text:
@testset "1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q" begin
    (A, B, a, b, c, d, e, f, g, h, x, ) = @variables A B a b c d e f g h x

    #= ::Package:: =#

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

    #= ::Section:: =#
    #=Integrands*of*the*form*(g+h*x)*(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*with*b^2-4*a*c=0=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(g+h*x)*(a+b*x+c*x^2)^p*(d+e*x+f*x^2)^q*with*e=0=#

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

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

    @test_int [(A + B*x)*(a + b*x + c*x^2)^1/(d + f*x^2), x, 5, ((b*B + A*c)*x)/f + (B*c*x^2)/(2*f) - ((b*B*d + A*c*d - a*A*f)*atan((sqrt(f)*x)/sqrt(d)))/(sqrt(d)*f^(3/2)) - ((B*c*d - A*b*f - a*B*f)*log(d + f*x^2))/(2*f^2)]
    @test_int [(A + B*x)*(a + b*x + c*x^2)^2/(d + f*x^2), x, 5, ((A*b^2*f - A*c*(c*d - 2*a*f) - b*B*(2*c*d - 2*a*f))*x)/f^2 + ((2*A*b*c*f - B*(c^2*d - b^2*f - 2*a*c*f))*x^2)/(2*f^2) + (c*(2*b*B + A*c)*x^3)/(3*f) + (B*c^2*x^4)/(4*f) - ((A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2)*atan((sqrt(f)*x)/sqrt(d)))/(sqrt(d)*f^(5/2)) - ((2*A*b*f*(c*d - a*f) - B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*log(d + f*x^2))/(2*f^3)]
    @test_int [(A + B*x)*(a + b*x + c*x^2)^3/(d + f*x^2), x, 5, -(((b^3*B*d*f + 3*A*b^2*f*(c*d - a*f) - 3*b*B*(c*d - a*f)^2 - A*c*(c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2))*x)/f^3) - ((A*b*f*(3*c^2*d - b^2*f - 6*a*c*f) - B*(c^3*d^2 - 3*a*c^2*d*f + 3*a*b^2*f^2 - 3*c*f*(b^2*d - a^2*f)))*x^2)/(2*f^3) + ((b^3*B*f + 3*A*b^2*c*f - A*c^2*(c*d - 3*a*f) - 3*b*B*c*(c*d - 2*a*f))*x^3)/(3*f^2) + (c*(3*A*b*c*f - B*(c^2*d - 3*b^2*f - 3*a*c*f))*x^4)/(4*f^2) + (c^2*(3*b*B + A*c)*x^5)/(5*f) + (B*c^3*x^6)/(6*f) + ((b^3*B*d^2*f + 3*A*b^2*d*f*(c*d - a*f) - 3*b*B*d*(c*d - a*f)^2 - A*(c*d - a*f)^3)*atan((sqrt(f)*x)/sqrt(d)))/(sqrt(d)*f^(7/2)) + ((A*b*f*(3*c^2*d^2 - 6*a*c*d*f - f*(b^2*d - 3*a^2*f)) - B*(c*d - a*f)*(c^2*d^2 - 2*a*c*d*f - f*(3*b^2*d - a^2*f)))*log(d + f*x^2))/(2*f^4)]

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

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

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

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

    @test_int [(A + B*x)*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 9, -((B*sqrt(a + b*x + c*x^2))/f) - ((b*B + 2*A*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f) - ((B*sqrt(d) - A*sqrt(f))*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f^(3/2)) + ((B*sqrt(d) + A*sqrt(f))*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f^(3/2))]
    #= [(A + B*x)*(a + b*x + c*x^2)^(3/2)/(d + f*x^2), x, 10, -(((B*c*d - A*b*f - a*B*f)*sqrt(a + b*x + c*x^2))/f^2) + ((b*B + 2*A*c)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*f) + (B*(a + b*x + c*x^2)^(3/2))/(3*f) + ((b*f*(b*B*d + A*c*d - a*A*f) - (c*d - a*f)*(B*c*d - A*b*f - a*B*f) - (sqrt(f)*(A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2))/sqrt(-d))*atan((b*sqrt(-d) - 2*a*sqrt(f) + (2*c*sqrt(-d) - b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(-d)*sqrt(f) - a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2)*sqrt(c*d + b*sqrt(-d)*sqrt(f) - a*f)) - ((b*f*(b*B*d + A*c*d - a*A*f) - (c*d - a*f)*(B*c*d - A*b*f - a*B*f) + (sqrt(f)*(A*b^2*d*f - 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2))/sqrt(-d))*atan((b*sqrt(-d) + 2*a*sqrt(f) + (2*c*sqrt(-d) + b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(-d)*sqrt(f) - a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2)*sqrt(c*d - b*sqrt(-d)*sqrt(f) - a*f)) - ((b^2 - 4*a*c)*(b*B + 2*A*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*f) + ((A*b^2*f - 2*A*c*(c*d - a*f) - b*B*(3*c*d - a*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^2)] =#

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

    @test_int [(A + B*x)/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 5, -(((B - (A*sqrt(f))/sqrt(d))*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(f)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f))) + ((B + (A*sqrt(f))/sqrt(d))*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(f)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [(A + B*x)/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 6, -((2*(a*B*(2*c^2*d - b^2*f + 2*a*c*f) + A*(b^3*f - b*c*(c*d + 3*a*f)) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2))) - ((B*sqrt(d) - A*sqrt(f))*sqrt(f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + ((B*sqrt(d) + A*sqrt(f))*sqrt(f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [(A + B*x)/((a + b*x + c*x^2)^(5/2)*(d - f*x^2)), x, 7, -((2*(a*B*(2*c^2*d - b^2*f + 2*a*c*f) + A*(b^3*f - b*c*(c*d + 3*a*f)) + c*(A*b^2*f + b*B*(c*d - a*f) - 2*A*c*(c*d + a*f))*x))/(3*(b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*(a + b*x + c*x^2)^(3/2))) - (2*(3*b^6*B*d*f^2 + 24*a^2*B*c^2*f*(c*d + a*f)^2 - A*b^5*f^2*(7*c*d + 6*a*f) - b^4*B*f*(7*c^2*d^2 + 14*a*c*d*f - 3*a^2*f^2) + A*b^3*c*f*(15*c^2*d^2 + 46*a*c*d*f + 43*a^2*f^2) + 2*b^2*B*c*(2*c^3*d^3 + 5*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 11*a^3*f^3) - 4*A*b*c^2*(2*c^3*d^3 + 9*a*c^2*d^2*f + 24*a^2*c*d*f^2 + 17*a^3*f^3) + c*(3*b^5*B*d*f^2 - 2*A*b^4*f^2*(4*c*d + 3*a*f) - 8*A*c^2*(c*d + a*f)^2*(2*c*d + 5*a*f) - b^3*B*f*(17*c^2*d^2 + 10*a*c*d*f - 3*a^2*f^2) + 2*A*b^2*c*f*(15*c^2*d^2 + 22*a*c*d*f + 19*a^2*f^2) + 4*b*B*c*(2*c^3*d^3 + 11*a*c^2*d^2*f + 4*a^2*c*d*f^2 - 5*a^3*f^3))*x))/(3*(b^2 - 4*a*c)^2*(c^2*d^2 + 2*a*c*d*f - f*(b^2*d - a^2*f))^2*sqrt(a + b*x + c*x^2)) - ((B*sqrt(d) - A*sqrt(f))*f^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(5/2)) + ((B*sqrt(d) + A*sqrt(f))*f^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(5/2))]

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

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

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

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

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

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

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

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

    @test_int [(A + B*x)/((a + b*x + c*x^2)^1*(d + e*x + f*x^2)), x, 9, -(((A*b^2*f + 2*c*(A*c*d + a*B*e - a*A*f) - b*(B*c*d + A*c*e + a*B*f))*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/(sqrt(b^2 - 4*a*c)*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f))))) + ((B*(c*d*e - 2*b*d*f + a*e*f) - A*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2))*atanh((e + 2*f*x)/sqrt(e^2 - 4*d*f)))/(sqrt(e^2 - 4*d*f)*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))) + ((B*c*d - A*c*e + A*b*f - a*B*f)*log(a + b*x + c*x^2))/(2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))) - ((B*c*d - A*c*e + A*b*f - a*B*f)*log(d + e*x + f*x^2))/(2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f))))]
    @test_int [(A + B*x)/((a + b*x + c*x^2)^2*(d + e*x + f*x^2)), x, 10, -((A*c*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(A*b^2*f + 2*c*(A*c*d + a*B*e - a*A*f) - b*(B*c*d + A*c*e + a*B*f))*x)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(a + b*x + c*x^2))) - (1/((b^2 - 4*a*c)^(3/2)*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))^2))*((b^5*(B*d - A*e)*f^2 - 2*b^4*f*(B*c*d*e - A*(c*e^2 - c*d*f + a*f^2)) - 4*c^2*(A*(c^3*d^3 - 3*a^3*f^3 - a^2*c*f*(e^2 - 7*d*f) + a*c^2*d*(3*e^2 - 5*d*f)) - a*B*e*(c^2*d^2 - 3*a^2*f^2 - a*c*(e^2 - 2*d*f))) - 4*b^2*c*(B*c^2*d^2*e + A*f*(2*c^2*d^2 + 3*a^2*f^2 + 3*a*c*(e^2 - d*f))) + 2*b*c*(B*(c^3*d^3 + 3*a^3*f^3 + a*c^2*d*(e^2 - 7*d*f) + 3*a^2*c*f*(e^2 + d*f)) + A*c*e*(3*c^2*d^2 + 3*a^2*f^2 + a*c*(3*e^2 + 2*d*f))) - b^3*(A*c*e*(c*e^2 - 2*c*d*f - 4*a*f^2) + B*(4*a*c*d*f^2 + a^2*f^3 - c^2*d*(e^2 + 5*d*f))))*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c))) + ((B*(c^2*d*e*(e^2 - 3*d*f) - 2*c*d*f*(b*e^2 - 2*b*d*f - a*e*f) + f^2*(b^2*d*e - 4*a*b*d*f + a^2*e*f)) - A*(c^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2) - f^2*(2*a*b*e*f - 2*a^2*f^2 - b^2*(e^2 - 2*d*f)) + 2*c*f*(a*f*(e^2 - 2*d*f) - b*(e^3 - 3*d*e*f))))*atanh((e + 2*f*x)/sqrt(e^2 - 4*d*f)))/(sqrt(e^2 - 4*d*f)*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))^2) + ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*log(a + b*x + c*x^2))/(2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))^2) - ((A*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) - B*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f)))*log(d + e*x + f*x^2))/(2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))^2)]

    @test_int [(g + h*x)/((a + b*x + c*x^2)*(a*d + b*d*x + c*d*x^2)^2), x, 5, -((b*g - 2*a*h + (2*c*g - b*h)*x)/(2*(b^2 - 4*a*c)*d^2*(a + b*x + c*x^2)^2)) + (3*(2*c*g - b*h)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*d^2*(a + b*x + c*x^2)) - (6*c*(2*c*g - b*h)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*d^2)]
    @test_int [(g + h*x)/((a + b*x + c*x^2)^2*(a*d + b*d*x + c*d*x^2)), x, 5, -((b*g - 2*a*h + (2*c*g - b*h)*x)/(2*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^2)) + (3*(2*c*g - b*h)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^2*d*(a + b*x + c*x^2)) - (6*c*(2*c*g - b*h)*atanh((b + 2*c*x)/sqrt(b^2 - 4*a*c)))/((b^2 - 4*a*c)^(5/2)*d)]

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

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

    @test_int [(A + B*x)*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 9, (B*sqrt(a + b*x + c*x^2))/f - ((2*B*c*e - b*B*f - 2*A*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^2) + ((2*f*(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e - sqrt(e^2 - 4*d*f))*(A*f*(c*e - b*f) + B*(f*(b*e - a*f) - c*(e^2 - d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) - ((2*f*(A*f*(c*d - a*f) - B*d*(c*e - b*f)) - (e + sqrt(e^2 - 4*d*f))*(A*f*(c*e - b*f) + B*(f*(b*e - a*f) - c*(e^2 - d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    @test_int [(A + B*x)*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 10, -((1/(8*c*f^3))*((2*A*c*f*(4*c*e - 5*b*f) - B*(b^2*f^2 - 2*c*f*(5*b*e - 4*a*f) + 8*c^2*(e^2 - d*f)) + 2*c*f*(2*B*c*e - b*B*f - 2*A*c*f)*x)*sqrt(a + b*x + c*x^2))) + (B*(a + b*x + c*x^2)^(3/2))/(3*f) + (1/(16*c^(3/2)*f^4))*((2*A*c*f*(3*b^2*f^2 - 12*c*f*(b*e - a*f) + 8*c^2*(e^2 - d*f)) - B*(b^3*f^3 + 6*b*c*f^2*(b*e - 2*a*f) - 24*c^2*f*(b*e^2 - b*d*f - a*e*f) + 16*c^3*(e^3 - 2*d*e*f)))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))) - ((2*c*f*(B*d*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f))) - c*(e - sqrt(e^2 - 4*d*f))*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f)))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*c*f^4*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) + ((2*f*(B*d*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + A*f*(2*c*d*f*(b*e - a*f) - f^2*(b^2*d - a^2*f) - c^2*d*(e^2 - d*f))) - (e + sqrt(e^2 - 4*d*f))*(A*f*(c*e - b*f)*(f*(b*e - 2*a*f) - c*(e^2 - 2*d*f)) + B*(c^2*(e^4 - 3*d*e^2*f + d^2*f^2) - f^2*(2*a*b*e*f - a^2*f^2 - b^2*(e^2 - d*f)) + 2*c*f*(a*f*(e^2 - d*f) - b*(e^3 - 2*d*e*f)))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^4*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]

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

    @test_int [(A + B*x)/((a + b*x + c*x^2)*sqrt(d + e*x + f*x^2)), x, 5, ((b*B - 2*A*c - B*sqrt(b^2 - 4*a*c))*atanh((4*c*d - (b - sqrt(b^2 - 4*a*c))*e + 2*(c*e - (b - sqrt(b^2 - 4*a*c))*f)*x)/(2*sqrt(2)*sqrt(2*c^2*d - b*c*e + b^2*f - 2*a*c*f + sqrt(b^2 - 4*a*c)*(c*e - b*f))*sqrt(d + e*x + f*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2*d - b*c*e + b^2*f - 2*a*c*f + sqrt(b^2 - 4*a*c)*(c*e - b*f))) + ((2*A*c - B*(b + sqrt(b^2 - 4*a*c)))*atanh((4*c*d - (b + sqrt(b^2 - 4*a*c))*e + 2*(c*e - (b + sqrt(b^2 - 4*a*c))*f)*x)/(2*sqrt(2)*sqrt(2*c^2*d - b*c*e + b^2*f - 2*a*c*f - sqrt(b^2 - 4*a*c)*(c*e - b*f))*sqrt(d + e*x + f*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2*d - b*c*e + b^2*f - 2*a*c*f - sqrt(b^2 - 4*a*c)*(c*e - b*f)))]
    @test_int [(A + B*x)/((a + 0*x + c*x^2)*sqrt(d + e*x + f*x^2)), x, 5, (sqrt(a*B*e + A*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt((-A)*c*e + B*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*atanh((sqrt(e)*(a*(A*c*e - B*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)))) - c*(a*B*e + A*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*x))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(a*B*e + A*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt((-A)*c*e + B*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt(d + e*x + f*x^2))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(e)*sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))) - (sqrt((-A)*c*e + B*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt(a*B*e + A*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*atanh((sqrt(e)*(a*(A*c*e - B*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)))) - c*(a*B*e + A*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*x))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt((-A)*c*e + B*(c*d - a*f - sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt(a*B*e + A*(c*d - a*f + sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))))*sqrt(d + e*x + f*x^2))))/(sqrt(2)*sqrt(a)*sqrt(c)*sqrt(e)*sqrt(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f)))]
    @test_int [(A + B*x)/((a + b*x + c*x^2)*sqrt(d + 0*x + f*x^2)), x, 5, ((b*B - 2*A*c - B*sqrt(b^2 - 4*a*c))*atanh((2*c*d - (b - sqrt(b^2 - 4*a*c))*f*x)/(sqrt(2)*sqrt(2*c^2*d - 2*a*c*f + b*(b - sqrt(b^2 - 4*a*c))*f)*sqrt(d + f*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2*d - 2*a*c*f + b*(b - sqrt(b^2 - 4*a*c))*f)) + ((2*A*c - B*(b + sqrt(b^2 - 4*a*c)))*atanh((2*c*d - (b + sqrt(b^2 - 4*a*c))*f*x)/(sqrt(2)*sqrt(2*c^2*d - 2*a*c*f + b*(b + sqrt(b^2 - 4*a*c))*f)*sqrt(d + f*x^2))))/(sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(2*c^2*d - 2*a*c*f + b*(b + sqrt(b^2 - 4*a*c))*f))]
    @test_int [(A + B*x)/((a + 0*x + c*x^2)*sqrt(d + 0*x + f*x^2)), x, 6, (A*atan((sqrt(c*d - a*f)*x)/(sqrt(a)*sqrt(d + f*x^2))))/(sqrt(a)*sqrt(c*d - a*f)) - (B*atanh((sqrt(c)*sqrt(d + f*x^2))/sqrt(c*d - a*f)))/(sqrt(c)*sqrt(c*d - a*f))]

    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(1/2)), x, 5, (1/2)*sqrt(-(13/5) + sqrt(10))*atan((3*(4 - sqrt(10)) + (1 + 4*sqrt(10))*x)/(2*sqrt(1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2))) + (1/2)*sqrt(13/5 + sqrt(10))*atanh((3*(4 + sqrt(10)) + (1 - 4*sqrt(10))*x)/(2*sqrt(-1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2)))]
    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(3/2)), x, 7, -((2*(15 + 14*x))/(17*sqrt(1 + 3*x - 2*x^2))) - (9/2)*sqrt((1/5)*(-3 + sqrt(10)))*atan((3*(4 - sqrt(10)) + (1 + 4*sqrt(10))*x)/(2*sqrt(1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2))) + (9/2)*sqrt((1/5)*(3 + sqrt(10)))*atanh((3*(4 + sqrt(10)) + (1 - 4*sqrt(10))*x)/(2*sqrt(-1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2)))]
    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x - 2*x^2)^(5/2)), x, 7, -((2*(15 + 14*x))/(51*(1 + 3*x - 2*x^2)^(3/2))) - (2*(291 + 4814*x))/(867*sqrt(1 + 3*x - 2*x^2)) + (9/2)*sqrt((1/5)*(-53 + 17*sqrt(10)))*atan((3*(4 - sqrt(10)) + (1 + 4*sqrt(10))*x)/(2*sqrt(1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2))) + (9/2)*sqrt((1/5)*(53 + 17*sqrt(10)))*atanh((3*(4 + sqrt(10)) + (1 - 4*sqrt(10))*x)/(2*sqrt(-1 + sqrt(10))*sqrt(1 + 3*x - 2*x^2)))]

    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(1/2)), x, 5, (-(1/2))*sqrt(1 + (7*sqrt(2/5))/5)*atanh((3*(4 - sqrt(10)) + (17 - 4*sqrt(10))*x)/(2*sqrt(55 - 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2))) + (1/2)*sqrt(1 - (7*sqrt(2/5))/5)*atanh((3*(4 + sqrt(10)) + (17 + 4*sqrt(10))*x)/(2*sqrt(55 + 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2)))]
    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(3/2)), x, 6, (2*(21 + 22*x))/(5*sqrt(1 + 3*x + 2*x^2)) - (1/10)*sqrt((3/5)*(2065 + 653*sqrt(10)))*atanh((3*(4 - sqrt(10)) + (17 - 4*sqrt(10))*x)/(2*sqrt(55 - 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2))) + (1/10)*sqrt((3/5)*(2065 - 653*sqrt(10)))*atanh((3*(4 + sqrt(10)) + (17 + 4*sqrt(10))*x)/(2*sqrt(55 + 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2)))]
    @test_int [(2 + x)/((2 + 4*x - 3*x^2)*(1 + 3*x + 2*x^2)^(5/2)), x, 7, (2*(21 + 22*x))/(15*(1 + 3*x + 2*x^2)^(3/2)) + (2*(273 + 230*x))/(15*sqrt(1 + 3*x + 2*x^2)) - (1/50)*sqrt((1/3)*(4885115 + 1544809*sqrt(10)))*atanh((3*(4 - sqrt(10)) + (17 - 4*sqrt(10))*x)/(2*sqrt(55 - 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2))) + (1/50)*sqrt((1/3)*(4885115 - 1544809*sqrt(10)))*atanh((3*(4 + sqrt(10)) + (17 + 4*sqrt(10))*x)/(2*sqrt(55 + 17*sqrt(10))*sqrt(1 + 3*x + 2*x^2)))]

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

    @test_int [(1 + 2*x)/((3 + x + x^2)*sqrt(5 + x + x^2)), x, 2, (-sqrt(2))*atanh(sqrt(5 + x + x^2)/sqrt(2))]
    @test_int [x/((3 + x + x^2)*sqrt(5 + x + x^2)), x, 5, -(atan((sqrt(2/11)*(1 + 2*x))/sqrt(5 + x + x^2))/sqrt(22)) - atanh(sqrt(5 + x + x^2)/sqrt(2))/sqrt(2)]

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

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

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

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

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

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

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

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

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

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

    @test_int [x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 6, -((a*c*x*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(8*d*(a + b*x))) + (b*x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((8*b*c - 15*a*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + d*x^2)^(3/2))/(60*d^2*(a + b*x)) - (a*c^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(3/2)*(a + b*x))]
    @test_int [x^1*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 5, -((b*c*x*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(8*d*(a + b*x))) + ((4*a + 3*b*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + d*x^2)^(3/2))/(12*d*(a + b*x)) - (b*c^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(8*d^(3/2)*(a + b*x))]
    @test_int [x^0*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 5, (a*x*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(2*(a + b*x)) + (b*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + d*x^2)^(3/2))/(3*d*(a + b*x)) + (a*c*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*sqrt(d)*(a + b*x))]
    @test_int [1/x^1*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 8, ((2*a + b*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(2*(a + b*x)) + (b*c*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(2*sqrt(d)*(a + b*x)) - (a*sqrt(c)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(a + b*x)]
    @test_int [1/x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 8, -(((a - b*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(x*(a + b*x))) + (a*sqrt(d)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(a + b*x) - (b*sqrt(c)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(a + b*x)]
    @test_int [1/x^3*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2), x, 8, -(((a + 2*b*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + d*x^2))/(2*x^2*(a + b*x))) + (b*sqrt(d)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((sqrt(d)*x)/sqrt(c + d*x^2)))/(a + b*x) - (a*d*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh(sqrt(c + d*x^2)/sqrt(c)))/(2*sqrt(c)*(a + b*x))]

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

    @test_int [x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 6, -(((2*a*d*(4*c*d - 5*e^2) - b*(12*c*d*e - 7*e^3))*(e + 2*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(128*d^4*(a + b*x))) + (b*x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + e*x + d*x^2)^(3/2))/(5*d*(a + b*x)) - ((32*b*c*d + 50*a*d*e - 35*b*e^2 - 6*d*(10*a*d - 7*b*e)*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + e*x + d*x^2)^(3/2))/(240*d^3*(a + b*x)) - ((4*c*d - e^2)*(8*a*c*d^2 - 12*b*c*d*e - 10*a*d*e^2 + 7*b*e^3)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(256*d^(9/2)*(a + b*x))]
    @test_int [x^1*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 5, -(((4*b*c*d + 8*a*d*e - 5*b*e^2)*(e + 2*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(64*d^3*(a + b*x))) + ((8*a*d - 5*b*e + 6*b*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + e*x + d*x^2)^(3/2))/(24*d^2*(a + b*x)) - ((4*c*d - e^2)*(4*b*c*d + 8*a*d*e - 5*b*e^2)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(128*d^(7/2)*(a + b*x))]
    @test_int [x^0*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 5, ((2*a*d - b*e)*(e + 2*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(8*d^2*(a + b*x)) + (b*sqrt(a^2 + 2*a*b*x + b^2*x^2)*(c + e*x + d*x^2)^(3/2))/(3*d*(a + b*x)) + ((2*a*d - b*e)*(4*c*d - e^2)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(16*d^(5/2)*(a + b*x))]
    @test_int [1/x^1*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 7, ((4*a*d + b*e + 2*b*d*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(4*d*(a + b*x)) + ((4*b*c*d + 4*a*d*e - b*e^2)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(8*d^(3/2)*(a + b*x)) - (a*sqrt(c)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((2*c + e*x)/(2*sqrt(c)*sqrt(c + e*x + d*x^2))))/(a + b*x)]
    @test_int [1/x^2*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 7, -(((a - b*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(x*(a + b*x))) + ((2*a*d + b*e)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(2*sqrt(d)*(a + b*x)) - ((2*b*c + a*e)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((2*c + e*x)/(2*sqrt(c)*sqrt(c + e*x + d*x^2))))/(2*sqrt(c)*(a + b*x))]
    @test_int [1/x^3*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2), x, 7, -(((2*a*c + (4*b*c + a*e)*x)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*sqrt(c + e*x + d*x^2))/(4*c*x^2*(a + b*x))) + (b*sqrt(d)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((e + 2*d*x)/(2*sqrt(d)*sqrt(c + e*x + d*x^2))))/(a + b*x) - ((4*a*c*d + 4*b*c*e - a*e^2)*sqrt(a^2 + 2*a*b*x + b^2*x^2)*atanh((2*c + e*x)/(2*sqrt(c)*sqrt(c + e*x + d*x^2))))/(8*c^(3/2)*(a + b*x))]

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

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

    @test_int [x^2*sqrt(a + c*x^2)/(d + e*x + f*x^2), x, 9, -(((2*e - f*x)*sqrt(a + c*x^2))/(2*f^2)) + ((a*f^2 + 2*c*(e^2 - d*f))*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*sqrt(c)*f^3) - ((e*(e - sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((e*(e + sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - 2*d*f)) - 2*d*f*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^1*sqrt(a + c*x^2)/(d + e*x + f*x^2), x, 9, sqrt(a + c*x^2)/f - (sqrt(c)*e*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/f^2 - ((2*c*d*e*f - (e - sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((2*c*d*e*f - (e + sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^0*sqrt(a + c*x^2)/(d + e*x + f*x^2), x, 8, (sqrt(c)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/f - (sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f)) + (sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f))]
    @test_int [sqrt(a + c*x^2)/(x^1*(d + e*x + f*x^2)), x, 12, ((2*a*e*f + (c*d - a*f)*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - ((2*a*e*f + (c*d - a*f)*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - (sqrt(a)*atanh(sqrt(a + c*x^2)/sqrt(a)))/d]
    @test_int [sqrt(a + c*x^2)/(x^2*(d + e*x + f*x^2)), x, 18, -(sqrt(a + c*x^2)/(d*x)) - (f*(2*c*d^2 + a*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + (f*(2*c*d^2 + a*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) + (sqrt(a)*e*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^2]
    @test_int [sqrt(a + c*x^2)/(x^3*(d + e*x + f*x^2)), x, 22, -(sqrt(a + c*x^2)/(2*d*x^2)) + (e*sqrt(a + c*x^2))/(d^2*x) + (f*(c*d^2*(e + sqrt(e^2 - 4*d*f)) + a*(e^3 - 3*d*e*f + e^2*sqrt(e^2 - 4*d*f) - d*f*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(c*d^2*(e - sqrt(e^2 - 4*d*f)) + a*(e^3 - 3*d*e*f - e^2*sqrt(e^2 - 4*d*f) + d*f*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - (c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*sqrt(a)*d) - (sqrt(a)*(e^2 - d*f)*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^3]

    @test_int [x^2*(a + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 10, -(((8*e*(a*f^2 + c*(e^2 - 2*d*f)) - f*(3*a*f^2 + 4*c*(e^2 - d*f))*x)*sqrt(a + c*x^2))/(8*f^4)) - ((4*e - 3*f*x)*(a + c*x^2)^(3/2))/(12*f^2) + ((3*a^2*f^4 + 12*a*c*f^2*(e^2 - d*f) + 8*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(8*sqrt(c)*f^5) - ((a^2*f^4*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*sqrt(e^2 - 4*d*f) + 2*d*e*f*sqrt(e^2 - 4*d*f)) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 - e^5*sqrt(e^2 - 4*d*f) + 4*d*e^3*f*sqrt(e^2 - 4*d*f) - 3*d^2*e*f^2*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^5*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((a^2*f^4*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + 2*a*c*f^2*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*sqrt(e^2 - 4*d*f) - 2*d*e*f*sqrt(e^2 - 4*d*f)) + c^2*(e^6 - 6*d*e^4*f + 9*d^2*e^2*f^2 - 2*d^3*f^3 + e^5*sqrt(e^2 - 4*d*f) - 4*d*e^3*f*sqrt(e^2 - 4*d*f) + 3*d^2*e*f^2*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^5*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^1*(a + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 10, ((2*(a*f^2 + c*(e^2 - d*f)) - c*e*f*x)*sqrt(a + c*x^2))/(2*f^3) + (a + c*x^2)^(3/2)/(3*f) - (sqrt(c)*e*(3*a*f^2 + 2*c*(e^2 - 2*d*f))*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*f^4) - ((2*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f)) - (e - sqrt(e^2 - 4*d*f))*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^4*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((2*c*d*e*f*(2*a*f^2 + c*(e^2 - 2*d*f)) - (e + sqrt(e^2 - 4*d*f))*(a^2*f^4 + 2*a*c*f^2*(e^2 - d*f) + c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^4*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^0*(a + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 9, -((c*(2*e - f*x)*sqrt(a + c*x^2))/(2*f^2)) + (sqrt(c)*(3*a*f^2 + 2*c*(e^2 - d*f))*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*f^3) - ((c*e*(e - sqrt(e^2 - 4*d*f))*(2*a*f^2 + c*(e^2 - 2*d*f)) - 2*f*(2*a*c*d*f^2 - a^2*f^3 + c^2*d*(e^2 - d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((c*e*(e + sqrt(e^2 - 4*d*f))*(2*a*f^2 + c*(e^2 - 2*d*f)) - 2*f*(2*a*c*d*f^2 - a^2*f^3 + c^2*d*(e^2 - d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [(a + c*x^2)^(3/2)/(x^1*(d + e*x + f*x^2)), x, 17, (a*sqrt(a + c*x^2))/d + ((c*d - a*f)*sqrt(a + c*x^2))/(d*f) - (c^(3/2)*e*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/f^2 - ((2*e*f*(c^2*d^2 - a^2*f^2) - (c^2*d*e^2 - f*(c*d - a*f)^2)*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((2*e*f*(c^2*d^2 - a^2*f^2) - (c^2*d*e^2 - f*(c*d - a*f)^2)*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - (a^(3/2)*atanh(sqrt(a + c*x^2)/sqrt(a)))/d]
    @test_int [(a + c*x^2)^(3/2)/(x^2*(d + e*x + f*x^2)), x, 21, -((a*e*sqrt(a + c*x^2))/d^2) + (3*c*x*sqrt(a + c*x^2))/(2*d) + ((2*a*e - c*d*x)*sqrt(a + c*x^2))/(2*d^2) - (a + c*x^2)^(3/2)/(d*x) + (3*a*sqrt(c)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*d) + (sqrt(c)*(2*c*d - 3*a*f)*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(2*d*f) - ((4*a*c*d^2*f^2 + c^2*d^2*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + a^2*f^2*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*f*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((4*a*c*d^2*f^2 + a^2*f^2*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + c^2*d^2*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*f*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) + (a^(3/2)*e*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^2]
    @test_int [(a + c*x^2)^(3/2)/(x^3*(d + e*x + f*x^2)), x, 26, (3*c*sqrt(a + c*x^2))/(2*d) + (a*(e^2 - d*f)*sqrt(a + c*x^2))/d^3 - (3*c*e*x*sqrt(a + c*x^2))/(2*d^2) - ((2*(c*d^2 + a*(e^2 - d*f)) - c*d*e*x)*sqrt(a + c*x^2))/(2*d^3) - (a + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + c*x^2)^(3/2))/(d^2*x) + ((c^2*d^3*(e - sqrt(e^2 - 4*d*f)) + 2*a*c*d^2*f*(e + sqrt(e^2 - 4*d*f)) + a^2*f*(e^3 - 3*d*e*f + e^2*sqrt(e^2 - 4*d*f) - d*f*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - ((2*a*c*d^2*f*(e - sqrt(e^2 - 4*d*f)) + c^2*d^3*(e + sqrt(e^2 - 4*d*f)) + a^2*f*(e^3 - 3*d*e*f - e^2*sqrt(e^2 - 4*d*f) + d*f*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - (3*sqrt(a)*c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*d) - (a^(3/2)*(e^2 - d*f)*atanh(sqrt(a + c*x^2)/sqrt(a)))/d^3]

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

    @test_int [x^3/(sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 10, sqrt(a + c*x^2)/(c*f) - (e*atanh((sqrt(c)*x)/sqrt(a + c*x^2)))/(sqrt(c)*f^2) - ((2*d*e*f - (e^2 - d*f)*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((2*d*e*f - (e^2 - d*f)*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^2/(sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 8, atanh((sqrt(c)*x)/sqrt(a + c*x^2))/(sqrt(c)*f) - ((e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - ((2*d*f - e*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^1/(sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 5, ((e - sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - ((e + sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^0/(sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 5, -((sqrt(2)*f*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f))))) + (sqrt(2)*f*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [1/(x^1*sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 10, (f*(e + sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(e - sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - atanh(sqrt(a + c*x^2)/sqrt(a))/(sqrt(a)*d)]
    @test_int [1/(x^2*sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 11, -(sqrt(a + c*x^2)/(a*d*x)) - (f*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + (f*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) + (e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^2)]
    @test_int [1/(x^3*sqrt(a + c*x^2)*(d + e*x + f*x^2)), x, 15, -sqrt(a + c*x^2)/(2*a*d*x^2) + (e*sqrt(a + c*x^2))/(a*d^2*x) + (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(2*e^3 - 4*d*e*f - (e^2 - d*f)*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^3*sqrt(e^2 - 4*d*f)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) + (c*atanh(sqrt(a + c*x^2)/sqrt(a)))/(2*a^(3/2)*d) - ((e^2 - d*f)*atanh(sqrt(a + c*x^2)/sqrt(a)))/(sqrt(a)*d^3)]

    @test_int [x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 10, -(1/(c*f*sqrt(a + c*x^2))) - (e*x)/(a*f^2*sqrt(a + c*x^2)) + (a*f*(c*d^2 + a*(e^2 - d*f)) + c*e*(c*d^2 + a*(e^2 - 2*d*f))*x)/(a*f^2*(a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2)) - ((2*a*d*e*f - (e - sqrt(e^2 - 4*d*f))*(c*d^2 + a*(e^2 - d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + ((2*a*d*e*f - (e + sqrt(e^2 - 4*d*f))*(c*d^2 + a*(e^2 - d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^2/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, -((a*e + (c*d - a*f)*x)/((a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2))) - (f*(2*d*(c*d - a*f) + a*e*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + (f*(2*d*(c*d - a*f) + a*e*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^1/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, -((c*d - a*f - c*e*x)/((a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2))) + (f*(2*c*d*e - (c*d - a*f)*(e - sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(2*c*d*e - (c*d - a*f)*(e + sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [x^0/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, (c*(a*e + (c*d - a*f)*x))/(a*(a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2)) - (f*(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) + (f*(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f))))]
    @test_int [1/(x^1*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 12, 1/(a*d*sqrt(a + c*x^2)) - (a*(a*f^2 + c*(e^2 - d*f)) + c^2*d*e*x)/(a*d*(a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2)) + (f*(2*e*(a*f^2 + c*(e^2 - 2*d*f)) - (e - sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(2*e*(a*f^2 + c*(e^2 - 2*d*f)) - (e + sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - d*f)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) - atanh(sqrt(a + c*x^2)/sqrt(a))/(a^(3/2)*d)]
    @test_int [1/(x^2*(a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 14, -(e/(a*d^2*sqrt(a + c*x^2))) - 1/(a*d*x*sqrt(a + c*x^2)) - (2*c*x)/(a^2*d*sqrt(a + c*x^2)) + (a*e*(a*f^2 + c*(e^2 - 2*d*f)) + c*d*(a*f^2 + c*(e^2 - d*f))*x)/(a*d^2*(a*c*e^2 + (c*d - a*f)^2)*sqrt(a + c*x^2)) + (f*(e*(e - sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - 2*d*f)) - 2*(a*f^2*(e^2 - d*f) + c*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((2*a*f - c*(e - sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))) - (f*(e*(e + sqrt(e^2 - 4*d*f))*(a*f^2 + c*(e^2 - 2*d*f)) - 2*(a*f^2*(e^2 - d*f) + c*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((2*a*f - c*(e + sqrt(e^2 - 4*d*f))*x)/(sqrt(2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*sqrt(a + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*(a*c*e^2 + (c*d - a*f)^2)*sqrt(2*a*f^2 + c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))) + (e*atanh(sqrt(a + c*x^2)/sqrt(a)))/(a^(3/2)*d^2)]

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

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

    @test_int [x^3*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 15, -((d*sqrt(a + b*x + c*x^2))/f^2) + (b*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c^2*f) - (a + b*x + c*x^2)^(3/2)/(3*c*f) - (b*d*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^2) - (b*(b^2 - 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(5/2)*f) - (d*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2)) + (d*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2))]
    @test_int [x^2*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 9, -(((b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*c*f)) - ((8*c^2*d - b^2*f + 4*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(3/2)*f^2) + (sqrt(d)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^2) + (sqrt(d)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^2)]
    @test_int [x^1*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 9, -(sqrt(a + b*x + c*x^2)/f) - (b*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f) - (sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(3/2)) + (sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(3/2))]
    @test_int [x^0*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 8, -((sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/f) + (sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f) + (sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f)]
    @test_int [1/x^1*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 17, -((sqrt(a)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d) - (sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*sqrt(f)) + (sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*sqrt(f))]
    @test_int [1/x^2*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 16, -(sqrt(a + b*x + c*x^2)/(d*x)) - (b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*sqrt(a)*d) + (sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)) + (sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2))]
    @test_int [1/x^3*(a + b*x + c*x^2)^(1/2)/(d - f*x^2), x, 20, -(((2*a + b*x)*sqrt(a + b*x + c*x^2))/(4*a*d*x^2)) + ((b^2 - 4*a*c)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(8*a^(3/2)*d) - (sqrt(a)*f*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^2 - (sqrt(f)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2) + (sqrt(f)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2)]

    @test_int [x^3*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 17, -((3*b*(b^2 - 4*a*c)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(128*c^3*f)) - (d*(8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*sqrt(a + b*x + c*x^2))/(8*c*f^3) - (d*(a + b*x + c*x^2)^(3/2))/(3*f^2) + (b*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(16*c^2*f) - (a + b*x + c*x^2)^(5/2)/(5*c*f) + (3*b*(b^2 - 4*a*c)^2*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(256*c^(7/2)*f) - (b*d*(24*c^2*d - b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*f^3) - (d*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(7/2)) + (d*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(7/2))]
    @test_int [x^2*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 10, -(((b*(80*c^2*d - 3*b^2*f + 12*a*c*f) + 2*c*(16*c^2*d - 3*b^2*f + 12*a*c*f)*x)*sqrt(a + b*x + c*x^2))/(64*c^2*f^2)) - ((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c*f) - ((128*c^4*d^2 + 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 48*c^2*f*(b^2*d + a^2*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(128*c^(5/2)*f^3) + (sqrt(d)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^3) + (sqrt(d)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^3)]
    @test_int [x^1*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 10, -(((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*sqrt(a + b*x + c*x^2))/(8*c*f^2)) - (a + b*x + c*x^2)^(3/2)/(3*f) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*f^2) - ((c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2)) + ((c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(5/2))]
    @test_int [x^0*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 9, -(((5*b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*f)) - ((8*c^2*d + 3*b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*f^2) + ((c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f^2) + ((c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*f^2)]
    @test_int [1/x^1*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 19, ((b^2 + 8*a*c + 2*b*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*sqrt(a + b*x + c*x^2))/(8*c*d*f) - (a^(3/2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d - (b*(b^2 - 12*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d*f) - ((c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*f^(3/2)) + ((c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*f^(3/2))]
    @test_int [1/x^2*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 18, (3*(3*b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d) - ((5*b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d) - (a + b*x + c*x^2)^(3/2)/(d*x) - (3*sqrt(a)*b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*d) + (3*(b^2 + 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*d) - ((8*c^2*d + 3*b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*d*f) + ((c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*f) + ((c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*f)]
    @test_int [1/x^3*(a + b*x + c*x^2)^(3/2)/(d - f*x^2), x, 26, -((3*(b - 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d*x)) + (f*(b^2 + 8*a*c + 2*b*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d^2) - ((8*c^2*d + b^2*f + 8*a*c*f + 2*b*c*f*x)*sqrt(a + b*x + c*x^2))/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) - (3*(b^2 + 4*a*c)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(8*sqrt(a)*d) - (a^(3/2)*f*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^2 + (3*b*sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*d) - (b*(b^2 - 12*a*c)*f*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^2) - (b*(24*c^2*d - b^2*f + 12*a*c*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^2) - ((c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2*sqrt(f)) + ((c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2*sqrt(f))]

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

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

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

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

    @test_int [x^4/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 13, (3*b*sqrt(a + b*x + c*x^2))/(4*c^2*f) - (x*sqrt(a + b*x + c*x^2))/(2*c*f) - (d*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(sqrt(c)*f^2) - ((3*b^2 - 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(5/2)*f) + (d^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (d^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [x^3/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 10, -(sqrt(a + b*x + c*x^2)/(c*f)) + (b*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*c^(3/2)*f) - (d*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(3/2)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (d*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f^(3/2)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [x^2/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 8, -(atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))/(sqrt(c)*f)) + (sqrt(d)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (sqrt(d)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [x^1/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 5, -(atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2)))/(2*sqrt(f)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f))) + atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2)))/(2*sqrt(f)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [x^0/((a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 5, atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2)))/(2*sqrt(d)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2)))/(2*sqrt(d)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [1/(x^1*(a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 9, -(atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2)))/(sqrt(a)*d)) - (sqrt(f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (sqrt(f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [1/(x^2*(a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 10, -(sqrt(a + b*x + c*x^2)/(a*d*x)) + (b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*a^(3/2)*d) + (f*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (f*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]
    @test_int [1/(x^3*(a + b*x + c*x^2)^(1/2)*(d - f*x^2)), x, 13, -(sqrt(a + b*x + c*x^2)/(2*a*d*x^2)) + (3*b*sqrt(a + b*x + c*x^2))/(4*a^2*d*x) - ((3*b^2 - 4*a*c)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(8*a^(5/2)*d) - (f*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(sqrt(a)*d^2) - (f^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)) + (f^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f))]

    @test_int [x^4/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 13, -((2*x*(2*a + b*x))/((b^2 - 4*a*c)*f*sqrt(a + b*x + c*x^2))) + (2*d*(b + 2*c*x))/((b^2 - 4*a*c)*f^2*sqrt(a + b*x + c*x^2)) - (2*d^2*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*f^2*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2)) + (2*b*sqrt(a + b*x + c*x^2))/(c*(b^2 - 4*a*c)*f) - atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2)))/(c^(3/2)*f) + (d^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (d^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*f*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [x^3/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 9, -((2*(2*a + b*x))/((b^2 - 4*a*c)*f*sqrt(a + b*x + c*x^2))) - (2*d*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*f*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2)) - (d*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(f)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (d*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(f)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [x^2/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 6, (2*(a*b*(c*d - a*f) + c*(b^2*d - 2*a*(c*d + a*f))*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2)) + (sqrt(d)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (sqrt(d)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [x^1/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 6, -((2*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2))) - (sqrt(f)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (sqrt(f)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [x^0/((a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 6, -((2*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2))) + (f*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (f*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*sqrt(d)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [1/(x^1*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 12, (2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*sqrt(a + b*x + c*x^2)) - (2*f*(a*(2*c^2*d - b^2*f + 2*a*c*f) + b*c*(c*d - a*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2)) - atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2)))/(a^(3/2)*d) - (f^(3/2)*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (f^(3/2)*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]
    @test_int [1/(x^2*(a + b*x + c*x^2)^(3/2)*(d - f*x^2)), x, 12, (2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*sqrt(a + b*x + c*x^2)) - (2*f*(b*(b^2*f - c*(c*d + 3*a*f)) - c*(2*c^2*d - b^2*f + 2*a*c*f)*x))/((b^2 - 4*a*c)*d*(b^2*d*f - (c*d + a*f)^2)*sqrt(a + b*x + c*x^2)) - ((3*b^2 - 8*a*c)*sqrt(a + b*x + c*x^2))/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*a^(5/2)*d) + (f^2*atanh((b*sqrt(d) - 2*a*sqrt(f) + (2*c*sqrt(d) - b*sqrt(f))*x)/(2*sqrt(c*d - b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*(c*d - b*sqrt(d)*sqrt(f) + a*f)^(3/2)) + (f^2*atanh((b*sqrt(d) + 2*a*sqrt(f) + (2*c*sqrt(d) + b*sqrt(f))*x)/(2*sqrt(c*d + b*sqrt(d)*sqrt(f) + a*f)*sqrt(a + b*x + c*x^2))))/(2*d^(3/2)*(c*d + b*sqrt(d)*sqrt(f) + a*f)^(3/2))]

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

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

    @test_int [x^2*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 9, If(13>=8, -(((4*c*e - b*f - 2*c*f*x)*sqrt(a + b*x + c*x^2))/(4*c*f^2)) - ((b^2*f^2 + 4*c*f*(b*e - a*f) - 8*c^2*(e^2 - d*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*sqrt(e^2 - 4*d*f) + 2*d*e*f*sqrt(e^2 - 4*d*f)) + f*(a*f*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) - b*(e^3 - 3*d*e*f - e^2*sqrt(e^2 - 4*d*f) + d*f*sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*sqrt(e^2 - 4*d*f) - 2*d*e*f*sqrt(e^2 - 4*d*f)) + f*(a*f*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) - b*(e^3 - 3*d*e*f + e^2*sqrt(e^2 - 4*d*f) - d*f*sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))), -(((4*c*e - b*f - 2*c*f*x)*sqrt(a + b*x + c*x^2))/(4*c*f^2)) - ((b^2*f^2 + 4*c*f*(b*e - a*f) - 8*c^2*(e^2 - d*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*c^(3/2)*f^3) - ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 - e^3*sqrt(e^2 - 4*d*f) + 2*d*e*f*sqrt(e^2 - 4*d*f)) + f*(a*f*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) - b*(e^3 - 3*d*e*f - e^2*sqrt(e^2 - 4*d*f) + d*f*sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + ((c*(e^4 - 4*d*e^2*f + 2*d^2*f^2 + e^3*sqrt(e^2 - 4*d*f) - 2*d*e*f*sqrt(e^2 - 4*d*f)) + f*(a*f*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) - b*(e^3 - 3*d*e*f + e^2*sqrt(e^2 - 4*d*f) - d*f*sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^3*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))))]
    @test_int [x^1*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 9, sqrt(a + b*x + c*x^2)/f - ((2*c*e - b*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^2) - ((2*d*f*(c*e - b*f) + (e - sqrt(e^2 - 4*d*f))*(f*(b*e - a*f) - c*(e^2 - d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) + ((2*d*f*(c*e - b*f) + (e + sqrt(e^2 - 4*d*f))*(f*(b*e - a*f) - c*(e^2 - d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    @test_int [x^0*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 8, (sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/f - (sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f)) + (sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*f*sqrt(e^2 - 4*d*f))]
    @test_int [1/x^1*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 17, If(13>=8, -((sqrt(a)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d) + ((c*d*(e - sqrt(e^2 - 4*d*f)) - f*(2*b*d - a*(e + sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) - ((c*d*(e + sqrt(e^2 - 4*d*f)) - f*(2*b*d - a*(e - sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))), -((sqrt(a)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d) + ((c*d*(e - sqrt(e^2 - 4*d*f)) - f*(2*b*d - a*(e + sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) - ((c*d*(e + sqrt(e^2 - 4*d*f)) - f*(2*b*d - a*(e - sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))))]
    @test_int [1/x^2*(a + b*x + c*x^2)^(1/2)/(d + e*x + f*x^2), x, 23, If(13>=8, -(sqrt(a + b*x + c*x^2)/(d*x)) - (b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*sqrt(a)*d) + (sqrt(a)*e*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^2 + (sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/d - (b*e*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^2) - ((2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^2) - (f*(2*c*d^2 - b*d*(e + sqrt(e^2 - 4*d*f)) + a*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + (f*(2*c*d^2 - b*d*(e - sqrt(e^2 - 4*d*f)) + a*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))), -(sqrt(a + b*x + c*x^2)/(d*x)) - (b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*sqrt(a)*d) + (sqrt(a)*e*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^2 + (sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/d - (b*e*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^2) - ((2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^2) - (f*(2*c*d^2 - b*d*(e + sqrt(e^2 - 4*d*f)) + a*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + (f*(2*c*d^2 - b*d*(e - sqrt(e^2 - 4*d*f)) + a*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))))]

    #= [x^2*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 15, ((d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))*sqrt(a + b*x + c*x^2))/f^4 - (3*(b^2 - 4*a*c)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(64*c^2*f) + ((2*c*e^2 - 2*c*d*f - b*e*f)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*f^3) - (e*(a + b*x + c*x^2)^(3/2))/(3*f^2) + ((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(8*c*f) + (3*(b^2 - 4*a*c)^2*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(128*c^(5/2)*f) - ((b^2 - 4*a*c)*(2*c*e^2 - 2*c*d*f - b*e*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*f^3) - ((3*b*c*e^3*f - e*(6*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + a*b*e)*f^3 - 2*c^2*(e^4 - 3*d*e^2*f + d^2*f^2))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^5) + (((f*(c*d - a*f) - e*(c*e - b*f))*(e - sqrt(e^2 - 4*d*f))*(2*d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f))) - 2*d*f*(f*(c*d - a*f)*(f*(c*d - a*f) - e*(c*e - b*f)) - (c*e - b*f)*(d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^5*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)) - (((f*(c*d - a*f) - e*(c*e - b*f))*(e + sqrt(e^2 - 4*d*f))*(2*d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f))) - 2*d*f*(f*(c*d - a*f)*(f*(c*d - a*f) - e*(c*e - b*f)) - (c*e - b*f)*(d*f*(c*e - b*f) + e*(f*(c*d - a*f) - e*(c*e - b*f)))))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^5*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2))]
    @test_int [x^1*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 10, ((c*e^2 - c*d*f - b*e*f + a*f^2)*sqrt(a + b*x + c*x^2))/f^3 - ((2*c*e - b*f)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*f^2) + (a + b*x + c*x^2)^(3/2)/(3*f) + ((b^2 - 4*a*c)*(2*c*e - b*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*f^2) + ((3*b*c*e^2*f - (3*b*c*d + b^2*e + 2*a*c*e)*f^2 + a*b*f^3 - 2*c^2*e*(e^2 - 2*d*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^4) + ((2*d*f*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + (e - sqrt(e^2 - 4*d*f))*(2*b*c*e^3*f - e*(4*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + 2*a*b*e)*f^3 - a^2*f^4 - c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^4*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)) - ((2*d*f*(c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2) + (e + sqrt(e^2 - 4*d*f))*(2*b*c*e^3*f - e*(4*b*c*d + (b^2 + 2*a*c)*e)*f^2 + (b^2*d + 2*a*c*d + 2*a*b*e)*f^3 - a^2*f^4 - c^2*(e^4 - 3*d*e^2*f + d^2*f^2)))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^4*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2))]
    @test_int [x^0*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 9, -(((c*e - b*f)*sqrt(a + b*x + c*x^2))/f^2) + ((b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*f) - ((b^2 - 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*f) - ((3*b*c*e*f - (b^2 + 2*a*c)*f^2 - 2*c^2*(e^2 - d*f))*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*f^3) + (((c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2)*(e - sqrt(e^2 - 4*d*f)) + 2*f*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^3*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)) - (((c*e - b*f)*(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2)*(e + sqrt(e^2 - 4*d*f)) + 2*f*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(f^3*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2))]
    @test_int [1/x^1*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 17, ((c*d - a*f)*sqrt(a + b*x + c*x^2))/(d*f) - (b*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d) + ((b^2 + 8*a*c + 2*b*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d) - (a^(3/2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d - (b*(b^2 - 12*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d) + (b*(b^2 - 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d) - ((2*c^2*d*e - 3*b*c*d*f + a*b*f^2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d*f^2) + ((2*f*(c*d - a*f)*(c*d*e - 2*b*d*f + a*e*f) + (e - sqrt(e^2 - 4*d*f))*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(d*f^2*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)) - ((2*f*(c*d - a*f)*(c*d*e - 2*b*d*f + a*e*f) + (e + sqrt(e^2 - 4*d*f))*(2*b*c*d*e*f - (b^2 + 2*a*c)*d*f^2 + a^2*f^3 - c^2*d*(e^2 - d*f)))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(d*f^2*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2))]
    @test_int [1/x^2*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 22, -(((b*d - a*e)*sqrt(a + b*x + c*x^2))/d^2) - ((2*c*d - b*e)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d^2) + (3*(3*b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d) - (e*(b^2 + 8*a*c + 2*b*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/(d*x) - (3*sqrt(a)*b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*d) + (a^(3/2)*e*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^2 + (3*(b^2 + 4*a*c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*d) + (b*(b^2 - 12*a*c)*e*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^2) + ((b^2 - 4*a*c)*(2*c*d - b*e)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^2) + ((2*c^2*d^2 - b^2*d*f - 2*a*c*d*f + a*b*e*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^2*f) - ((2*a^2*d*f^3 + a*f^2*(2*b*d*e + 2*b*d*sqrt(e^2 - 4*d*f) - a*e*(e + sqrt(e^2 - 4*d*f))) + d^2*(2*f*(b*c*e - (b^2 + 2*a*c)*f - b*c*sqrt(e^2 - 4*d*f)) + c^2*(2*d*f - e*(e - sqrt(e^2 - 4*d*f)))))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(d^2*f*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)) + ((2*a^2*d*f^3 + a*f^2*(2*b*d*e - 2*b*d*sqrt(e^2 - 4*d*f) - a*e*(e - sqrt(e^2 - 4*d*f))) + d^2*(2*f*(b*c*e - (b^2 + 2*a*c)*f + b*c*sqrt(e^2 - 4*d*f)) + c^2*(2*d*f - e*(e + sqrt(e^2 - 4*d*f)))))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(d^2*f*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2))]
    @test_int [1/x^3*(a + b*x + c*x^2)^(3/2)/(d + e*x + f*x^2), x, 27, ((e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))*sqrt(a + b*x + c*x^2))/(d^3*f^2) - (3*(b - 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d*x) + ((2*c*d*e - b*e^2 + b*d*f)*(b + 2*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d^3) - (3*e*(3*b + 2*c*x)*sqrt(a + b*x + c*x^2))/(4*d^2) + ((e^2 - d*f)*(b^2 + 8*a*c + 2*b*c*x)*sqrt(a + b*x + c*x^2))/(8*c*d^3) - (a + b*x + c*x^2)^(3/2)/(2*d*x^2) + (e*(a + b*x + c*x^2)^(3/2))/(d^2*x) - (3*(b^2 + 4*a*c)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(8*sqrt(a)*d) + (3*sqrt(a)*b*e*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*d^2) - (a^(3/2)*(e^2 - d*f)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/d^3 + (3*b*sqrt(c)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*d) - (3*(b^2 + 4*a*c)*e*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(8*sqrt(c)*d^2) - (b*(b^2 - 12*a*c)*(e^2 - d*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^3) - ((b^2 - 4*a*c)*(2*c*d*e - b*e^2 + b*d*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(16*c^(3/2)*d^3) - ((3*b*c*d^2 - b^2*d*e - 2*a*c*d*e + a*b*e^2 - a*b*d*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x^2))))/(2*sqrt(c)*d^3) - (1/(d^3*f^3*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e - sqrt(e^2 - 4*d*f)) + c*(e - sqrt(e^2 - 4*d*f))^2)))*((2*f*(f*(c*d - a*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) - d*(c*e - b*f)*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))) - (e - sqrt(e^2 - 4*d*f))*(f*(c*e - b*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) + (f*(c*d - a*f) - e*(c*e - b*f))*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))))*atanh((b*e - 4*a*f - b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f - c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2)))) + (1/(d^3*f^3*sqrt(e^2 - 4*d*f)*sqrt(4*a*f^2 - 2*b*f*(e + sqrt(e^2 - 4*d*f)) + c*(e + sqrt(e^2 - 4*d*f))^2)))*((2*f*(f*(c*d - a*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) - d*(c*e - b*f)*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))) - (e + sqrt(e^2 - 4*d*f))*(f*(c*e - b*f)*(e*(c*d - a*f)*(e^2 - 2*d*f) - d*(c*e - b*f)*(e^2 - d*f)) + (f*(c*d - a*f) - e*(c*e - b*f))*(e*(c*e - b*f)*(e^2 - 2*d*f) + (e^2 - d*f)*(f*(c*d - a*f) - e*(c*e - b*f)))))*atanh((b*e - 4*a*f + b*sqrt(e^2 - 4*d*f) + 2*(c*e - b*f + c*sqrt(e^2 - 4*d*f))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))] =#

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

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

    @test_int [x^3/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 10, (2*(2*a + b*x))/((b^2 - 4*a*c)*f*sqrt(a + b*x + c*x^2)) + (2*e*(b + 2*c*x))/((b^2 - 4*a*c)*f^2*sqrt(a + b*x + c*x^2)) + (2*(c*d*e*(b*c*d - 2*a*c*e + a*b*f) - (b*d*e - a*e^2 + a*d*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*((b*c*d - 2*a*c*e + a*b*f)*(e^2 - d*f) - d*e*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*x))/((b^2 - 4*a*c)*f^2*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2)) + ((2*d*(b*d - a*e)*f + (e - sqrt(e^2 - 4*d*f))*(c*d^2 - b*d*e + a*(e^2 - d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) - ((2*d*(b*d - a*e)*f + (e + sqrt(e^2 - 4*d*f))*(c*d^2 - b*d*e + a*(e^2 - d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    @test_int [x^2/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, -((2*(a*(b*c*d - 2*a*c*e + a*b*f) + c*(b^2*d - a*b*e - 2*a*(c*d - a*f))*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2))) - (f*(2*d*(c*d - a*f) - (b*d - a*e)*(e - sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) + (f*(2*d*(c*d - a*f) - (b*d - a*e)*(e + sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    @test_int [x^1/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, (2*(a*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f) + c*(b*c*d - 2*a*c*e + a*b*f)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2)) + (f*(2*d*(c*e - b*f) - (c*d - a*f)*(e - sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) - (f*(2*d*(c*e - b*f) - (c*d - a*f)*(e + sqrt(e^2 - 4*d*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    @test_int [x^0/((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 6, If(13>=8, (2*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f) - c*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2)) - (f*(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + (f*(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))), (2*(b^2*c*e - 2*a*c^2*e - b^3*f - b*c*(c*d - 3*a*f) - c*(2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x))/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2)) - (f*(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))) + (f*(c*(e^2 - 2*d*f - e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e - sqrt(e^2 - 4*d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*(e^2 - 2*d*f + e*sqrt(e^2 - 4*d*f)) + f*(2*a*f - b*(e + sqrt(e^2 - 4*d*f))))))]
    @test_int [1/(x^1*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 12, (2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*sqrt(a + b*x + c*x^2)) + (2*(c*e*(2*a*c*e - b*(c*d + a*f)) + (b*e - a*f)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) + c*(2*c^2*d*e + b*f*(b*e - a*f) - b*c*(e^2 + d*f))*x))/((b^2 - 4*a*c)*d*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(a + b*x + c*x^2)) - atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2)))/(a^(3/2)*d) + (f*((e - sqrt(e^2 - 4*d*f))*(f*(b*e - a*f) - c*(e^2 - d*f)) - 2*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))) - (f*((e + sqrt(e^2 - 4*d*f))*(f*(b*e - a*f) - c*(e^2 - d*f)) - 2*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d*sqrt(e^2 - 4*d*f)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f)))]
    #= [1/(x^2*(a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2)), x, 13, -((2*e*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d^2*sqrt(a + b*x + c*x^2))) + (2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*d*x*sqrt(a + b*x + c*x^2)) + (2*(2*a*c*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - b*(a*e*f*(c*e - b*f) + (e^2 - d*f)*(c^2*d + b^2*f - c*(b*e + a*f))) + c*(b*c*e^3 - 2*a*c*d*f^2 - 2*c^2*d*(e^2 - d*f) - b*f*(b*e^2 - b*d*f - a*e*f))*x))/((b^2 - 4*a*c)*d^2*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))*sqrt(a + b*x + c*x^2)) - ((3*b^2 - 8*a*c)*sqrt(a + b*x + c*x^2))/(a^2*(b^2 - 4*a*c)*d*x) + (3*b*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(2*a^(5/2)*d) + (e*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x^2))))/(a^(3/2)*d^2) - (f*((e - sqrt(e^2 - 4*d*f))*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - 2*(d*e*f*(c*e - b*f) + (e^2 - d*f)*(f*(b*e - a*f) - c*(e^2 - d*f))))*atanh((4*a*f - b*(e - sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e - sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b*f)*sqrt(e^2 - 4*d*f))*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f)))) + (f*((e + sqrt(e^2 - 4*d*f))*(f*(b*e^2 - b*d*f - a*e*f) - c*(e^3 - 2*d*e*f)) - 2*(d*e*f*(c*e - b*f) + (e^2 - d*f)*(f*(b*e - a*f) - c*(e^2 - d*f))))*atanh((4*a*f - b*(e + sqrt(e^2 - 4*d*f)) + 2*(b*f - c*(e + sqrt(e^2 - 4*d*f)))*x)/(2*sqrt(2)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*sqrt(a + b*x + c*x^2))))/(sqrt(2)*d^2*sqrt(e^2 - 4*d*f)*sqrt(c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 + (c*e - b*f)*sqrt(e^2 - 4*d*f))*(c^2*d^2 + f*(b^2*d - a*b*e + a^2*f) - c*(b*d*e - a*(e^2 - 2*d*f))))] =#

    @test_int [x^4/(sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 24, (5/2)*sqrt(-3 - 4*x - x^2) - (1/4)*x*sqrt(-3 - 4*x - x^2) + (11/2)*asin(2 + x) + atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/(2*sqrt(2)) - atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/(2*sqrt(2)) - (5/4)*atanh(x/sqrt(-3 - 4*x - x^2))]
    @test_int [x^3/(sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 20, (-(1/2))*sqrt(-3 - 4*x - x^2) - 2*asin(2 + x) + atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/(2*sqrt(2)) - atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/(2*sqrt(2)) + atanh(x/sqrt(-3 - 4*x - x^2))]
    @test_int [x^2/(sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 16, (1/2)*asin(2 + x) - atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/sqrt(2) + atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/sqrt(2) - (1/2)*atanh(x/sqrt(-3 - 4*x - x^2))]
    @test_int [x^1/(sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 6, -(atan((1 - (3*sqrt(-1 - x))/sqrt(3 + x))/sqrt(2))/sqrt(2)) + atan((1 + (3*sqrt(-1 - x))/sqrt(3 + x))/sqrt(2))/sqrt(2), atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/sqrt(2) - atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2))/sqrt(2)]
    @test_int [x^0/(sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 10, (-(1/3))*sqrt(2)*atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) + (1/3)*sqrt(2)*atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) + (1/3)*atanh(x/sqrt(-3 - 4*x - x^2))]
    @test_int [1/(x^1*sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 17, -(atan((3 + 2*x)/(sqrt(3)*sqrt(-3 - 4*x - x^2)))/(3*sqrt(3))) + (1/9)*sqrt(2)*atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) - (1/9)*sqrt(2)*atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) - (4/9)*atanh(x/sqrt(-3 - 4*x - x^2))]
    @test_int [1/(x^2*sqrt(-3 - 4*x - x^2)*(3 + 4*x + 2*x^2)), x, 20, sqrt(-3 - 4*x - x^2)/(9*x) + (2*atan((3 + 2*x)/(sqrt(3)*sqrt(-3 - 4*x - x^2))))/(3*sqrt(3)) + (2/27)*sqrt(2)*atan((1 - (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) - (2/27)*sqrt(2)*atan((1 + (3 + x)/sqrt(-3 - 4*x - x^2))/sqrt(2)) + (10/27)*atanh(x/sqrt(-3 - 4*x - x^2))]

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

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

    @test_int [(2 + 3*x)^2*(30 + 31*x - 12*x^2)^2*(6 + 17*x + 12*x^2)^(1/2), x, 8, (125455*(17 + 24*x)*sqrt(6 + 17*x + 12*x^2))/150994944 - (125455*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/4718592 + (25091*(17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/24576 - (873*(6 + 17*x + 12*x^2)^(7/2))/1792 - (1/32)*(10 - 3*x)*(6 + 17*x + 12*x^2)^(7/2) - (125455*atanh((17 + 24*x)/(4*sqrt(3)*sqrt(6 + 17*x + 12*x^2))))/(603979776*sqrt(3))]
    @test_int [(2 + 3*x)^1*(30 + 31*x - 12*x^2)^1*(6 + 17*x + 12*x^2)^(1/2), x, 6, -((97*(17 + 24*x)*sqrt(6 + 17*x + 12*x^2))/24576) + (97/768)*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2) - (1/20)*(6 + 17*x + 12*x^2)^(5/2) + (97*atanh((17 + 24*x)/(4*sqrt(3)*sqrt(6 + 17*x + 12*x^2))))/(98304*sqrt(3))]
    @test_int [(2 + 3*x)^(-1)*(30 + 31*x - 12*x^2)^(-1)*(6 + 17*x + 12*x^2)^(1/2), x, 3, (1/42)*atanh((206 + 291*x)/(84*sqrt(6 + 17*x + 12*x^2)))]
    @test_int [(2 + 3*x)^(-2)*(30 + 31*x - 12*x^2)^(-2)*(6 + 17*x + 12*x^2)^(1/2), x, 5, -((275 + 388*x)/(98*(10 - 3*x)*sqrt(6 + 17*x + 12*x^2))) + (3137*sqrt(6 + 17*x + 12*x^2))/(38416*(10 - 3*x)) + (97*atanh((206 + 291*x)/(84*sqrt(6 + 17*x + 12*x^2))))/3226944]
    @test_int [(2 + 3*x)^(-3)*(30 + 31*x - 12*x^2)^(-3)*(6 + 17*x + 12*x^2)^(1/2), x, 7, -((275 + 388*x)/(294*(10 - 3*x)^2*(6 + 17*x + 12*x^2)^(3/2))) + (738029 + 1042556*x)/(8232*(10 - 3*x)^2*sqrt(6 + 17*x + 12*x^2)) - (50555899*sqrt(6 + 17*x + 12*x^2))/(19361664*(10 - 3*x)^2) - (1634466587*sqrt(6 + 17*x + 12*x^2))/(7589772288*(10 - 3*x)) + (40325*atanh((206 + 291*x)/(84*sqrt(6 + 17*x + 12*x^2))))/637540872192]

    #= The*following*integrands*are*equal. =#
    @test_int [(-3 + 2*x)*(-3*x + x^2)^(2/3), x, 1, (3/5)*(-3*x + x^2)^(5/3)]
    @test_int [(-3 + 2*x)*(x*(-3 + x))^(2/3), x, 1, (3/5)*(-((3 - x)*x))^(5/3)]
    @test_int [(x*(9 - 9*x + 2*x^2))/(-3*x + x^2)^(1/3), x, 2, (3/5)*(-3*x + x^2)^(5/3)]
    @test_int [(x*(9 - 9*x + 2*x^2))/(x*(-3 + x))^(1/3), x, 3, (3/5)*(-3*x + x^2)^(5/3)]

    #= [(A + B*x)^2*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^2*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 8, (5*(B^2*d - A^2*f)^4*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))/(16384*A^6*B^4*d^2*f^7) - (5*(B^2*d - A^2*f)^2*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(3/2))/(6144*A^4*B^2*d^2*f^6) + ((9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(5/2))/(384*A^2*d^2*f^5) - (9*B*c*((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(7/2))/(112*f^3) + (B*c*(a*A*f + B*c*d*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(7/2))/(8*d*f^3) - (5*(B^2*d - A^2*f)^6*(9*B^4*c^2*d^4 + 2*A^2*B^2*c*d^2*f*(7*c*d - 16*a*f) + A^4*f^2*(9*c^2*d^2 - 32*a*c*d*f + 32*a^2*f^2))*atanh(((B*d)/A + (A*f)/B + 2*f*x)/(2*sqrt(f)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))))/(32768*A^8*B^6*d^2*f^(15/2))]
    @test_int [(A + B*x)^1*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^1*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 6, (3*(B^2*d - A^2*f)^2*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*((B*d)/A + (A*f)/B + 2*f*x)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))/(128*A^3*B^2*d*f^4) - ((A*c + (B^2*c*d)/(A*f) - (2*a*A*f)/d)*((B*d)/A + (A*f)/B + 2*f*x)*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(3/2))/(16*f^2) + (B*c*(d + ((B*d)/A + (A*f)/B)*x + f*x^2)^(5/2))/(5*f^2) - (3*(B^2*d - A^2*f)^4*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*atanh(((B*d)/A + (A*f)/B + 2*f*x)/(2*sqrt(f)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))))/(256*A^5*B^4*d*f^(9/2))]
    @test_int [(A + B*x)^(-1)*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^(-1)*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 3, (d*f*atanh((d*f*(a*B - (2*B*c*d)/f + (a*A^2*f)/(B*d) - ((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*x))/(2*sqrt(c*d - a*f)*sqrt(B^2*c*d^2 - a*A^2*f^2)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))))/(sqrt(c*d - a*f)*sqrt(B^2*c*d^2 - a*A^2*f^2))]
    @test_int [(A + B*x)^(-2)*(a + ((B^2*c*d^2 + a*A^2*f^2)/(A*B*d*f))*x + c*x^2)^(-2)*(d + ((B^2*d + A^2*f)/(A*B))*x + f*x^2)^(1/2), x, 6, (2*A*B^2*d^2*f^3*(A*d*((B^3*c*d^2)/(A^2*f) - a*B*f + (A^2*f*(c*d - a*f))/(B*d)) + (B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*x))/((c*d - a*f)*(B^2*d - A^2*f)^2*(B^2*c*d^2 - a*A^2*f^2)*(a*A*f + B*c*d*x)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2)) - (B^3*c*d^3*f^2*(3*B^4*c^2*d^4 - 2*A^2*B^2*c*d^2*f*(c*d + 2*a*f) + A^4*f^2*(3*c^2*d^2 - 4*a*c*d*f + 4*a^2*f^2))*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))/((c*d - a*f)^2*(B^2*d - A^2*f)^2*(B^2*c*d^2 - a*A^2*f^2)^2*(a*A*f + B*c*d*x)) + (3*B^2*c^2*d^4*f^2*(B^2*c*d^2 + A^2*f*(c*d - 2*a*f))*atanh((d*f*((2*B*c*d)/f - a*(B + (A^2*f)/(B*d)) + ((B^2*c*d)/(A*f) + A*(c - (2*a*f)/d))*x))/(2*sqrt(c*d - a*f)*sqrt(B^2*c*d^2 - a*A^2*f^2)*sqrt(d + ((B*d)/A + (A*f)/B)*x + f*x^2))))/(2*A*(c*d - a*f)^(5/2)*(B^2*c*d^2 - a*A^2*f^2)^(5/2))] =#

    @test_int [(g + h*x)/((-((c*g^2)/h^2) + 9*c*x^2)^(1/3)*(g^2 + 3*h^2*x^2)), x, 2, ((1 - (9*h^2*x^2)/g^2)^(1/3)*atan(1/sqrt(3) - (2^(2/3)*(1 - (3*h*x)/g)^(2/3))/(sqrt(3)*(1 + (3*h*x)/g)^(1/3))))/(2^(2/3)*sqrt(3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3)) + ((1 - (9*h^2*x^2)/g^2)^(1/3)*log(g^2 + 3*h^2*x^2))/(6*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3)) - ((1 - (9*h^2*x^2)/g^2)^(1/3)*log((1 - (3*h*x)/g)^(2/3) + 2^(1/3)*(1 + (3*h*x)/g)^(1/3)))/(2*2^(2/3)*h*(-((c*g^2)/h^2) + 9*c*x^2)^(1/3))]

    @test_int [(g + h*x)/((((-c^2)*g^2 + b*c*g*h + 2*b^2*h^2)/(9*c*h^2) + b*x + c*x^2)^(1/3)*((f*(b^2 - ((-c^2)*g^2 + b*c*g*h + 2*b^2*h^2)/(3*h^2)))/c^2 + (b*f*x)/c + f*x^2)), x, 2, (3*3^(1/6)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*atan(1/sqrt(3) - (2^(2/3)*(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3))/(sqrt(3)*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3))))/(f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) + (3^(2/3)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*log((f*(c^2*g^2 - b*c*g*h + b^2*h^2))/(3*c^2*h^2) + (b*f*x)/c + f*x^2))/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) - (3*3^(2/3)*h*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2))/(2*c*g - b*h)^2)^(1/3)*log((1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3) + 2^(1/3)*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3)))/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3))]
    end

max_stars_repo_path: test/1 Algebraic functions/1.2 Trinomial products/1.2.1 Quadratic/1.2.1.6 (g+h x)^m (a+b x+c x^2)^p (d+e x+f x^2)^q.jl