*d*(1 - n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(c^2*(b*c - a*d)^3*n)]
    @test_int [1/(a + b*x^n)^2/(c + d*x^n)^3, x, 6, (d*(2*b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*n*(c + d*x^n)^2) + (b*x)/(a*(b*c - a*d)*n*(a + b*x^n)*(c + d*x^n)^2) - (d*(a*b*c*d*(1 - 6*n) - a^2*d^2*(1 - 2*n) - 2*b^2*c^2*n)*x)/(2*a*c^2*(b*c - a*d)^3*n^2*(c + d*x^n)) + (b^3*(a*d*(1 - 4*n) - b*c*(1 - n))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((b*x^n)/a)))/(a^2*(b*c - a*d)^4*n) + (d^2*(a^2*d^2*(1 - 3*n + 2*n^2) - 2*a*b*c*d*(1 - 5*n + 4*n^2) + b^2*c^2*(1 - 7*n + 12*n^2))*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -((d*x^n)/c)))/(2*c^3*(b*c - a*d)^4*n^2)]

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

    @test_int [(a + b*x^n)^p*(c + d*x^n)^q, x, 3, (x*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1(1/n, -p, -q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)))/((1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)]

    @test_int [(a + b*x^n)^p*(c + d*x^n)^3, x, 5, (d*(a^2*d^2*(1 + 3*n + 2*n^2) - a*b*c*d*(2 + n^2*(7 + p) + n*(9 + 2*p)) + b^2*c^2*(1 + 2*n*(3 + p) + n^2*(11 + 6*p + p^2)))*x*(a + b*x^n)^(1 + p))/(b^3*(1 + n + n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))) - (d*(a*d*(1 + 2*n) - b*c*(1 + n*(5 + p)))*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b^2*(1 + n*(2 + p))*(1 + n*(3 + p))) + (d*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^2)/(b*(1 + 3*n + n*p)) - (1/(b^3*(1 + n + n*p)*(1 + n*(2 + p))*(1 + n*(3 + p))))*(((a^3*d^3*(1 + 3*n + 2*n^2) - 3*a^2*b*c*d^2*(1 + n)*(1 + n*(3 + p)) + 3*a*b^2*c^2*d*(1 + n*(5 + 2*p) + n^2*(6 + 5*p + p^2)) - b^3*c^3*(1 + 3*n*(2 + p) + n^2*(11 + 12*p + 3*p^2) + n^3*(6 + 11*p + 6*p^2 + p^3)))*x*(a + b*x^n)^p*HypergeometricFunctions._₂F₁(1/n, -p, 1 + 1/n, -((b*x^n)/a)))/(1 + (b*x^n)/a)^p)]
    @test_int [(a + b*x^n)^p*(c + d*x^n)^2, x, 4, -((d*(a*d*(1 + n) - b*c*(1 + (3 + p)*n))*x*(a + b*x^n)^(1 + p))/(b^2*(1 + n + p*n)*(1 + (2 + p)*n))) + (d*x*(a + b*x^n)^(1 + p)*(c + d*x^n))/(b*(1 + 2*n + p*n)) - ((b*c*(1 + n + p*n)*(a*d - b*c*(1 + (2 + p)*n)) - a*d*(a*d*(1 + n) - b*c*(1 + (3 + p)*n)))*x*(a + b*x^n)^p*HypergeometricFunctions._₂F₁(-p, 1/n, 1 + 1/n, -((b*x^n)/a)))/((1 + (b*x^n)/a)^p*(b^2*(1 + n + p*n)*(1 + (2 + p)*n)))]
    @test_int [(a + b*x^n)^p*(c + d*x^n)^1, x, 3, (d*x*(a + b*x^n)^(1 + p))/(b*(1 + n + p*n)) - ((a*d - b*c*(1 + n + p*n))*x*(a + b*x^n)^p*HypergeometricFunctions._₂F₁(-p, 1/n, 1 + 1/n, -((b*x^n)/a)))/((1 + (b*x^n)/a)^p*(b*(1 + n + p*n)))]
    @test_int [(a + b*x^n)^p*(c + d*x^n)^0, x, 2, (x*(a + b*x^n)^p*HypergeometricFunctions._₂F₁(-p, 1/n, 1 + 1/n, -((b*x^n)/a)))/(1 + (b*x^n)/a)^p]
    @test_int [(a + b*x^n)^p/(c + d*x^n)^1, x, 2, (x*(a + b*x^n)^p*AppellF1(1/n, -p, 1, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)))/((1 + (b*x^n)/a)^p*c)]
    @test_int [(a + b*x^n)^p/(c + d*x^n)^2, x, 2, (x*(a + b*x^n)^p*AppellF1(1/n, -p, 2, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)))/((1 + (b*x^n)/a)^p*c^2)]
    @test_int [(a + b*x^n)^p/(c + d*x^n)^3, x, 2, (x*(a + b*x^n)^p*AppellF1(1/n, -p, 3, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)))/((1 + (b*x^n)/a)^p*c^3)]

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

    @test_int [(a + b*x^n)^3/(c + d*x^n)^(1/n + 4), x, 4, If(13>=8, (x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(c*(1 + 3*n)) + (3*a*n*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c^2*(1 + 5*n + 6*n^2)) + (6*a^2*n^2*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)) + (6*a^3*n^3*x)/((c + d*x^n)^n^(-1)*(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n))), (x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(c*(1 + 3*n)) + (3*a*n*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c^2*(1 + 5*n + 6*n^2)) + (6*a^2*n^2*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^3*(1 + 6*n + 11*n^2 + 6*n^3)) + (6*a^3*n^3*x)/((c + d*x^n)^n^(-1)*(c^4*(1 + 6*n + 11*n^2 + 6*n^3))))]
    @test_int [(a + b*x^n)^2/(c + d*x^n)^(1/n + 3), x, 3, If(13>=8, (x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c*(1 + 2*n)) + (2*a*n*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^2*(1 + n)*(1 + 2*n)) + (2*a^2*n^2*x)/((c + d*x^n)^n^(-1)*(c^3*(1 + n)*(1 + 2*n))), (x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c*(1 + 2*n)) + (2*a*n*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^2*(1 + 3*n + 2*n^2)) + (2*a^2*n^2*x)/((c + d*x^n)^n^(-1)*(c^3*(1 + 3*n + 2*n^2))))]
    @test_int [(a + b*x^n)^1/(c + d*x^n)^(1/n + 2), x, 2, (x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c*(1 + n)) + (a*n*x)/((c + d*x^n)^n^(-1)*(c^2*(1 + n)))]
    @test_int [(a + b*x^n)^0/(c + d*x^n)^(1/n + 1), x, 1, x/((c + d*x^n)^n^(-1)*c)]
    @test_int [1/((a + b*x^n)^1*(c + d*x^n)^(1/n + 0)), x, 1, (x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*a)]
    @test_int [1/((a + b*x^n)^2*(c + d*x^n)^(1/n - 1)), x, 1, (c*x*HypergeometricFunctions._₂F₁(2, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*a^2)]
    @test_int [1/((a + b*x^n)^3*(c + d*x^n)^(1/n - 2)), x, 1, (c^2*x*HypergeometricFunctions._₂F₁(3, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*a^3)]

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

    @test_int [(a + b*x^n)^2/(c + d*x^n)^(1/n + 4), x, 5, If(13>=8, -((b*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(3*a*(b*c - a*d)*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(3*a*c*(b*c - a*d)*n*(1 + 3*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c^2*(b*c - a*d)*(1 + 5*n + 6*n^2)) - (2*a*n*(3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^3*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1 + 3*n)) - (2*a^2*n^2*(3*a*d*n - b*(c + 3*c*n))*x)/((c + d*x^n)^n^(-1)*(c^4*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1 + 3*n))), -((b*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(3*a*(b*c - a*d)*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - 1/n))/(3*a*c*(b*c - a*d)*n*(1 + 3*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - 1/n))/(c^2*(b*c - a*d)*(1 + 5*n + 6*n^2)) - (2*a*n*(3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)*(c + d*x^n)^(-1 - 1/n))/(c^3*(b*c - a*d)*(1 + 6*n + 11*n^2 + 6*n^3)) - (2*a^2*n^2*(3*a*d*n - b*(c + 3*c*n))*x)/((c + d*x^n)^n^(-1)*(c^4*(b*c - a*d)*(1 + 6*n + 11*n^2 + 6*n^3))))]
    @test_int [(a + b*x^n)^1/(c + d*x^n)^(1/n + 3), x, 3, -(((b*c - a*d)*x*(c + d*x^n)^(-2 - 1/n))/(c*d*(1 + 2*n))) + ((b*c + 2*a*d*n)*x*(c + d*x^n)^(-1 - 1/n))/(c^2*d*(1 + n)*(1 + 2*n)) + (n*(b*c + 2*a*d*n)*x)/((c + d*x^n)^n^(-1)*(c^3*d*(1 + n)*(1 + 2*n)))]
    @test_int [(a + b*x^n)^0/(c + d*x^n)^(1/n + 2), x, 2, (x*(c + d*x^n)^(-1 - 1/n))/(c*(1 + n)) + (n*x)/((c + d*x^n)^n^(-1)*(c^2*(1 + n)))]
    @test_int [1/((a + b*x^n)^1*(c + d*x^n)^(1/n + 1)), x, 2, -((d*x)/((c + d*x^n)^n^(-1)*(c*(b*c - a*d)))) + (b*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*(a*(b*c - a*d)))]
    @test_int [1/((a + b*x^n)^2*(c + d*x^n)^(1/n + 0)), x, 2, (b*x)/((c + d*x^n)^((1 - n)/n)*(a*(b*c - a*d)*n*(a + b*x^n))) - ((b*c*(1 - n) + a*d*n)*x*HypergeometricFunctions._₂F₁(1, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*(a^2*(b*c - a*d)*n))]
    @test_int [1/((a + b*x^n)^3*(c + d*x^n)^(1/n - 1)), x, 2, (b*x*(c + d*x^n)^(2 - 1/n))/(2*a*(b*c - a*d)*n*(a + b*x^n)^2) - (c*(b*c*(1 - 2*n) + 2*a*d*n)*x*HypergeometricFunctions._₂F₁(2, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*(2*a^3*(b*c - a*d)*n))]
    @test_int [1/((a + b*x^n)^4*(c + d*x^n)^(1/n - 2)), x, 2, (b*x*(c + d*x^n)^(3 - 1/n))/(3*a*(b*c - a*d)*n*(a + b*x^n)^3) - (c^2*(b*c*(1 - 3*n) + 3*a*d*n)*x*HypergeometricFunctions._₂F₁(3, 1/n, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))))/((c + d*x^n)^n^(-1)*(3*a^4*(b*c - a*d)*n))]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*u*(a1+b1*x^n)^p*(a2+b2*x^n)^p*with*a2*b1+a1*b2=0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x^n)^p*(a2+b2*x^n)^p*(c+d*x^(2*n))^q*with*a2*b1+a1*b2=0=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x)^(p/2)*(a2+b2*x)^(p/2)*(c+d*x^2)^q*with*a2*b1+a1*b2=0=#

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

    @test_int [x^5*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 6, (c^4*(b*c^2 + a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^8) + (c^2*(3*b*c^2 + 2*a*d^2)*(-c + d*x)^(5/2)*(c + d*x)^(5/2))/(5*d^8) + ((3*b*c^2 + a*d^2)*(-c + d*x)^(7/2)*(c + d*x)^(7/2))/(7*d^8) + (b*(-c + d*x)^(9/2)*(c + d*x)^(9/2))/(9*d^8), (8*c^4*(2*b*c^2 + 3*a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(315*d^8) + (4*c^2*(2*b*c^2 + 3*a*d^2)*x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(105*d^6) + ((2*b*c^2 + 3*a*d^2)*x^4*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(21*d^4) + (b*x^6*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(9*d^2)]
    @test_int [x^3*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 4, (c^2*(b*c^2 + a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^6) + ((2*b*c^2 + a*d^2)*(-c + d*x)^(5/2)*(c + d*x)^(5/2))/(5*d^6) + (b*(-c + d*x)^(7/2)*(c + d*x)^(7/2))/(7*d^6), (2*c^2*(4*b*c^2 + 7*a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(105*d^6) + ((4*b*c^2 + 7*a*d^2)*x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(35*d^4) + (b*x^4*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(7*d^2)]
    @test_int [x^1*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 2, ((b*c^2 + a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^4) + (b*(-c + d*x)^(5/2)*(c + d*x)^(5/2))/(5*d^4), ((2*b*c^2 + 5*a*d^2)*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(15*d^4) + (b*x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(5*d^2)]
    @test_int [(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x)/x^1, x, 5, a*sqrt(-c + d*x)*sqrt(c + d*x) + (b*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*d^2) - a*c*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c)]
    @test_int [(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x)/x^3, x, 5, b*sqrt(-c + d*x)*sqrt(c + d*x) - (a*sqrt(-c + d*x)*sqrt(c + d*x))/(2*x^2) - ((2*b*c^2 - a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(2*c), (1/2)*(2*b - (a*d^2)/c^2)*sqrt(-c + d*x)*sqrt(c + d*x) + (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(2*c^2*x^2) - ((2*b*c^2 - a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(2*c)]
    @test_int [(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x)/x^5, x, 5, (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(4*c^2*x^4) - ((4*b*c^2 + a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(8*c^2*x^2) + (d^2*(4*b*c^2 + a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(8*c^3), (d*(4*b*c^2 + a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(8*c^3*x) - ((4*b*c^2 + a*d^2)*sqrt(-c + d*x)*(c + d*x)^(3/2))/(8*c^3*x^2) + (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(4*c^2*x^4) + (d^2*(4*b*c^2 + a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(8*c^3)]

    @test_int [x^4*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 9, (c^4*(5*b*c^2 + 8*a*d^2)*x*sqrt(-c + d*x)*sqrt(c + d*x))/(128*d^6) + (c^2*(5*b*c^2 + 8*a*d^2)*x*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(64*d^6) + ((5*b*c^2 + 8*a*d^2)*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(48*d^4) + (b*x^5*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(8*d^2) - (c^6*(5*b*c^2 + 8*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/(64*d^7)]
    @test_int [x^2*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 7, (c^2*(b*c^2 + 2*a*d^2)*x*sqrt(-c + d*x)*sqrt(c + d*x))/(16*d^4) + ((b*c^2 + 2*a*d^2)*x*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(8*d^4) + (b*x^3*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(6*d^2) - (c^4*(b*c^2 + 2*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/(8*d^5)]
    @test_int [x^0*(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x), x, 5, ((b*c^2 + 4*a*d^2)*x*sqrt(-c + d*x)*sqrt(c + d*x))/(8*d^2) + (b*x*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(4*d^2) - (c^2*(b*c^2 + 4*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/(4*d^3)]
    @test_int [(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x)/x^2, x, 5, (1/2)*(b - (2*a*d^2)/c^2)*x*sqrt(-c + d*x)*sqrt(c + d*x) + (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(c^2*x) - ((b*c^2 - 2*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/d]
    @test_int [(a + b*x^2)*sqrt(-c + d*x)*sqrt(c + d*x)/x^4, x, 6, -((b*sqrt(-c + d*x)*sqrt(c + d*x))/x) + (a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(3*c^2*x^3) + 2*b*d*atanh(sqrt(-c + d*x)/sqrt(c + d*x))]

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

    @test_int [x^4*(a + b*x^2)/(sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 5, ((5*b + 6*a*c^2)*x*sqrt(-1 + c*x)*sqrt(1 + c*x))/(16*c^6) + ((5*b + 6*a*c^2)*x^3*sqrt(-1 + c*x)*sqrt(1 + c*x))/(24*c^4) + (b*x^5*sqrt(-1 + c*x)*sqrt(1 + c*x))/(6*c^2) + ((5*b + 6*a*c^2)*acosh(c*x))/(16*c^7)]
    @test_int [x^3*(a + b*x^2)/(sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 4, (2*(4*b + 5*a*c^2)*sqrt(-1 + c*x)*sqrt(1 + c*x))/(15*c^6) + ((4*b + 5*a*c^2)*x^2*sqrt(-1 + c*x)*sqrt(1 + c*x))/(15*c^4) + (b*x^4*sqrt(-1 + c*x)*sqrt(1 + c*x))/(5*c^2)]
    @test_int [x^2*(a + b*x^2)/(sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 3, ((3*b + 4*a*c^2)*x*sqrt(-1 + c*x)*sqrt(1 + c*x))/(8*c^4) + (b*x^3*sqrt(-1 + c*x)*sqrt(1 + c*x))/(4*c^2) + ((3*b + 4*a*c^2)*acosh(c*x))/(8*c^5)]
    @test_int [x^1*(a + b*x^2)/(sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 2, ((2*b + 3*a*c^2)*sqrt(-1 + c*x)*sqrt(1 + c*x))/(3*c^4) + (b*x^2*sqrt(-1 + c*x)*sqrt(1 + c*x))/(3*c^2)]
    @test_int [x^0*(a + b*x^2)/(sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 2, (b*x*sqrt(-1 + c*x)*sqrt(1 + c*x))/(2*c^2) + ((b + 2*a*c^2)*acosh(c*x))/(2*c^3)]
    @test_int [(a + b*x^2)/(x^1*sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 3, (b*sqrt(-1 + c*x)*sqrt(1 + c*x))/c^2 + a*atan(sqrt(-1 + c*x)*sqrt(1 + c*x))]
    @test_int [(a + b*x^2)/(x^2*sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 2, (a*sqrt(-1 + c*x)*sqrt(1 + c*x))/x + (b*acosh(c*x))/c]
    @test_int [(a + b*x^2)/(x^3*sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 3, (a*sqrt(-1 + c*x)*sqrt(1 + c*x))/(2*x^2) + (1/2)*(2*b + a*c^2)*atan(sqrt(-1 + c*x)*sqrt(1 + c*x))]
    @test_int [(a + b*x^2)/(x^4*sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 2, (a*sqrt(-1 + c*x)*sqrt(1 + c*x))/(3*x^3) + ((3*b + 2*a*c^2)*sqrt(-1 + c*x)*sqrt(1 + c*x))/(3*x)]
    @test_int [(a + b*x^2)/(x^5*sqrt(-1 + c*x)*sqrt(1 + c*x)), x, 5, (a*sqrt(-1 + c*x)*sqrt(1 + c*x))/(4*x^4) + ((4*b + 3*a*c^2)*sqrt(-1 + c*x)*sqrt(1 + c*x))/(8*x^2) + (1/8)*c^2*(4*b + 3*a*c^2)*atan(sqrt(-1 + c*x)*sqrt(1 + c*x))]

    @test_int [x^4*(a + b*x^2)/(sqrt(-c + d*x)*sqrt(c + d*x)), x, 8, (c^2*(5*b*c^2 + 6*a*d^2)*x*sqrt(-c + d*x)*sqrt(c + d*x))/(16*d^6) + ((5*b*c^2 + 6*a*d^2)*x^3*sqrt(-c + d*x)*sqrt(c + d*x))/(24*d^4) + (b*x^5*sqrt(-c + d*x)*sqrt(c + d*x))/(6*d^2) + (c^4*(5*b*c^2 + 6*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/(8*d^7)]
    @test_int [x^3*(a + b*x^2)/(sqrt(-c + d*x)*sqrt(c + d*x)), x, 4, (2*c^2*(4*b*c^2 + 5*a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(15*d^6) + ((4*b*c^2 + 5*a*d^2)*x^2*sqrt(-c + d*x)*sqrt(c + d*x))/(15*d^4) + (b*x^4*sqrt(-c + d*x)*sqrt(c + d*x))/(5*d^2)]
    @test_int [x^2*(a + b*x^2)/(sqrt(-c + d*x)*sqrt(c + d*x)), x, 6, ((3*b*c^2 + 4*a*d^2)*x*sqrt(-c + d*x)*sqrt(c + d*x))/(8*d^4) + (b*x^3*sqrt(-c + d*x)*sqrt(c + d*x))/(4*d^2) + (c^2*(3*b*c^2 + 4*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/(4*d^5)]
    @test_int [x^1*(a + b*x^2)/(sqrt(-c + d*x)*sqrt(c + d*x)), x, 2, ((2*b*c^2 + 3*a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(3*d^4) + (b*x^2*sqrt(-c + d*x)*sqrt(c + d*x))/(3*d^2)]
    @test_int [x^0*(a + b*x^2)/(sqrt(-c + d*x)*sqrt(c + d*x)), x, 4, (b*x*sqrt(-c + d*x)*sqrt(c + d*x))/(2*d^2) + ((b*c^2 + 2*a*d^2)*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/d^3]
    @test_int [(a + b*x^2)/(x^1*sqrt(-c + d*x)*sqrt(c + d*x)), x, 3, (b*sqrt(-c + d*x)*sqrt(c + d*x))/d^2 + (a*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/c]
    @test_int [(a + b*x^2)/(x^2*sqrt(-c + d*x)*sqrt(c + d*x)), x, 4, (a*sqrt(-c + d*x)*sqrt(c + d*x))/(c^2*x) + (2*b*atanh(sqrt(-c + d*x)/sqrt(c + d*x)))/d]
    @test_int [(a + b*x^2)/(x^3*sqrt(-c + d*x)*sqrt(c + d*x)), x, 3, (a*sqrt(-c + d*x)*sqrt(c + d*x))/(2*c^2*x^2) + ((2*b*c^2 + a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(2*c^3)]
    @test_int [(a + b*x^2)/(x^4*sqrt(-c + d*x)*sqrt(c + d*x)), x, 2, (a*sqrt(-c + d*x)*sqrt(c + d*x))/(3*c^2*x^3) + ((3*b*c^2 + 2*a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(3*c^4*x)]
    @test_int [(a + b*x^2)/(x^5*sqrt(-c + d*x)*sqrt(c + d*x)), x, 5, (a*sqrt(-c + d*x)*sqrt(c + d*x))/(4*c^2*x^4) + ((4*b*c^2 + 3*a*d^2)*sqrt(-c + d*x)*sqrt(c + d*x))/(8*c^4*x^2) + (d^2*(4*b*c^2 + 3*a*d^2)*atan((sqrt(-c + d*x)*sqrt(c + d*x))/c))/(8*c^5)]

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x)^(p/2)*(a2+b2*x)^(p/2)*(c+d*x^2)^q*with*a2*b1+a1*b2=0*with*m*symbolic=#

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x^(1/2))^p*(a2+b2*x^(1/2))^p*(c+d*x)^q*with*a2*b1+a1*b2=0=#

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x^n)^p*(a2+b2*x^n)^p*(c+d*x^(2*n))^q*with*a2*b1+a1*b2=0*and*n*symbolic=#

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

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

    @test_int [(c + d*x^(2*n))^p/((a + b*x^n)*(a - b*x^n)), x, 3, (x*(c + d*x^(2*n))^p*AppellF1(1/(2*n), 1, -p, (1/2)*(2 + 1/n), (b^2*x^(2*n))/a^2, -((d*x^(2*n))/c)))/((1 + (d*x^(2*n))/c)^p*a^2)]

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

    #= ::Section:: =#
    #=Integrands*of*the*form*(e*x)^m*(a1+b1*x^n)^p*(a2+b2*x^n)^p*(c+d*x^(2*n)+e*x^(4*n))^q*with*a2*b1+a1*b2=0=#
    end

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.3 General/1.1.3.3 (a+b x^n)^p (c+d x^n)^q.jl