text:
@testset "1.1.2.3 (a+b x^2)^p (c+d x^2)^q" begin
    (a, b, c, d, m, p, q, x, ) = @variables a b c d m p q x

    #= ::Package:: =#

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

    #= ::Section::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)^4, x, 2, a*c^4*x + (1/3)*c^3*(b*c + 4*a*d)*x^3 + (2/5)*c^2*d*(2*b*c + 3*a*d)*x^5 + (2/7)*c*d^2*(3*b*c + 2*a*d)*x^7 + (1/9)*d^3*(4*b*c + a*d)*x^9 + (1/11)*b*d^4*x^11]
    @test_int [(a + b*x^2)*(c + d*x^2)^3, x, 2, a*c^3*x + (1/3)*c^2*(b*c + 3*a*d)*x^3 + (3/5)*c*d*(b*c + a*d)*x^5 + (1/7)*d^2*(3*b*c + a*d)*x^7 + (1/9)*b*d^3*x^9]
    @test_int [(a + b*x^2)*(c + d*x^2)^2, x, 2, a*c^2*x + (1/3)*c*(b*c + 2*a*d)*x^3 + (1/5)*d*(2*b*c + a*d)*x^5 + (1/7)*b*d^2*x^7]
    @test_int [(a + b*x^2)*(c + d*x^2)^1, x, 2, a*c*x + (1/3)*(b*c + a*d)*x^3 + (1/5)*b*d*x^5]
    @test_int [(a + b*x^2)/(c + d*x^2)^1, x, 2, (b*x)/d - ((b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*d^(3/2))]
    @test_int [(a + b*x^2)/(c + d*x^2)^2, x, 2, -(((b*c - a*d)*x)/(2*c*d*(c + d*x^2))) + ((b*c + a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*d^(3/2))]
    @test_int [(a + b*x^2)/(c + d*x^2)^3, x, 3, -(((b*c - a*d)*x)/(4*c*d*(c + d*x^2)^2)) + ((b*c + 3*a*d)*x)/(8*c^2*d*(c + d*x^2)) + ((b*c + 3*a*d)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*d^(3/2))]

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

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

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

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

    @test_int [1/(a + b*x^2)^2*(c + d*x^2)^5, x, 4, (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^3)/(3*b^4) + (d^4*(5*b*c - 2*a*d)*x^5)/(5*b^3) + (d^5*x^7)/(7*b^2) + ((b*c - a*d)^5*x)/(2*a*b^5*(a + b*x^2)) + ((b*c - a*d)^4*(b*c + 9*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(11/2))]
    @test_int [1/(a + b*x^2)^2*(c + d*x^2)^4, x, 4, (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^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(9/2))]
    @test_int [1/(a + b*x^2)^2*(c + d*x^2)^3, x, 4, (d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*c + 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(7/2))]
    @test_int [1/(a + b*x^2)^2*(c + d*x^2)^2, x, 4, (d^2*x)/b^2 + ((b*c - a*d)^2*x)/(2*a*b^2*(a + b*x^2)) + ((b*c - a*d)*(b*c + 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(5/2))]
    @test_int [1/(a + b*x^2)^2*(c + d*x^2)^1, x, 2, ((b*c - a*d)*x)/(2*a*b*(a + b*x^2)) + ((b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*b^(3/2))]
    @test_int [1/(a + b*x^2)^2/(c + d*x^2)^1, x, 4, (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)) + (sqrt(b)*(b*c - 3*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^2) + (d^(3/2)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*(b*c - a*d)^2)]
    @test_int [1/(a + b*x^2)^2/(c + d*x^2)^2, x, 5, (d*(b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*(c + d*x^2)) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(b*c - 5*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^3) + (d^(3/2)*(5*b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*(b*c - a*d)^3)]
    @test_int [1/(a + b*x^2)^2/(c + d*x^2)^3, x, 6, (d*(2*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x)/(2*a*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) + (d*(4*b*c - a*d)*(b*c + 3*a*d)*x)/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) + (b^(5/2)*(b*c - 7*a*d)*atan((sqrt(b)*x)/sqrt(a)))/(2*a^(3/2)*(b*c - a*d)^4) + (d^(3/2)*(35*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*(b*c - a*d)^4)]

    @test_int [1/(a + b*x^2)^3*(c + d*x^2)^5, x, 5, (d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c - a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(11/2))]
    @test_int [1/(a + b*x^2)^3*(c + d*x^2)^4, x, 5, (d^3*(4*b*c - 3*a*d)*x)/b^4 + (d^4*x^3)/(3*b^3) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^2)^2) + ((b*c - a*d)^3*(3*b*c + 13*a*d)*x)/(8*a^2*b^4*(a + b*x^2)) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(9/2))]
    @test_int [1/(a + b*x^2)^3*(c + d*x^2)^3, x, 5, (d^3*x)/b^3 + ((b*c - a*d)^3*x)/(4*a*b^3*(a + b*x^2)^2) + (3*(b*c - a*d)^2*(b*c + 3*a*d)*x)/(8*a^2*b^3*(a + b*x^2)) + (3*(b*c - a*d)*(4*a^2*d^2 + (b*c + a*d)^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(7/2))]
    @test_int [1/(a + b*x^2)^3*(c + d*x^2)^2, x, 3, (3*(c^2/a^2 - d^2/b^2)*x)/(8*(a + b*x^2)) + ((b*c - a*d)*x*(c + d*x^2))/(4*a*b*(a + b*x^2)^2) + ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(5/2))]
    @test_int [1/(a + b*x^2)^3*(c + d*x^2)^1, x, 3, ((b*c - a*d)*x)/(4*a*b*(a + b*x^2)^2) + ((3*b*c + a*d)*x)/(8*a^2*b*(a + b*x^2)) + ((3*b*c + a*d)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*b^(3/2))]
    @test_int [1/(a + b*x^2)^3/(c + d*x^2)^1, x, 5, (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (b*(3*b*c - 7*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)) + (sqrt(b)*(3*b^2*c^2 - 10*a*b*c*d + 15*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*(b*c - a*d)^3) - (d^(5/2)*atan((sqrt(d)*x)/sqrt(c)))/(sqrt(c)*(b*c - a*d)^3)]
    @test_int [1/(a + b*x^2)^3/(c + d*x^2)^2, x, 6, (d*(b*c - 4*a*d)*(3*b*c + a*d)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)) + (3*b*(b*c - 3*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)) + (b^(3/2)*(3*b^2*c^2 - 14*a*b*c*d + 35*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*(b*c - a*d)^4) - (d^(5/2)*(7*b*c - a*d)*atan((sqrt(d)*x)/sqrt(c)))/(2*c^(3/2)*(b*c - a*d)^4)]
    @test_int [1/(a + b*x^2)^3/(c + d*x^2)^3, x, 7, (d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2) + (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*atan((sqrt(b)*x)/sqrt(a)))/(8*a^(5/2)*(b*c - a*d)^5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*atan((sqrt(d)*x)/sqrt(c)))/(8*c^(5/2)*(b*c - a*d)^5)]

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

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

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

    @test_int [(a + b*x^2)^(1/2)*(c + d*x^2)^3, x, 6, ((64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*x*sqrt(a + b*x^2))/(128*b^3) + (d*(72*b^2*c^2 - 52*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^3) + (d*(12*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(48*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2)^2)/(8*b) + (a*(64*b^3*c^3 - 48*a*b^2*c^2*d + 24*a^2*b*c*d^2 - 5*a^3*d^3)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(7/2))]
    @test_int [(a + b*x^2)^(1/2)*(c + d*x^2)^2, x, 5, ((8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*x*sqrt(a + b*x^2))/(16*b^2) + (d*(8*b*c - 3*a*d)*x*(a + b*x^2)^(3/2))/(24*b^2) + (d*x*(a + b*x^2)^(3/2)*(c + d*x^2))/(6*b) + (a*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(5/2))]
    @test_int [(a + b*x^2)^(1/2)*(c + d*x^2)^1, x, 4, ((4*b*c - a*d)*x*sqrt(a + b*x^2))/(8*b) + (d*x*(a + b*x^2)^(3/2))/(4*b) + (a*(4*b*c - a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(3/2))]
    @test_int [(a + b*x^2)^(1/2)*(c + d*x^2)^0, x, 3, (1/2)*x*sqrt(a + b*x^2) + (a*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*sqrt(b))]
    @test_int [(a + b*x^2)^(1/2)/(c + d*x^2)^1, x, 5, (sqrt(b)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/d - (sqrt(b*c - a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(sqrt(c)*d)]
    @test_int [(a + b*x^2)^(1/2)/(c + d*x^2)^2, x, 3, (x*sqrt(a + b*x^2))/(2*c*(c + d*x^2)) + (a*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*sqrt(b*c - a*d))]
    @test_int [(a + b*x^2)^(1/2)/(c + d*x^2)^3, x, 4, -((d*x*(a + b*x^2)^(3/2))/(4*c*(b*c - a*d)*(c + d*x^2)^2)) + ((4*b*c - 3*a*d)*x*sqrt(a + b*x^2))/(8*c^2*(b*c - a*d)*(c + d*x^2)) + (a*(4*b*c - 3*a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*(b*c - a*d)^(3/2))]
    @test_int [(a + b*x^2)^(1/2)/(c + d*x^2)^4, x, 6, (x*sqrt(a + b*x^2))/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*sqrt(a + b*x^2))/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*sqrt(a + b*x^2))/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(16*c^(7/2)*(b*c - a*d)^(5/2))]

    @test_int [(a + b*x^2)^(3/2)*(c + d*x^2)^3, x, 7, (3*a*(4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*sqrt(a + b*x^2))/(256*b^3) + ((4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x*(a + b*x^2)^(3/2))/(128*b^3) + (d*(36*b^2*c^2 - 20*a*b*c*d + 5*a^2*d^2)*x*(a + b*x^2)^(5/2))/(160*b^3) + (d*(14*b*c - 5*a*d)*x*(a + b*x^2)^(5/2)*(c + d*x^2))/(80*b^2) + (d*x*(a + b*x^2)^(5/2)*(c + d*x^2)^2)/(10*b) + (3*a^2*(4*b*c - a*d)*(8*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*b^(7/2))]
    @test_int [(a + b*x^2)^(3/2)*(c + d*x^2)^2, x, 6, (a*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*x*sqrt(a + b*x^2))/(128*b^2) + ((48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(192*b^2) + (d*(10*b*c - 3*a*d)*x*(a + b*x^2)^(5/2))/(48*b^2) + (d*x*(a + b*x^2)^(5/2)*(c + d*x^2))/(8*b) + (a^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(5/2))]
    @test_int [(a + b*x^2)^(3/2)*(c + d*x^2)^1, x, 5, (a*(6*b*c - a*d)*x*sqrt(a + b*x^2))/(16*b) + ((6*b*c - a*d)*x*(a + b*x^2)^(3/2))/(24*b) + (d*x*(a + b*x^2)^(5/2))/(6*b) + (a^2*(6*b*c - a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(3/2))]
    @test_int [(a + b*x^2)^(3/2)*(c + d*x^2)^0, x, 4, (3/8)*a*x*sqrt(a + b*x^2) + (1/4)*x*(a + b*x^2)^(3/2) + (3*a^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*sqrt(b))]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^1, x, 6, (b*x*sqrt(a + b*x^2))/(2*d) - (sqrt(b)*(2*b*c - 3*a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*d^2) + ((b*c - a*d)^(3/2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(sqrt(c)*d^2)]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^2, x, 6, -(((b*c - a*d)*x*sqrt(a + b*x^2))/(2*c*d*(c + d*x^2))) + (b^(3/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/d^2 - (sqrt(b*c - a*d)*(2*b*c + a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*d^2)]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^3, x, 4, (x*(a + b*x^2)^(3/2))/(4*c*(c + d*x^2)^2) + (3*a*x*sqrt(a + b*x^2))/(8*c^2*(c + d*x^2)) + (3*a^2*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*sqrt(b*c - a*d))]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^4, x, 5, -((d*x*(a + b*x^2)^(5/2))/(6*c*(b*c - a*d)*(c + d*x^2)^3)) + ((6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2))/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + (a*(6*b*c - 5*a*d)*x*sqrt(a + b*x^2))/(16*c^3*(b*c - a*d)*(c + d*x^2)) + (a^2*(6*b*c - 5*a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(16*c^(7/2)*(b*c - a*d)^(3/2))]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^5, x, 7, -(((b*c - a*d)*x*sqrt(a + b*x^2))/(8*c*d*(c + d*x^2)^4)) + ((2*b*c + 7*a*d)*x*sqrt(a + b*x^2))/(48*c^2*d*(c + d*x^2)^3) + ((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*sqrt(a + b*x^2))/(192*c^3*d*(b*c - a*d)*(c + d*x^2)^2) + ((16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*sqrt(a + b*x^2))/(384*c^4*d*(b*c - a*d)^2*(c + d*x^2)) + (a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(128*c^(9/2)*(b*c - a*d)^(5/2))]

    @test_int [(a + b*x^2)^(5/2)*(c + d*x^2)^3, x, 8, (a^2*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*sqrt(a + b*x^2))/(1024*b^3) + (a*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(3/2))/(1536*b^3) + ((320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(5/2))/(1920*b^3) + (d*(152*b^2*c^2 - 68*a*b*c*d + 15*a^2*d^2)*x*(a + b*x^2)^(7/2))/(960*b^3) + (d*(16*b*c - 5*a*d)*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(120*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2)^2)/(12*b) + (a^3*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(1024*b^(7/2))]
    @test_int [(a + b*x^2)^(5/2)*(c + d*x^2)^2, x, 7, (a^2*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*sqrt(a + b*x^2))/(256*b^2) + (a*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(3/2))/(384*b^2) + ((80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*x*(a + b*x^2)^(5/2))/(480*b^2) + (3*d*(4*b*c - a*d)*x*(a + b*x^2)^(7/2))/(80*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2))/(10*b) + (a^3*(80*b^2*c^2 - 20*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(256*b^(5/2))]
    @test_int [(a + b*x^2)^(5/2)*(c + d*x^2)^1, x, 6, (5*a^2*(8*b*c - a*d)*x*sqrt(a + b*x^2))/(128*b) + (5*a*(8*b*c - a*d)*x*(a + b*x^2)^(3/2))/(192*b) + ((8*b*c - a*d)*x*(a + b*x^2)^(5/2))/(48*b) + (d*x*(a + b*x^2)^(7/2))/(8*b) + (5*a^3*(8*b*c - a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(128*b^(3/2))]
    @test_int [(a + b*x^2)^(5/2)*(c + d*x^2)^0, x, 5, (5/16)*a^2*x*sqrt(a + b*x^2) + (5/24)*a*x*(a + b*x^2)^(3/2) + (1/6)*x*(a + b*x^2)^(5/2) + (5*a^3*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*sqrt(b))]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^1, x, 7, -((b*(4*b*c - 7*a*d)*x*sqrt(a + b*x^2))/(8*d^2)) + (b*x*(a + b*x^2)^(3/2))/(4*d) + (sqrt(b)*(8*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*d^3) - ((b*c - a*d)^(5/2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(sqrt(c)*d^3)]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^2, x, 7, (b*(2*b*c - a*d)*x*sqrt(a + b*x^2))/(2*c*d^2) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(2*c*d*(c + d*x^2)) - (b^(3/2)*(4*b*c - 5*a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*d^3) + ((b*c - a*d)^(3/2)*(4*b*c + a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*d^3)]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^3, x, 7, -(((b*c - a*d)*x*(a + b*x^2)^(3/2))/(4*c*d*(c + d*x^2)^2)) - ((b*c - a*d)*(4*b*c + 3*a*d)*x*sqrt(a + b*x^2))/(8*c^2*d^2*(c + d*x^2)) + (b^(5/2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/d^3 - (sqrt(b*c - a*d)*(8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*d^3)]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^4, x, 5, (x*(a + b*x^2)^(5/2))/(6*c*(c + d*x^2)^3) + (5*a*x*(a + b*x^2)^(3/2))/(24*c^2*(c + d*x^2)^2) + (5*a^2*x*sqrt(a + b*x^2))/(16*c^3*(c + d*x^2)) + (5*a^3*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(16*c^(7/2)*sqrt(b*c - a*d))]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^5, x, 6, -((d*x*(a + b*x^2)^(7/2))/(8*c*(b*c - a*d)*(c + d*x^2)^4)) + ((8*b*c - 7*a*d)*x*(a + b*x^2)^(5/2))/(48*c^2*(b*c - a*d)*(c + d*x^2)^3) + (5*a*(8*b*c - 7*a*d)*x*(a + b*x^2)^(3/2))/(192*c^3*(b*c - a*d)*(c + d*x^2)^2) + (5*a^2*(8*b*c - 7*a*d)*x*sqrt(a + b*x^2))/(128*c^4*(b*c - a*d)*(c + d*x^2)) + (5*a^3*(8*b*c - 7*a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(128*c^(9/2)*(b*c - a*d)^(3/2))]

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

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

    @test_int [1/(a + b*x^2)^(1/2)*(c + d*x^2)^3, x, 5, (d*(44*b^2*c^2 - 44*a*b*c*d + 15*a^2*d^2)*x*sqrt(a + b*x^2))/(48*b^3) + (5*d*(2*b*c - a*d)*x*sqrt(a + b*x^2)*(c + d*x^2))/(24*b^2) + (d*x*sqrt(a + b*x^2)*(c + d*x^2)^2)/(6*b) + ((2*b*c - a*d)*(8*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(7/2))]
    @test_int [1/(a + b*x^2)^(1/2)*(c + d*x^2)^2, x, 4, (3*d*(2*b*c - a*d)*x*sqrt(a + b*x^2))/(8*b^2) + (d*x*sqrt(a + b*x^2)*(c + d*x^2))/(4*b) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(5/2))]
    @test_int [1/(a + b*x^2)^(1/2)*(c + d*x^2)^1, x, 3, (d*x*sqrt(a + b*x^2))/(2*b) + ((2*b*c - a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(3/2))]
    @test_int [1/(a + b*x^2)^(1/2)*(c + d*x^2)^0, x, 2, atanh((sqrt(b)*x)/sqrt(a + b*x^2))/sqrt(b)]
    @test_int [1/(a + b*x^2)^(1/2)/(c + d*x^2)^1, x, 2, atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2)))/(sqrt(c)*sqrt(b*c - a*d))]
    @test_int [1/(a + b*x^2)^(1/2)/(c + d*x^2)^2, x, 3, -((d*x*sqrt(a + b*x^2))/(2*c*(b*c - a*d)*(c + d*x^2))) + ((2*b*c - a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*(b*c - a*d)^(3/2))]
    @test_int [1/(a + b*x^2)^(1/2)/(c + d*x^2)^3, x, 5, -((d*x*sqrt(a + b*x^2))/(4*c*(b*c - a*d)*(c + d*x^2)^2)) - (3*d*(2*b*c - a*d)*x*sqrt(a + b*x^2))/(8*c^2*(b*c - a*d)^2*(c + d*x^2)) + ((8*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*(b*c - a*d)^(5/2))]

    @test_int [1/(a + b*x^2)^(3/2)*(c + d*x^2)^4, x, 6, -((d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*sqrt(a + b*x^2))/(48*a*b^4)) - (d*(24*b^2*c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*sqrt(a + b*x^2)*(c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*sqrt(a + b*x^2)*(c + d*x^2)^2)/(6*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*sqrt(a + b*x^2)) + (d*(64*b^3*c^3 - 144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(16*b^(9/2))]
    @test_int [1/(a + b*x^2)^(3/2)*(c + d*x^2)^3, x, 5, -((d*(2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*sqrt(a + b*x^2))/(8*a*b^3)) - (d*(4*b*c - 5*a*d)*x*sqrt(a + b*x^2)*(c + d*x^2))/(4*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^2)/(a*b*sqrt(a + b*x^2)) + (3*d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(7/2))]
    @test_int [1/(a + b*x^2)^(3/2)*(c + d*x^2)^2, x, 4, ((b*c - a*d)^2*x)/(a*b^2*sqrt(a + b*x^2)) + (d^2*x*sqrt(a + b*x^2))/(2*b^2) + (d*(4*b*c - 3*a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(5/2)), -((d*(2*b*c - 3*a*d)*x*sqrt(a + b*x^2))/(2*a*b^2)) + ((b*c - a*d)*x*(c + d*x^2))/(a*b*sqrt(a + b*x^2)) + (d*(4*b*c - 3*a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(5/2))]
    @test_int [1/(a + b*x^2)^(3/2)*(c + d*x^2)^1, x, 3, ((b*c - a*d)*x)/(a*b*sqrt(a + b*x^2)) + (d*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/b^(3/2)]
    @test_int [1/(a + b*x^2)^(3/2)*(c + d*x^2)^0, x, 1, x/(a*sqrt(a + b*x^2))]
    @test_int [1/(a + b*x^2)^(3/2)/(c + d*x^2)^1, x, 3, (b*x)/(a*(b*c - a*d)*sqrt(a + b*x^2)) - (d*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(sqrt(c)*(b*c - a*d)^(3/2))]
    @test_int [1/(a + b*x^2)^(3/2)/(c + d*x^2)^2, x, 5, (b*(2*b*c + a*d)*x)/(2*a*c*(b*c - a*d)^2*sqrt(a + b*x^2)) - (d*x)/(2*c*(b*c - a*d)*sqrt(a + b*x^2)*(c + d*x^2)) - (d*(4*b*c - a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*(b*c - a*d)^(5/2))]
    @test_int [1/(a + b*x^2)^(3/2)/(c + d*x^2)^3, x, 6, -((d*x)/(4*c*(b*c - a*d)*sqrt(a + b*x^2)*(c + d*x^2)^2)) + (b*(4*b*c + a*d)*x)/(4*a*c*(b*c - a*d)^2*sqrt(a + b*x^2)*(c + d*x^2)) + (d*(4*b*c - a*d)*(2*b*c + 3*a*d)*x*sqrt(a + b*x^2))/(8*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*(b*c - a*d)^(7/2))]

    @test_int [1/(a + b*x^2)^(5/2)*(c + d*x^2)^4, x, 6, -((d*(16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*sqrt(a + b*x^2))/(24*a^2*b^4)) - (d*(8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x*sqrt(a + b*x^2)*(c + d*x^2))/(12*a^2*b^3) + ((b*c - a*d)*(2*b*c + 7*a*d)*x*(c + d*x^2)^2)/(3*a^2*b^2*sqrt(a + b*x^2)) + ((b*c - a*d)*x*(c + d*x^2)^3)/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(8*b^(9/2))]
    @test_int [1/(a + b*x^2)^(5/2)*(c + d*x^2)^3, x, 5, -((d*(4*b^2*c^2 + 8*a*b*c*d - 15*a^2*d^2)*x*sqrt(a + b*x^2))/(6*a^2*b^3)) + ((b*c - a*d)*(2*b*c + 5*a*d)*x*(c + d*x^2))/(3*a^2*b^2*sqrt(a + b*x^2)) + ((b*c - a*d)*x*(c + d*x^2)^2)/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*(6*b*c - 5*a*d)*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/(2*b^(7/2))]
    @test_int [1/(a + b*x^2)^(5/2)*(c + d*x^2)^2, x, 4, ((b*c - a*d)*(2*b*c + 3*a*d)*x)/(3*a^2*b^2*sqrt(a + b*x^2)) + ((b*c - a*d)*x*(c + d*x^2))/(3*a*b*(a + b*x^2)^(3/2)) + (d^2*atanh((sqrt(b)*x)/sqrt(a + b*x^2)))/b^(5/2)]
    @test_int [1/(a + b*x^2)^(5/2)*(c + d*x^2)^1, x, 2, (2*c*x)/(3*a^2*sqrt(a + b*x^2)) + (x*(c + d*x^2))/(3*a*(a + b*x^2)^(3/2))]
    @test_int [1/(a + b*x^2)^(5/2)*(c + d*x^2)^0, x, 2, x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*sqrt(a + b*x^2))]
    @test_int [1/(a + b*x^2)^(5/2)/(c + d*x^2)^1, x, 5, (b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (b*(2*b*c - 5*a*d)*x)/(3*a^2*(b*c - a*d)^2*sqrt(a + b*x^2)) + (d^2*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(sqrt(c)*(b*c - a*d)^(5/2))]
    @test_int [1/(a + b*x^2)^(5/2)/(c + d*x^2)^2, x, 6, (b*(2*b*c + 3*a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (b*(4*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*x)/(6*a^2*c*(b*c - a*d)^3*sqrt(a + b*x^2)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)) + (d^2*(6*b*c - a*d)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(2*c^(3/2)*(b*c - a*d)^(7/2))]
    @test_int [1/(a + b*x^2)^(5/2)/(c + d*x^2)^3, x, 7, -((d*x)/(4*c*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^2)) + (b*(4*b*c + 3*a*d)*x)/(12*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/2)*(c + d*x^2)) + (b*(8*b^2*c^2 - 40*a*b*c*d - 3*a^2*d^2)*x)/(12*a^2*c*(b*c - a*d)^3*sqrt(a + b*x^2)*(c + d*x^2)) + (d*(16*b^3*c^3 - 88*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 9*a^3*d^3)*x*sqrt(a + b*x^2))/(24*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (d^2*(48*b^2*c^2 - 16*a*b*c*d + 3*a^2*d^2)*atanh((sqrt(b*c - a*d)*x)/(sqrt(c)*sqrt(a + b*x^2))))/(8*c^(5/2)*(b*c - a*d)^(9/2))]

    @test_int [(a + b*x^2)^3/(c + d*x^2)^(11/2), x, 5, -((d*x*(a + b*x^2)^4)/(9*c*(b*c - a*d)*(c + d*x^2)^(9/2))) + ((9*b*c - 8*a*d)*x*(a + b*x^2)^3)/(63*c^2*(b*c - a*d)*(c + d*x^2)^(7/2)) + (2*a*(9*b*c - 8*a*d)*x*(a + b*x^2)^2)/(105*c^3*(b*c - a*d)*(c + d*x^2)^(5/2)) + (8*a^2*(9*b*c - 8*a*d)*x*(a + b*x^2))/(315*c^4*(b*c - a*d)*(c + d*x^2)^(3/2)) + (16*a^3*(9*b*c - 8*a*d)*x)/(315*c^5*(b*c - a*d)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^2/(c + d*x^2)^(9/2), x, 4, -((d*x*(a + b*x^2)^3)/(7*c*(b*c - a*d)*(c + d*x^2)^(7/2))) + ((7*b*c - 6*a*d)*x*(a + b*x^2)^2)/(35*c^2*(b*c - a*d)*(c + d*x^2)^(5/2)) + (4*a*(7*b*c - 6*a*d)*x*(a + b*x^2))/(105*c^3*(b*c - a*d)*(c + d*x^2)^(3/2)) + (8*a^2*(7*b*c - 6*a*d)*x)/(105*c^4*(b*c - a*d)*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^1/(c + d*x^2)^(7/2), x, 3, -(((b*c - a*d)*x)/(5*c*d*(c + d*x^2)^(5/2))) + ((b*c + 4*a*d)*x)/(15*c^2*d*(c + d*x^2)^(3/2)) + (2*(b*c + 4*a*d)*x)/(15*c^3*d*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^0/(c + d*x^2)^(5/2), x, 2, x/(3*c*(c + d*x^2)^(3/2)) + (2*x)/(3*c^2*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^1*(c + d*x^2)^(3/2)), x, 3, -((d*x)/(c*(b*c - a*d)*sqrt(c + d*x^2))) + (b*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(sqrt(a)*(b*c - a*d)^(3/2))]
    @test_int [1/((a + b*x^2)^2*(c + d*x^2)^(1/2)), x, 3, (b*x*sqrt(c + d*x^2))/(2*a*(b*c - a*d)*(a + b*x^2)) + ((b*c - 2*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(2*a^(3/2)*(b*c - a*d)^(3/2))]
    @test_int [(c + d*x^2)^(1/2)/(a + b*x^2)^3, x, 4, ((3*b*c - 4*a*d)*x*sqrt(c + d*x^2))/(8*a^2*(b*c - a*d)*(a + b*x^2)) + (b*x*(c + d*x^2)^(3/2))/(4*a*(b*c - a*d)*(a + b*x^2)^2) + (c*(3*b*c - 4*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(8*a^(5/2)*(b*c - a*d)^(3/2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^4, x, 5, (c*(5*b*c - 6*a*d)*x*sqrt(c + d*x^2))/(16*a^3*(b*c - a*d)*(a + b*x^2)) + ((5*b*c - 6*a*d)*x*(c + d*x^2)^(3/2))/(24*a^2*(b*c - a*d)*(a + b*x^2)^2) + (b*x*(c + d*x^2)^(5/2))/(6*a*(b*c - a*d)*(a + b*x^2)^3) + (c^2*(5*b*c - 6*a*d)*atan((sqrt(b*c - a*d)*x)/(sqrt(a)*sqrt(c + d*x^2))))/(16*a^(7/2)*(b*c - a*d)^(3/2))]

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

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

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

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

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

    @test_int [(a - b*x^2)^(2/3)*(3*a + b*x^2)^3, x, 8, (18144*a^3*x*(a - b*x^2)^(2/3))/1235 - (23544*a^2*x*(a - b*x^2)^(5/3))/6175 - (378/475)*a*x*(a - b*x^2)^(5/3)*(3*a + b*x^2) - (3/25)*x*(a - b*x^2)^(5/3)*(3*a + b*x^2)^2 - (72576*a^4*x)/(1235*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (36288*3^(1/4)*sqrt(2 + sqrt(3))*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1235*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (24192*sqrt(2)*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1235*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)*(3*a + b*x^2)^2, x, 7, (7776*a^2*x*(a - b*x^2)^(2/3))/1729 - (252/247)*a*x*(a - b*x^2)^(5/3) - (3/19)*x*(a - b*x^2)^(5/3)*(3*a + b*x^2) - (31104*a^3*x)/(1729*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (15552*3^(1/4)*sqrt(2 + sqrt(3))*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (10368*sqrt(2)*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)*(3*a + b*x^2)^1, x, 6, (18/13)*a*x*(a - b*x^2)^(2/3) - (3/13)*x*(a - b*x^2)^(5/3) - (72*a^2*x)/(13*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(13*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (24*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(13*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)/(3*a + b*x^2)^1, x, 6, (3*x)/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3)) + (2^(1/3)*a^(1/6)*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) + (2^(1/3)*a^(1/6)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) - (2^(1/3)*a^(1/6)*atanh((sqrt(b)*x)/sqrt(a)))/(3*sqrt(b)) + (2^(1/3)*a^(1/6)*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/sqrt(b) + (3*3^(1/4)*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (sqrt(2)*3^(3/4)*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)/(3*a + b*x^2)^2, x, 6, (x*(a - b*x^2)^(2/3))/(6*a*(3*a + b*x^2)) - x/(6*a*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*3^(3/4)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3*sqrt(2)*3^(1/4)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)/(3*a + b*x^2)^3, x, 8, (x*(a - b*x^2)^(2/3))/(12*a*(3*a + b*x^2)^2) + (x*(a - b*x^2)^(2/3))/(36*a^2*(3*a + b*x^2)) - x/(36*a^2*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(72*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(72*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(216*2^(2/3)*a^(11/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(72*2^(2/3)*a^(11/6)*sqrt(b)) - (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(24*3^(3/4)*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(18*sqrt(2)*3^(1/4)*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(2/3)/(3*a + b*x^2)^4, x, 9, (x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)^3) + (x*(a - b*x^2)^(2/3))/(54*a^2*(3*a + b*x^2)^2) + (x*(a - b*x^2)^(2/3))/(144*a^3*(3*a + b*x^2)) - x/(144*a^3*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (7*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(1296*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) + (7*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(1296*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) - (7*atanh((sqrt(b)*x)/sqrt(a)))/(3888*2^(2/3)*a^(17/6)*sqrt(b)) + (7*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/(1296*2^(2/3)*a^(17/6)*sqrt(b)) - (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(96*3^(3/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(72*sqrt(2)*3^(1/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]

    @test_int [(a - b*x^2)^(5/3)*(3*a + b*x^2)^3, x, 9, (2809728*a^4*x*(a - b*x^2)^(2/3))/267995 + (1404864*a^3*x*(a - b*x^2)^(5/3))/191425 - (33264*a^2*x*(a - b*x^2)^(8/3))/14725 - (432/775)*a*x*(a - b*x^2)^(8/3)*(3*a + b*x^2) - (3/31)*x*(a - b*x^2)^(8/3)*(3*a + b*x^2)^2 - (11238912*a^5*x)/(267995*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (5619456*3^(1/4)*sqrt(2 + sqrt(3))*a^(16/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(267995*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (3746304*sqrt(2)*3^(3/4)*a^(16/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(267995*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(5/3)*(3*a + b*x^2)^2, x, 8, (28512*a^3*x*(a - b*x^2)^(2/3))/8645 + (14256*a^2*x*(a - b*x^2)^(5/3))/6175 - (306/475)*a*x*(a - b*x^2)^(8/3) - (3/25)*x*(a - b*x^2)^(8/3)*(3*a + b*x^2) - (114048*a^4*x)/(8645*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (57024*3^(1/4)*sqrt(2 + sqrt(3))*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8645*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (38016*sqrt(2)*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8645*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(5/3)*(3*a + b*x^2)^1, x, 7, (1800*a^2*x*(a - b*x^2)^(2/3))/1729 + (180/247)*a*x*(a - b*x^2)^(5/3) - (3/19)*x*(a - b*x^2)^(8/3) - (7200*a^3*x)/(1729*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (3600*3^(1/4)*sqrt(2 + sqrt(3))*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (2400*sqrt(2)*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(5/3)/(3*a + b*x^2)^1, x, 7, (-(3/7))*x*(a - b*x^2)^(2/3) + (96*a*x)/(7*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (4*2^(1/3)*a^(7/6)*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) + (4*2^(1/3)*a^(7/6)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) - (4*2^(1/3)*a^(7/6)*atanh((sqrt(b)*x)/sqrt(a)))/(3*sqrt(b)) + (4*2^(1/3)*a^(7/6)*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/sqrt(b) + (48*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (32*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(5/3)/(3*a + b*x^2)^2, x, 7, (2*x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)) - (11*x)/(3*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (2^(1/3)*a^(1/6)*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) - (2^(1/3)*a^(1/6)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(sqrt(3)*sqrt(b)) + (2^(1/3)*a^(1/6)*atanh((sqrt(b)*x)/sqrt(a)))/(3*sqrt(b)) - (2^(1/3)*a^(1/6)*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/sqrt(b) - (11*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*3^(3/4)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (11*sqrt(2)*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(3*3^(1/4)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(a - b*x^2)^(5/3)/(3*a + b*x^2)^3, x, 9, (x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)^2) - (x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)) + x/(18*a*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(18*2^(2/3)*sqrt(3)*a^(5/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(18*2^(2/3)*sqrt(3)*a^(5/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(54*2^(2/3)*a^(5/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(18*2^(2/3)*a^(5/6)*sqrt(b)) + (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(12*3^(3/4)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(9*sqrt(2)*3^(1/4)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]

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

    @test_int [(3*a + b*x^2)^4/(a - b*x^2)^(1/3), x, 8, -((1552608*a^3*x*(a - b*x^2)^(2/3))/43225) - (36288*a^2*x*(a - b*x^2)^(2/3)*(3*a + b*x^2))/6175 - (18/19)*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2 - (3/25)*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^3 - (3794688*a^4*x)/(8645*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (1897344*3^(1/4)*sqrt(2 + sqrt(3))*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8645*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (1264896*sqrt(2)*3^(3/4)*a^(13/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8645*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^3/(a - b*x^2)^(1/3), x, 7, -((15768*a^2*x*(a - b*x^2)^(2/3))/1729) - (324/247)*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2) - (3/19)*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2 - (215136*a^3*x)/(1729*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (107568*3^(1/4)*sqrt(2 + sqrt(3))*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (71712*sqrt(2)*3^(3/4)*a^(10/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(1729*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^2/(a - b*x^2)^(1/3), x, 6, (-(198/91))*a*x*(a - b*x^2)^(2/3) - (3/13)*x*(a - b*x^2)^(2/3)*(3*a + b*x^2) - (3240*a^2*x)/(91*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (1620*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (1080*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^1/(a - b*x^2)^(1/3), x, 5, (-(3/7))*x*(a - b*x^2)^(2/3) - (72*a*x)/(7*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (24*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^1*(a - b*x^2)^(1/3)), x, 1, atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(2*2^(2/3)*sqrt(3)*a^(5/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(2*2^(2/3)*sqrt(3)*a^(5/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(6*2^(2/3)*a^(5/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(2*2^(2/3)*a^(5/6)*sqrt(b))]
    @test_int [1/((3*a + b*x^2)^2*(a - b*x^2)^(1/3)), x, 7, (x*(a - b*x^2)^(2/3))/(24*a^2*(3*a + b*x^2)) - x/(24*a^2*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(8*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(8*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(24*2^(2/3)*a^(11/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(8*2^(2/3)*a^(11/6)*sqrt(b)) - (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*3^(3/4)*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(12*sqrt(2)*3^(1/4)*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^3*(a - b*x^2)^(1/3)), x, 8, (x*(a - b*x^2)^(2/3))/(48*a^2*(3*a + b*x^2)^2) + (5*x*(a - b*x^2)^(2/3))/(288*a^3*(3*a + b*x^2)) - (5*x)/(288*a^3*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (5*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(144*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) + (5*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(144*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) - (5*atanh((sqrt(b)*x)/sqrt(a)))/(432*2^(2/3)*a^(17/6)*sqrt(b)) + (5*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/(144*2^(2/3)*a^(17/6)*sqrt(b)) - (5*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(192*3^(3/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (5*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(144*sqrt(2)*3^(1/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]

    @test_int [(3*a + b*x^2)^3/(a - b*x^2)^(4/3), x, 7, (2538/91)*a*x*(a - b*x^2)^(2/3) + (81/13)*x*(a - b*x^2)^(2/3)*(3*a + b*x^2) + (6*x*(3*a + b*x^2)^2)/(a - b*x^2)^(1/3) + (20088*a^2*x)/(91*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (10044*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (6696*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^2/(a - b*x^2)^(4/3), x, 6, (45/7)*x*(a - b*x^2)^(2/3) + (6*x*(3*a + b*x^2))/(a - b*x^2)^(1/3) + (324*a*x)/(7*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (162*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (108*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^1/(a - b*x^2)^(4/3), x, 5, (6*x)/(a - b*x^2)^(1/3) + (9*x)/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3)) + (9*3^(1/4)*sqrt(2 + sqrt(3))*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (3*sqrt(2)*3^(3/4)*a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^1*(a - b*x^2)^(4/3)), x, 7, (3*x)/(8*a^2*(a - b*x^2)^(1/3)) + (3*x)/(8*a^2*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(8*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(8*2^(2/3)*sqrt(3)*a^(11/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(24*2^(2/3)*a^(11/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(8*2^(2/3)*a^(11/6)*sqrt(b)) + (3*3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(16*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(4*sqrt(2)*a^(5/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^2*(a - b*x^2)^(4/3)), x, 8, x/(12*a^3*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)) + x/(12*a^3*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(16*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(16*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(48*2^(2/3)*a^(17/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(16*2^(2/3)*a^(17/6)*sqrt(b)) + (sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8*3^(3/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - ((a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(6*sqrt(2)*3^(1/4)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^3*(a - b*x^2)^(4/3)), x, 9, x/(48*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)^2) + (17*x)/(192*a^3*(a - b*x^2)^(1/3)*(3*a + b*x^2)) - (19*x*(a - b*x^2)^(2/3))/(1152*a^4*(3*a + b*x^2)) + (19*x)/(1152*a^4*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (7*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(288*2^(2/3)*sqrt(3)*a^(23/6)*sqrt(b)) + (7*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(288*2^(2/3)*sqrt(3)*a^(23/6)*sqrt(b)) - (7*atanh((sqrt(b)*x)/sqrt(a)))/(864*2^(2/3)*a^(23/6)*sqrt(b)) + (7*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/(288*2^(2/3)*a^(23/6)*sqrt(b)) + (19*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(768*3^(3/4)*a^(11/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (19*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(576*sqrt(2)*3^(1/4)*a^(11/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]

    @test_int [(3*a + b*x^2)^4/(a - b*x^2)^(7/3), x, 8, (-(3240/91))*a*x*(a - b*x^2)^(2/3) - (81/13)*x*(a - b*x^2)^(2/3)*(3*a + b*x^2) - (9*x*(3*a + b*x^2)^2)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2)^3)/(2*(a - b*x^2)^(4/3)) - (36936*a^2*x)/(91*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (18468*3^(1/4)*sqrt(2 + sqrt(3))*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (12312*sqrt(2)*3^(3/4)*a^(7/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(91*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^3/(a - b*x^2)^(7/3), x, 7, (-(27/14))*x*(a - b*x^2)^(2/3) + (3*x*(3*a + b*x^2)^2)/(2*(a - b*x^2)^(4/3)) - (324*a*x)/(7*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) - (162*3^(1/4)*sqrt(2 + sqrt(3))*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) + (108*sqrt(2)*3^(3/4)*a^(4/3)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(7*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [(3*a + b*x^2)^2/(a - b*x^2)^(7/3), x, 2, (9*x)/(2*(a - b*x^2)^(1/3)) + (3*x*(3*a + b*x^2))/(2*(a - b*x^2)^(4/3))]
    @test_int [(3*a + b*x^2)^1/(a - b*x^2)^(7/3), x, 6, (3*x)/(2*(a - b*x^2)^(4/3)) + (9*x)/(4*a*(a - b*x^2)^(1/3)) + (9*x)/(4*a*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (9*3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(8*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (3*3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(2*sqrt(2)*a^(2/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^1*(a - b*x^2)^(7/3)), x, 8, (3*x)/(32*a^2*(a - b*x^2)^(4/3)) + (21*x)/(64*a^3*(a - b*x^2)^(1/3)) + (21*x)/(64*a^3*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + atan((sqrt(3)*sqrt(a))/(sqrt(b)*x))/(32*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) + atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x))/(32*2^(2/3)*sqrt(3)*a^(17/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(96*2^(2/3)*a^(17/6)*sqrt(b)) + atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3))))/(32*2^(2/3)*a^(17/6)*sqrt(b)) + (21*3^(1/4)*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(128*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (7*3^(3/4)*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(32*sqrt(2)*a^(8/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]
    @test_int [1/((3*a + b*x^2)^2*(a - b*x^2)^(7/3)), x, 9, (5*x)/(384*a^3*(a - b*x^2)^(4/3)) + (79*x)/(768*a^4*(a - b*x^2)^(1/3)) + x/(24*a^2*(a - b*x^2)^(4/3)*(3*a + b*x^2)) + (79*x)/(768*a^4*((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))) + (sqrt(3)*atan((sqrt(3)*sqrt(a))/(sqrt(b)*x)))/(128*2^(2/3)*a^(23/6)*sqrt(b)) + (sqrt(3)*atan((sqrt(3)*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(sqrt(b)*x)))/(128*2^(2/3)*a^(23/6)*sqrt(b)) - atanh((sqrt(b)*x)/sqrt(a))/(128*2^(2/3)*a^(23/6)*sqrt(b)) + (3*atanh((sqrt(b)*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))))/(128*2^(2/3)*a^(23/6)*sqrt(b)) + (79*sqrt(2 + sqrt(3))*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.E(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(512*3^(3/4)*a^(11/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2))) - (79*(a^(1/3) - (a - b*x^2)^(1/3))*sqrt((a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)*Elliptic.F(asin(((1 + sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))), -7 + 4*sqrt(3)))/(384*sqrt(2)*3^(1/4)*a^(11/3)*b*x*sqrt(-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - sqrt(3))*a^(1/3) - (a - b*x^2)^(1/3))^2)))]

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

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

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

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

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

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*x^2)^(p/3)*(c+d*x^2)^q*when*b*c-9*a*d=0=#

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

    @test_int [1/((2 + b*x^2)^(1/3)*((18*d)/b + d*x^2)), x, 1, (sqrt(b)*atan((sqrt(b)*x)/(3*sqrt(2))))/(12*2^(5/6)*d) + (sqrt(b)*atan((2^(1/3) - (2 + b*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(b)*x)))/(12*2^(5/6)*d) - (sqrt(b)*atanh((2^(1/6)*sqrt(3)*(2^(1/3) - (2 + b*x^2)^(1/3)))/(sqrt(b)*x)))/(4*2^(5/6)*sqrt(3)*d)]
    @test_int [1/((-2 + b*x^2)^(1/3)*(-((18*d)/b) + d*x^2)), x, 1, (sqrt(b)*atan((2^(1/6)*sqrt(3)*(2^(1/3) + (-2 + b*x^2)^(1/3)))/(sqrt(b)*x)))/(4*2^(5/6)*sqrt(3)*d) + (sqrt(b)*atanh((sqrt(b)*x)/(3*sqrt(2))))/(12*2^(5/6)*d) - (sqrt(b)*atanh((2^(1/3) + (-2 + b*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(b)*x)))/(12*2^(5/6)*d)]

    @test_int [1/((2 + 3*x^2)^(1/3)*((18*d)/3 + d*x^2)), x, 1, atan(x/sqrt(6))/(4*2^(5/6)*sqrt(3)*d) + atan((2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(3)*x))/(4*2^(5/6)*sqrt(3)*d) - atanh((2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x)/(4*2^(5/6)*d)]
    @test_int [1/((2 - 3*x^2)^(1/3)*(-((18*d)/3) + d*x^2)), x, 1, -(atan((2^(1/6)*(2^(1/3) - (2 - 3*x^2)^(1/3)))/x)/(4*2^(5/6)*d)) - atanh(x/sqrt(6))/(4*2^(5/6)*sqrt(3)*d) + atanh((2^(1/3) - (2 - 3*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(3)*x))/(4*2^(5/6)*sqrt(3)*d)]
    @test_int [1/((-2 + 3*x^2)^(1/3)*(-((18*d)/3) + d*x^2)), x, 1, atan((2^(1/6)*(2^(1/3) + (-2 + 3*x^2)^(1/3)))/x)/(4*2^(5/6)*d) + atanh(x/sqrt(6))/(4*2^(5/6)*sqrt(3)*d) - atanh((2^(1/3) + (-2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(3)*x))/(4*2^(5/6)*sqrt(3)*d)]
    @test_int [1/((-2 - 3*x^2)^(1/3)*((18*d)/3 + d*x^2)), x, 1, -(atan(x/sqrt(6))/(4*2^(5/6)*sqrt(3)*d)) - atan((2^(1/3) + (-2 - 3*x^2)^(1/3))^2/(3*2^(1/6)*sqrt(3)*x))/(4*2^(5/6)*sqrt(3)*d) + atanh((2^(1/6)*(2^(1/3) + (-2 - 3*x^2)^(1/3)))/x)/(4*2^(5/6)*d)]

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

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

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

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

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

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

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

    @test_int [(a + b*x^2)^(3/2)*sqrt(c + d*x^2), x, 6, ((7*a*c - (2*b*c^2)/d + (3*a^2*d)/b)*x*sqrt(a + b*x^2))/(15*sqrt(c + d*x^2)) - (2*(b*c - 3*a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*d) + (b*x*sqrt(a + b*x^2)*(c + d*x^2)^(3/2))/(5*d) + (sqrt(c)*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*(b*c - 9*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(1/2)*sqrt(c + d*x^2), x, 5, ((b*c + a*d)*x*sqrt(a + b*x^2))/(3*b*sqrt(c + d*x^2)) + (1/3)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2) - (sqrt(c)*(b*c + a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (2*c^(3/2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2)^(1/2), x, 4, (d*x*sqrt(a + b*x^2))/(b*sqrt(c + d*x^2)) - (sqrt(c)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(b*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2)^(3/2), x, 1, (sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(sqrt(a)*sqrt(b)*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2))))]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2)^(5/2), x, 4, (x*sqrt(c + d*x^2))/(3*a*(a + b*x^2)^(3/2)) + ((2*b*c - a*d)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(3*a^(3/2)*sqrt(b)*(b*c - a*d)*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (c^(3/2)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*(b*c - a*d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [sqrt(c + d*x^2)/(a + b*x^2)^(7/2), x, 5, (x*sqrt(c + d*x^2))/(5*a*(a + b*x^2)^(5/2)) + ((4*b*c - 3*a*d)*x*sqrt(c + d*x^2))/(15*a^2*(b*c - a*d)*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(15*a^(5/2)*sqrt(b)*(b*c - a*d)^2*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (2*c^(3/2)*sqrt(d)*(2*b*c - 3*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*a^3*(b*c - a*d)^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2), x, 7, -((2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*x*sqrt(a + b*x^2))/(35*b^2*d*sqrt(c + d*x^2))) + (1/35)*(9*a*c + (b*c^2)/d - (2*a^2*d)/b)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2) + (2*(4*b*c - a*d)*x*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(35*b) + (d*x*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(7*b) + (2*sqrt(c)*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(35*b^2*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (c^(3/2)*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(35*b*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2), x, 6, ((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*sqrt(a + b*x^2))/(15*b^2*sqrt(c + d*x^2)) + (2*(3*b*c - a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*b) + (d*x*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(5*b) - (sqrt(c)*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b^2*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*(9*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*b*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^(1/2), x, 5, (2*d*(2*b*c - a*d)*x*sqrt(a + b*x^2))/(3*b^2*sqrt(c + d*x^2)) + (d*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*b) - (2*sqrt(c)*sqrt(d)*(2*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*b^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*(3*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a*b*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^(3/2), x, 5, -((d*(b*c - 2*a*d)*x*sqrt(a + b*x^2))/(a*b^2*sqrt(c + d*x^2))) + ((b*c - a*d)*x*sqrt(c + d*x^2))/(a*b*sqrt(a + b*x^2)) + (sqrt(c)*sqrt(d)*(b*c - 2*a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*b^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (c^(3/2)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*b*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^(5/2), x, 4, ((b*c - a*d)*x*sqrt(c + d*x^2))/(3*a*b*(a + b*x^2)^(3/2)) + (2*(b*c + a*d)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(3*a^(3/2)*b^(3/2)*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (c^(3/2)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*b*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(c + d*x^2)^(3/2)/(a + b*x^2)^(7/2), x, 5, ((b*c - a*d)*x*sqrt(c + d*x^2))/(5*a*b*(a + b*x^2)^(5/2)) + (2*(2*b*c + a*d)*x*sqrt(c + d*x^2))/(15*a^2*b*(a + b*x^2)^(3/2)) + ((8*b^2*c^2 - 3*a*b*c*d - 2*a^2*d^2)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(15*a^(5/2)*b^(3/2)*(b*c - a*d)*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (c^(3/2)*sqrt(d)*(4*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*a^3*b*(b*c - a*d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [sqrt(2 + b*x^2)*sqrt(3 + d*x^2), x, 5, ((3*b + 2*d)*x*sqrt(2 + b*x^2))/(3*b*sqrt(3 + d*x^2)) + (1/3)*x*sqrt(2 + b*x^2)*sqrt(3 + d*x^2) - (sqrt(2)*(3*b + 2*d)*sqrt(2 + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(3*b*sqrt(d)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2)) + (2*sqrt(2)*sqrt(2 + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(sqrt(d)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2))]
    @test_int [sqrt(2 + 4*x^2)*sqrt(3 - 6*x^2), x, 3, sqrt(2/3)*x*sqrt(1 - 4*x^4) + (2*Elliptic.F(asin(sqrt(2)*x), -1))/sqrt(3)]
    @test_int [sqrt(2 + 4*x^2)*sqrt(3 + 6*x^2), x, 2, sqrt(6)*x + 2*sqrt(2/3)*x^3]

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

    @test_int [sqrt(1 - x^2)/sqrt(2 - 3*x^2), x, 1, Elliptic.E(asin(sqrt(3/2)*x), 2/3)/sqrt(3)]
    @test_int [sqrt(4 - x^2)/sqrt(2 - 3*x^2), x, 1, (2*Elliptic.E(asin(sqrt(3/2)*x), 1/6))/sqrt(3)]
    @test_int [sqrt(1 - 4*x^2)/sqrt(2 - 3*x^2), x, 1, Elliptic.E(asin(sqrt(3/2)*x), 8/3)/sqrt(3)]

    @test_int [sqrt(1 + x^2)/sqrt(1 - x^2), x, 1, Elliptic.E(asin(x), -1)]
    @test_int [sqrt(1 + x^2)/sqrt(2 - 3*x^2), x, 1, Elliptic.E(asin(sqrt(3/2)*x), -2/3)/sqrt(3)]
    @test_int [sqrt(4 + x^2)/sqrt(2 - 3*x^2), x, 1, (2*Elliptic.E(asin(sqrt(3/2)*x), -1/6))/sqrt(3)]
    @test_int [sqrt(1 + 4*x^2)/sqrt(2 - 3*x^2), x, 1, Elliptic.E(asin(sqrt(3/2)*x), -8/3)/sqrt(3)]

    @test_int [sqrt(1 - x^2)/sqrt(1 + x^2), x, 4, -Elliptic.E(asin(x), -1) + 2*Elliptic.F(asin(x), -1)]
    @test_int [sqrt(1 - x^2)/sqrt(2 + 3*x^2), x, 3, (-(1/3))*sqrt(2)*Elliptic.E(asin(x), -(3/2)) + (5*Elliptic.F(asin(x), -(3/2)))/(3*sqrt(2))]
    @test_int [sqrt(4 - x^2)/sqrt(2 + 3*x^2), x, 3, (-(1/3))*sqrt(2)*Elliptic.E(asin(x/2), -6) + (7/3)*sqrt(2)*Elliptic.F(asin(x/2), -6)]
    @test_int [sqrt(1 - 4*x^2)/sqrt(2 + 3*x^2), x, 3, (-(2/3))*sqrt(2)*Elliptic.E(asin(2*x), -(3/8)) + (11*Elliptic.F(asin(2*x), -(3/8)))/(6*sqrt(2))]

    @test_int [sqrt(1 + x^2)/sqrt(2 + 3*x^2), x, 4, (x*sqrt(2 + 3*x^2))/(3*sqrt(1 + x^2)) - (sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.E(atan(x), -(1/2)))/(3*sqrt(1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2))) + (sqrt(2 + 3*x^2)*Elliptic.F(atan(x), -(1/2)))/(sqrt(2)*sqrt(1 + x^2)*sqrt((2 + 3*x^2)/(1 + x^2)))]
    @test_int [sqrt(4 + x^2)/sqrt(2 + 3*x^2), x, 4, (x*sqrt(2 + 3*x^2))/(3*sqrt(4 + x^2)) - (sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.E(atan(x/2), -5))/(3*sqrt(4 + x^2)*sqrt((2 + 3*x^2)/(4 + x^2))) + (2*sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.F(atan(x/2), -5))/(sqrt(4 + x^2)*sqrt((2 + 3*x^2)/(4 + x^2)))]
    @test_int [sqrt(1 + 4*x^2)/sqrt(2 + 3*x^2), x, 4, (4*x*sqrt(2 + 3*x^2))/(3*sqrt(1 + 4*x^2)) - (2*sqrt(2)*sqrt(2 + 3*x^2)*Elliptic.E(atan(2*x), 5/8))/(3*sqrt((2 + 3*x^2)/(1 + 4*x^2))*sqrt(1 + 4*x^2)) + (sqrt(2 + 3*x^2)*Elliptic.F(atan(2*x), 5/8))/(2*sqrt(2)*sqrt((2 + 3*x^2)/(1 + 4*x^2))*sqrt(1 + 4*x^2))]

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

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

    @test_int [(a + b*x^2)^(7/2)/sqrt(c + d*x^2), x, 7, -((8*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*x*sqrt(a + b*x^2))/(105*d^3*sqrt(c + d*x^2))) + (b*(24*b^2*c^2 - 71*a*b*c*d + 71*a^2*d^2)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(105*d^3) - (6*b*(b*c - 2*a*d)*x*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(35*d^2) + (b*x*(a + b*x^2)^(5/2)*sqrt(c + d*x^2))/(7*d) + (8*sqrt(c)*(b*c - 2*a*d)*(6*b^2*c^2 - 11*a*b*c*d + 11*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(105*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (sqrt(c)*(3*b*c - 7*a*d)*(8*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(105*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(5/2)/sqrt(c + d*x^2), x, 6, ((8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*x*sqrt(a + b*x^2))/(15*d^2*sqrt(c + d*x^2)) - (4*b*(b*c - 2*a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*d^2) + (b*x*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(5*d) - (sqrt(c)*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (sqrt(c)*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(3/2)/sqrt(c + d*x^2), x, 5, -((2*(b*c - 2*a*d)*x*sqrt(a + b*x^2))/(3*d*sqrt(c + d*x^2))) + (b*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*d) + (2*sqrt(c)*(b*c - 2*a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (sqrt(c)*(b*c - 3*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(1/2)/sqrt(c + d*x^2), x, 4, (x*sqrt(a + b*x^2))/sqrt(c + d*x^2) - (sqrt(c)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (sqrt(c)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(1/2)*sqrt(c + d*x^2)), x, 1, (sqrt(c)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(3/2)*sqrt(c + d*x^2)), x, 6, -((d*x*sqrt(a + b*x^2))/(a*(b*c - a*d)*sqrt(c + d*x^2))) + (b*x*sqrt(c + d*x^2))/(a*(b*c - a*d)*sqrt(a + b*x^2)) + (sqrt(c)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*(b*c - a*d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (sqrt(c)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*(b*c - a*d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(5/2)*sqrt(c + d*x^2)), x, 4, (b*x*sqrt(c + d*x^2))/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (2*sqrt(b)*(b*c - 2*a*d)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(3*a^(3/2)*(b*c - a*d)^2*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (sqrt(c)*sqrt(d)*(b*c - 3*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*(b*c - a*d)^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(7/2)*sqrt(c + d*x^2)), x, 5, (b*x*sqrt(c + d*x^2))/(5*a*(b*c - a*d)*(a + b*x^2)^(5/2)) + (4*b*(b*c - 2*a*d)*x*sqrt(c + d*x^2))/(15*a^2*(b*c - a*d)^2*(a + b*x^2)^(3/2)) + (sqrt(b)*(8*b^2*c^2 - 23*a*b*c*d + 23*a^2*d^2)*sqrt(c + d*x^2)*Elliptic.E(atan((sqrt(b)*x)/sqrt(a)), 1 - (a*d)/(b*c)))/(15*a^(5/2)*(b*c - a*d)^3*sqrt(a + b*x^2)*sqrt((a*(c + d*x^2))/(c*(a + b*x^2)))) - (sqrt(c)*sqrt(d)*(4*b^2*c^2 - 11*a*b*c*d + 15*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*a^3*(b*c - a*d)^3*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

    @test_int [(a + b*x^2)^(7/2)/(c + d*x^2)^(3/2), x, 7, ((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*x*sqrt(a + b*x^2))/(15*c*d^3*sqrt(c + d*x^2)) - ((b*c - a*d)*x*(a + b*x^2)^(5/2))/(c*d*sqrt(c + d*x^2)) - (b*(24*b^2*c^2 - 43*a*b*c*d + 15*a^2*d^2)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(15*c*d^3) + (b*(6*b*c - 5*a*d)*x*(a + b*x^2)^(3/2)*sqrt(c + d*x^2))/(5*c*d^2) - ((48*b^3*c^3 - 128*a*b^2*c^2*d + 103*a^2*b*c*d^2 - 15*a^3*d^3)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*sqrt(c)*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (b*sqrt(c)*(24*b^2*c^2 - 61*a*b*c*d + 45*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(15*d^(7/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(5/2)/(c + d*x^2)^(3/2), x, 6, -(((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*x*sqrt(a + b*x^2))/(3*c*d^2*sqrt(c + d*x^2))) - ((b*c - a*d)*x*(a + b*x^2)^(3/2))/(c*d*sqrt(c + d*x^2)) + (b*(4*b*c - 3*a*d)*x*sqrt(a + b*x^2)*sqrt(c + d*x^2))/(3*c*d^2) + ((8*b^2*c^2 - 13*a*b*c*d + 3*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*sqrt(c)*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (2*b*sqrt(c)*(2*b*c - 3*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*d^(5/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(3/2)/(c + d*x^2)^(3/2), x, 5, -(((b*c - a*d)*x*sqrt(a + b*x^2))/(c*d*sqrt(c + d*x^2))) + ((2*b*c - a*d)*x*sqrt(a + b*x^2))/(c*d*sqrt(c + d*x^2)) - ((2*b*c - a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) + (b*sqrt(c)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(d^(3/2)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [(a + b*x^2)^(1/2)/(c + d*x^2)^(3/2), x, 1, (sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*sqrt(d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2)), x, 6, -((sqrt(d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(sqrt(c)*(b*c - a*d)*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(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(d)*(b*c - a*d)*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x, 4, (b*x)/(a*(b*c - a*d)*sqrt(a + b*x^2)*sqrt(c + d*x^2)) + (sqrt(d)*(b*c + a*d)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*sqrt(c)*(b*c - a*d)^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (2*b*sqrt(c)*sqrt(d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(a*(b*c - a*d)^2*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]
    @test_int [1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)), x, 5, (b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*sqrt(c + d*x^2)) + (2*b*(b*c - 3*a*d)*x)/(3*a^2*(b*c - a*d)^2*sqrt(a + b*x^2)*sqrt(c + d*x^2)) + (sqrt(d)*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*sqrt(a + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*sqrt(c)*(b*c - a*d)^3*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2)) - (b*sqrt(c)*sqrt(d)*(b*c - 9*a*d)*sqrt(a + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(c)), 1 - (b*c)/(a*d)))/(3*a^2*(b*c - a*d)^3*sqrt((c*(a + b*x^2))/(a*(c + d*x^2)))*sqrt(c + d*x^2))]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    @test_int [sqrt(2 + b*x^2)/sqrt(3 + d*x^2), x, 4, (x*sqrt(2 + b*x^2))/sqrt(3 + d*x^2) - (sqrt(2)*sqrt(2 + b*x^2)*Elliptic.E(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(sqrt(d)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2)) + (sqrt(2)*sqrt(2 + b*x^2)*Elliptic.F(atan((sqrt(d)*x)/sqrt(3)), 1 - (3*b)/(2*d)))/(sqrt(d)*sqrt((2 + b*x^2)/(3 + d*x^2))*sqrt(3 + d*x^2))]
    @test_int [sqrt(-1 + 3*x^2)/sqrt(2 - 3*x^2), x, 1, -(Elliptic.E(acos(sqrt(3/2)*x), 2)/sqrt(3))]

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

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

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

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

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

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

    #= ::Subsubsection::Closed:: =#
    #=a>0=#

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

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

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

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

    #= ::Subsubsection::Closed:: =#
    #=a<0=#

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

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

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

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

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

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

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

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

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

    @test_int [(a + b*x^2)^(7/4)/(c + d*x^2)^2, x, 9, (b*(5*b*c - a*d)*x)/(2*c*d^2*(a + b*x^2)^(1/4)) - ((b*c - a*d)*x*(a + b*x^2)^(3/4))/(2*c*d*(c + d*x^2)) - (sqrt(a)*sqrt(b)*(5*b*c - a*d)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*d^2*(a + b*x^2)^(1/4)) + (a^(1/4)*sqrt((-b)*c + a*d)*(5*b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^(5/2)*x) - (a^(1/4)*sqrt((-b)*c + a*d)*(5*b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^(5/2)*x)]
    @test_int [(a + b*x^2)^(5/4)/(c + d*x^2)^2, x, 9, -(((b*c - a*d)*x*(a + b*x^2)^(1/4))/(2*c*d*(c + d*x^2))) + (sqrt(a)*sqrt(b)*(3*b*c + a*d)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*d^2*(a + b*x^2)^(3/4)) - (a^(1/4)*(3*b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^2*x) - (a^(1/4)*(3*b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^2*x)]
    @test_int [(a + b*x^2)^(3/4)/(c + d*x^2)^2, x, 9, -((b*x)/(2*c*d*(a + b*x^2)^(1/4))) + (x*(a + b*x^2)^(3/4))/(2*c*(c + d*x^2)) + (sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*d*(a + b*x^2)^(1/4)) + (a^(1/4)*(b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^(3/2)*sqrt((-b)*c + a*d)*x) - (a^(1/4)*(b*c + 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d^(3/2)*sqrt((-b)*c + a*d)*x)]
    @test_int [(a + b*x^2)^(1/4)/(c + d*x^2)^2, x, 9, (x*(a + b*x^2)^(1/4))/(2*c*(c + d*x^2)) + (sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*d*(a + b*x^2)^(3/4)) - (a^(1/4)*(b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d*(b*c - a*d)*x) - (a^(1/4)*(b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*d*(b*c - a*d)*x)]
    @test_int [1/((a + b*x^2)^(1/4)*(c + d*x^2)^2), x, 9, (b*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)) - (d*x*(a + b*x^2)^(3/4))/(2*c*(b*c - a*d)*(c + d*x^2)) - (sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)) - (a^(1/4)*(3*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*sqrt(d)*((-b)*c + a*d)^(3/2)*x) + (a^(1/4)*(3*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*sqrt(d)*((-b)*c + a*d)^(3/2)*x)]
    @test_int [1/((a + b*x^2)^(3/4)*(c + d*x^2)^2), x, 9, -((d*x*(a + b*x^2)^(1/4))/(2*c*(b*c - a*d)*(c + d*x^2))) - (sqrt(a)*sqrt(b)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*c*(b*c - a*d)*(a + b*x^2)^(3/4)) + (a^(1/4)*(5*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^2*x) + (a^(1/4)*(5*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^2*x)]
    @test_int [1/((a + b*x^2)^(5/4)*(c + d*x^2)^2), x, 10, -((d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(1/4)*(c + d*x^2))) + (sqrt(b)*(4*b*c + a*d)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(2*sqrt(a)*c*(b*c - a*d)^2*(a + b*x^2)^(1/4)) - (a^(1/4)*sqrt(d)*(7*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*((-b)*c + a*d)^(5/2)*x) + (a^(1/4)*sqrt(d)*(7*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*((-b)*c + a*d)^(5/2)*x)]
    @test_int [1/((a + b*x^2)^(7/4)*(c + d*x^2)^2), x, 10, (b*(4*b*c + 3*a*d)*x)/(6*a*c*(b*c - a*d)^2*(a + b*x^2)^(3/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(3/4)*(c + d*x^2)) + (sqrt(b)*(4*b*c + 3*a*d)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(6*sqrt(a)*c*(b*c - a*d)^2*(a + b*x^2)^(3/4)) - (a^(1/4)*d*(9*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^3*x) - (a^(1/4)*d*(9*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^3*x)]
    @test_int [1/((a + b*x^2)^(9/4)*(c + d*x^2)^2), x, 11, (b*(4*b*c + 5*a*d)*x)/(10*a*c*(b*c - a*d)^2*(a + b*x^2)^(5/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(5/4)*(c + d*x^2)) + (sqrt(b)*(12*b^2*c^2 - 52*a*b*c*d - 5*a^2*d^2)*(1 + (b*x^2)/a)^(1/4)*Elliptic.E((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(10*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(1/4)) - (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*((-b)*c + a*d)^(7/2)*x) + (a^(1/4)*d^(3/2)*(11*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*((-b)*c + a*d)^(7/2)*x)]
    @test_int [1/((a + b*x^2)^(11/4)*(c + d*x^2)^2), x, 11, (b*(4*b*c + 7*a*d)*x)/(14*a*c*(b*c - a*d)^2*(a + b*x^2)^(7/4)) + (b*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*x)/(42*a^2*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) - (d*x)/(2*c*(b*c - a*d)*(a + b*x^2)^(7/4)*(c + d*x^2)) + (sqrt(b)*(20*b^2*c^2 - 76*a*b*c*d - 21*a^2*d^2)*(1 + (b*x^2)/a)^(3/4)*Elliptic.F((1/2)*atan((sqrt(b)*x)/sqrt(a)), 2))/(42*a^(3/2)*c*(b*c - a*d)^3*(a + b*x^2)^(3/4)) + (a^(1/4)*d^2*(13*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi(-((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d)), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^4*x) + (a^(1/4)*d^2*(13*b*c - 2*a*d)*sqrt(-((b*x^2)/a))*Elliptic.Pi((sqrt(a)*sqrt(d))/sqrt((-b)*c + a*d), asin((a + b*x^2)^(1/4)/a^(1/4)), -1))/(4*c*(b*c - a*d)^4*x)]

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

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

    @test_int [(a + b*x^2)^p*(c + d*x^2)^3, x, 5, If(13>=8, (d*(15*a^2*d^2 - 8*a*b*c*d*(6 + p) + b^2*c^2*(57 + 28*p + 4*p^2))*x*(a + b*x^2)^(1 + p))/(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p)) - (d*(5*a*d - b*c*(11 + 2*p))*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^2*(5 + 2*p)*(7 + 2*p)) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*(7 + 2*p)) - ((15*a^3*d^3 - 9*a^2*b*c*d^2*(7 + 2*p) + 3*a*b^2*c^2*d*(35 + 24*p + 4*p^2) - b^3*c^3*(105 + 142*p + 60*p^2 + 8*p^3))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b^3*(3 + 2*p)*(5 + 2*p)*(7 + 2*p))), (d*(15*a^2*d^2 - 8*a*b*c*d*(6 + p) + b^2*c^2*(57 + 28*p + 4*p^2))*x*(a + b*x^2)^(1 + p))/(b^3*(105 + 142*p + 60*p^2 + 8*p^3)) - (d*(5*a*d - b*c*(11 + 2*p))*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^2*(35 + 24*p + 4*p^2)) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*(7 + 2*p)) - ((15*a^3*d^3 - 9*a^2*b*c*d^2*(7 + 2*p) + 3*a*b^2*c^2*d*(35 + 24*p + 4*p^2) - b^3*c^3*(105 + 142*p + 60*p^2 + 8*p^3))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p/(b^3*(105 + 142*p + 60*p^2 + 8*p^3)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^2, x, 4, If(13>=8, -((d*(3*a*d - b*c*(7 + 2*p))*x*(a + b*x^2)^(1 + p))/(b^2*(3 + 2*p)*(5 + 2*p))) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b*(5 + 2*p)) + ((3*a^2*d^2 - 2*a*b*c*d*(5 + 2*p) + b^2*c^2*(15 + 16*p + 4*p^2))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b^2*(3 + 2*p)*(5 + 2*p))), -((d*(3*a*d - b*c*(7 + 2*p))*x*(a + b*x^2)^(1 + p))/(b^2*(15 + 16*p + 4*p^2))) + (d*x*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b*(5 + 2*p)) + ((3*a^2*d^2 - 2*a*b*c*d*(5 + 2*p) + b^2*c^2*(15 + 16*p + 4*p^2))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b^2*(15 + 16*p + 4*p^2))))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^1, x, 3, (d*x*(a + b*x^2)^(1 + p))/(b*(3 + 2*p)) - ((a*d - b*c*(3 + 2*p))*x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/((1 + (b*x^2)/a)^p*(b*(3 + 2*p)))]
    @test_int [(a + b*x^2)^p*(c + d*x^2)^0, x, 2, (x*(a + b*x^2)^p*HypergeometricFunctions._₂F₁(1/2, -p, 3/2, -((b*x^2)/a)))/(1 + (b*x^2)/a)^p]
    @test_int [(a + b*x^2)^p/(c + d*x^2)^1, x, 2, (x*(a + b*x^2)^p*AppellF1(1/2, -p, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*c)]
    @test_int [(a + b*x^2)^p/(c + d*x^2)^2, x, 2, (x*(a + b*x^2)^p*AppellF1(1/2, -p, 2, 3/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*c^2)]
    @test_int [(a + b*x^2)^p/(c + d*x^2)^3, x, 2, (x*(a + b*x^2)^p*AppellF1(1/2, -p, 3, 3/2, -((b*x^2)/a), -((d*x^2)/c)))/((1 + (b*x^2)/a)^p*c^3)]

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

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.2 Quadratic/1.1.2.3 (a+b x^2)^p (c+d x^2)^q.jl