text:
@testset "1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q" begin
    (A, B, a, b, c, d, e, j, m, n, p, q, x, ) = @variables A B a b c d e j m n p q x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a*x^j+b*x^k)^p*(c+d*x^n)^q=#

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

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

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

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

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

    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^2, x, 3, (1/5)*A*b^3*x^5 + (1/7)*b^2*(b*B + 3*A*c)*x^7 + (1/3)*b*c*(b*B + A*c)*x^9 + (1/11)*c^2*(3*b*B + A*c)*x^11 + (1/13)*B*c^3*x^13]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^3, x, 4, (b*(b*B - A*c)*(b + c*x^2)^4)/(8*c^3) - ((2*b*B - A*c)*(b + c*x^2)^5)/(10*c^3) + (B*(b + c*x^2)^6)/(12*c^3)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^4, x, 3, (1/3)*A*b^3*x^3 + (1/5)*b^2*(b*B + 3*A*c)*x^5 + (3/7)*b*c*(b*B + A*c)*x^7 + (1/9)*c^2*(3*b*B + A*c)*x^9 + (1/11)*B*c^3*x^11]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^5, x, 4, -(((b*B - A*c)*(b + c*x^2)^4)/(8*c^2)) + (B*(b + c*x^2)^5)/(10*c^2)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^6, x, 3, A*b^3*x + (1/3)*b^2*(b*B + 3*A*c)*x^3 + (3/5)*b*c*(b*B + A*c)*x^5 + (1/7)*c^2*(3*b*B + A*c)*x^7 + (1/9)*B*c^3*x^9]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^7, x, 5, (3*A*b^2*c*x^2)/2 + (3*A*b*c^2*x^4)/4 + (A*c^3*x^6)/6 + (B*(b + c*x^2)^4)/(8*c) + A*b^3*log(x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^8, x, 3, -((A*b^3)/x) + b^2*(b*B + 3*A*c)*x + b*c*(b*B + A*c)*x^3 + (1/5)*c^2*(3*b*B + A*c)*x^5 + (1/7)*B*c^3*x^7]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^9, x, 4, -((A*b^3)/(2*x^2)) + (3/2)*b*c*(b*B + A*c)*x^2 + (1/4)*c^2*(3*b*B + A*c)*x^4 + (1/6)*B*c^3*x^6 + b^2*(b*B + 3*A*c)*log(x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^10, x, 3, -((A*b^3)/(3*x^3)) - (b^2*(b*B + 3*A*c))/x + 3*b*c*(b*B + A*c)*x + (1/3)*c^2*(3*b*B + A*c)*x^3 + (1/5)*B*c^3*x^5]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^11, x, 4, -((A*b^3)/(4*x^4)) - (b^2*(b*B + 3*A*c))/(2*x^2) + (1/2)*c^2*(3*b*B + A*c)*x^2 + (1/4)*B*c^3*x^4 + 3*b*c*(b*B + A*c)*log(x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^12, x, 3, -((A*b^3)/(5*x^5)) - (b^2*(b*B + 3*A*c))/(3*x^3) - (3*b*c*(b*B + A*c))/x + c^2*(3*b*B + A*c)*x + (1/3)*B*c^3*x^3]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^13, x, 4, -((A*b^3)/(6*x^6)) - (b^2*(b*B + 3*A*c))/(4*x^4) - (3*b*c*(b*B + A*c))/(2*x^2) + (1/2)*B*c^3*x^2 + c^2*(3*b*B + A*c)*log(x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^14, x, 3, -((A*b^3)/(7*x^7)) - (b^2*(b*B + 3*A*c))/(5*x^5) - (b*c*(b*B + A*c))/x^3 - (c^2*(3*b*B + A*c))/x + B*c^3*x]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^15, x, 5, -((b^3*B)/(6*x^6)) - (3*b^2*B*c)/(4*x^4) - (3*b*B*c^2)/(2*x^2) - (A*(b + c*x^2)^4)/(8*b*x^8) + B*c^3*log(x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^16, x, 3, -((A*b^3)/(9*x^9)) - (b^2*(b*B + 3*A*c))/(7*x^7) - (3*b*c*(b*B + A*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(3*x^3) - (B*c^3)/x]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^17, x, 4, -((A*(b + c*x^2)^4)/(10*b*x^10)) - ((5*b*B - A*c)*(b + c*x^2)^4)/(40*b^2*x^8)]

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

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

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

    @test_int [(x^14*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 6, (3*b*(2*b*B - A*c)*x)/c^5 - ((3*b*B - A*c)*x^3)/(3*c^4) + (B*x^5)/(5*c^3) - (b^3*(b*B - A*c)*x)/(4*c^5*(b + c*x^2)^2) + (b^2*(17*b*B - 13*A*c)*x)/(8*c^5*(b + c*x^2)) - (7*b^(3/2)*(9*b*B - 5*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*c^(11/2))]
    @test_int [(x^13*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -(((3*b*B - A*c)*x^2)/(2*c^4)) + (B*x^4)/(4*c^3) - (b^3*(b*B - A*c))/(4*c^5*(b + c*x^2)^2) + (b^2*(4*b*B - 3*A*c))/(2*c^5*(b + c*x^2)) + (3*b*(2*b*B - A*c)*log(b + c*x^2))/(2*c^5)]
    @test_int [(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 6, -(((3*b*B - A*c)*x)/c^4) + (B*x^3)/(3*c^3) + (b^2*(b*B - A*c)*x)/(4*c^4*(b + c*x^2)^2) - (b*(13*b*B - 9*A*c)*x)/(8*c^4*(b + c*x^2)) + (5*sqrt(b)*(7*b*B - 3*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*c^(9/2))]
    @test_int [(x^11*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, (B*x^2)/(2*c^3) + (b^2*(b*B - A*c))/(4*c^4*(b + c*x^2)^2) - (b*(3*b*B - 2*A*c))/(2*c^4*(b + c*x^2)) - ((3*b*B - A*c)*log(b + c*x^2))/(2*c^4)]
    @test_int [(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 5, (B*x)/c^3 - (b*(b*B - A*c)*x)/(4*c^3*(b + c*x^2)^2) + ((9*b*B - 5*A*c)*x)/(8*c^3*(b + c*x^2)) - (3*(5*b*B - A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*sqrt(b)*c^(7/2))]
    @test_int [(x^9*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -((b*(b*B - A*c))/(4*c^3*(b + c*x^2)^2)) + (2*b*B - A*c)/(2*c^3*(b + c*x^2)) + (B*log(b + c*x^2))/(2*c^3)]
    @test_int [(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, ((b*B - A*c)*x)/(4*c^2*(b + c*x^2)^2) - ((5*b*B - A*c)*x)/(8*b*c^2*(b + c*x^2)) + ((3*b*B + A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(3/2)*c^(5/2))]
    @test_int [(x^7*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 3, (A + B*x^2)^2/(4*(b*B - A*c)*(b + c*x^2)^2)]
    @test_int [(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -(((b*B - A*c)*x)/(4*b*c*(b + c*x^2)^2)) + ((b*B + 3*A*c)*x)/(8*b^2*c*(b + c*x^2)) + ((b*B + 3*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(5/2)*c^(3/2))]
    @test_int [(x^5*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -((b*B - A*c)/(4*b*c*(b + c*x^2)^2)) + A/(2*b^2*(b + c*x^2)) + (A*log(x))/b^3 - (A*log(b + c*x^2))/(2*b^3)]
    @test_int [(x^4*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 5, -(A/(b^3*x)) + ((b*B - A*c)*x)/(4*b^2*(b + c*x^2)^2) + ((3*b*B - 7*A*c)*x)/(8*b^3*(b + c*x^2)) + (3*(b*B - 5*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(7/2)*sqrt(c))]
    @test_int [(x^3*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -(A/(2*b^3*x^2)) + (b*B - A*c)/(4*b^2*(b + c*x^2)^2) + (b*B - 2*A*c)/(2*b^3*(b + c*x^2)) + ((b*B - 3*A*c)*log(x))/b^4 - ((b*B - 3*A*c)*log(b + c*x^2))/(2*b^4)]
    @test_int [(x^2*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 6, -(A/(3*b^3*x^3)) - (b*B - 3*A*c)/(b^4*x) - (c*(b*B - A*c)*x)/(4*b^3*(b + c*x^2)^2) - (c*(7*b*B - 11*A*c)*x)/(8*b^4*(b + c*x^2)) - (5*sqrt(c)*(3*b*B - 7*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(9/2))]
    @test_int [(x*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 4, -(A/(4*b^3*x^4)) - (b*B - 3*A*c)/(2*b^4*x^2) - (c*(b*B - A*c))/(4*b^3*(b + c*x^2)^2) - (c*(2*b*B - 3*A*c))/(2*b^4*(b + c*x^2)) - (3*c*(b*B - 2*A*c)*log(x))/b^5 + (3*c*(b*B - 2*A*c)*log(b + c*x^2))/(2*b^5)]
    @test_int [(A + B*x^2)/(b*x^2 + c*x^4)^3, x, 6, -(A/(5*b^3*x^5)) - (b*B - 3*A*c)/(3*b^4*x^3) + (3*c*(b*B - 2*A*c))/(b^5*x) + (c^2*(b*B - A*c)*x)/(4*b^4*(b + c*x^2)^2) + (c^2*(11*b*B - 15*A*c)*x)/(8*b^5*(b + c*x^2)) + (7*c^(3/2)*(5*b*B - 9*A*c)*atan((sqrt(c)*x)/sqrt(b)))/(8*b^(11/2))]
    @test_int [(A + B*x^2)/(x*(b*x^2 + c*x^4)^3), x, 4, -(A/(6*b^3*x^6)) - (b*B - 3*A*c)/(4*b^4*x^4) + (3*c*(b*B - 2*A*c))/(2*b^5*x^2) + (c^2*(b*B - A*c))/(4*b^4*(b + c*x^2)^2) + (c^2*(3*b*B - 4*A*c))/(2*b^5*(b + c*x^2)) + (2*c^2*(3*b*B - 5*A*c)*log(x))/b^6 - (c^2*(3*b*B - 5*A*c)*log(b + c*x^2))/b^6]

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

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

    @test_int [x^7*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 8, (7*b^3*(3*b*B - 4*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(1024*c^5) - (7*b^2*(3*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(384*c^4) + (7*b*(3*b*B - 4*A*c)*x^2*(b*x^2 + c*x^4)^(3/2))/(320*c^3) - ((3*b*B - 4*A*c)*x^4*(b*x^2 + c*x^4)^(3/2))/(40*c^2) + (B*x^6*(b*x^2 + c*x^4)^(3/2))/(12*c) - (7*b^5*(3*b*B - 4*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(1024*c^(11/2))]
    @test_int [x^5*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 7, -((b^2*(7*b*B - 10*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(256*c^4)) + (b*(7*b*B - 10*A*c)*(b*x^2 + c*x^4)^(3/2))/(96*c^3) - ((7*b*B - 10*A*c)*x^2*(b*x^2 + c*x^4)^(3/2))/(80*c^2) + (B*x^4*(b*x^2 + c*x^4)^(3/2))/(10*c) + (b^4*(7*b*B - 10*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(256*c^(9/2))]
    @test_int [x^3*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 5, (b*(5*b*B - 8*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(128*c^3) - ((5*b*B - 8*A*c - 6*B*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(48*c^2) - (b^3*(5*b*B - 8*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(128*c^(7/2))]
    @test_int [x*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 5, -(((b*B - 2*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(16*c^2)) + (B*(b*x^2 + c*x^4)^(3/2))/(6*c) + (b^2*(b*B - 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(5/2))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x, x, 5, -((b*B - 4*A*c)*sqrt(b*x^2 + c*x^4))/(8*c) + (B*(b*x^2 + c*x^4)^(3/2))/(4*c*x^2) - (b*(b*B - 4*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(3/2))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^3, x, 5, ((b*B + 2*A*c)*sqrt(b*x^2 + c*x^4))/(2*b) - (A*(b*x^2 + c*x^4)^(3/2))/(b*x^4) + ((b*B + 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*sqrt(c))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^5, x, 5, -((B*sqrt(b*x^2 + c*x^4))/x^2) - (A*(b*x^2 + c*x^4)^(3/2))/(3*b*x^6) + B*sqrt(c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^7, x, 3, -((A*(b*x^2 + c*x^4)^(3/2))/(5*b*x^8)) - ((5*b*B - 2*A*c)*(b*x^2 + c*x^4)^(3/2))/(15*b^2*x^6)]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^9, x, 4, -((A*(b*x^2 + c*x^4)^(3/2))/(7*b*x^10)) - ((7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(35*b^2*x^8) + (2*c*(7*b*B - 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^6)]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^11, x, 5, -((A*(b*x^2 + c*x^4)^(3/2))/(9*b*x^12)) - ((3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(3/2))/(21*b^2*x^10) + (4*c*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(3/2))/(105*b^3*x^8) - (8*c^2*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(3/2))/(315*b^4*x^6)]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^13, x, 6, -((A*(b*x^2 + c*x^4)^(3/2))/(11*b*x^14)) - ((11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(99*b^2*x^12) + (2*c*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(231*b^3*x^10) - (8*c^2*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(1155*b^4*x^8) + (16*c^3*(11*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(3465*b^5*x^6)]

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

    @test_int [x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 8, (b^4*(9*b*B - 14*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(2048*c^5) - (b^2*(9*b*B - 14*A*c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(768*c^4) + (b*(9*b*B - 14*A*c)*(b*x^2 + c*x^4)^(5/2))/(240*c^3) - ((9*b*B - 14*A*c)*x^2*(b*x^2 + c*x^4)^(5/2))/(168*c^2) + (B*x^4*(b*x^2 + c*x^4)^(5/2))/(14*c) - (b^6*(9*b*B - 14*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2048*c^(11/2))]
    @test_int [x^3*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 6, -((b^3*(7*b*B - 12*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(1024*c^4)) + (b*(7*b*B - 12*A*c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(384*c^3) - ((7*b*B - 12*A*c - 10*B*c*x^2)*(b*x^2 + c*x^4)^(5/2))/(120*c^2) + (b^5*(7*b*B - 12*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(1024*c^(9/2))]
    @test_int [x*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 6, (3*b^2*(b*B - 2*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(256*c^3) - ((b*B - 2*A*c)*(b + 2*c*x^2)*(b*x^2 + c*x^4)^(3/2))/(32*c^2) + (B*(b*x^2 + c*x^4)^(5/2))/(10*c) - (3*b^4*(b*B - 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(256*c^(7/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x, x, 6, -((b*(3*b*B - 8*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(128*c^2)) - ((3*b*B - 8*A*c)*(b*x^2 + c*x^4)^(3/2))/(48*c) + (B*(b*x^2 + c*x^4)^(5/2))/(8*c*x^2) + (b^3*(3*b*B - 8*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(128*c^(5/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^3, x, 6, ((b*B - 6*A*c)*(b + 2*c*x^2)*sqrt(b*x^2 + c*x^4))/(16*c) + ((b*B - 6*A*c)*(b*x^2 + c*x^4)^(3/2))/(6*b) + (A*(b*x^2 + c*x^4)^(5/2))/(b*x^4) - (b^2*(b*B - 6*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(3/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^5, x, 6, (3/8)*(b*B + 4*A*c)*sqrt(b*x^2 + c*x^4) + ((b*B + 4*A*c)*(b*x^2 + c*x^4)^(3/2))/(4*b*x^2) - (A*(b*x^2 + c*x^4)^(5/2))/(b*x^6) + (3*b*(b*B + 4*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*sqrt(c))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^7, x, 6, (c*(3*b*B + 2*A*c)*sqrt(b*x^2 + c*x^4))/(2*b) - ((3*b*B + 2*A*c)*(b*x^2 + c*x^4)^(3/2))/(3*b*x^4) - (A*(b*x^2 + c*x^4)^(5/2))/(3*b*x^8) + (1/2)*sqrt(c)*(3*b*B + 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^9, x, 6, -((B*c*sqrt(b*x^2 + c*x^4))/x^2) - (B*(b*x^2 + c*x^4)^(3/2))/(3*x^6) - (A*(b*x^2 + c*x^4)^(5/2))/(5*b*x^10) + B*c^(3/2)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^11, x, 3, -((A*(b*x^2 + c*x^4)^(5/2))/(7*b*x^12)) - ((7*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(35*b^2*x^10)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^13, x, 4, -((A*(b*x^2 + c*x^4)^(5/2))/(9*b*x^14)) - ((9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/(63*b^2*x^12) + (2*c*(9*b*B - 4*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*b^3*x^10)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^15, x, 5, -((A*(b*x^2 + c*x^4)^(5/2))/(11*b*x^16)) - ((11*b*B - 6*A*c)*(b*x^2 + c*x^4)^(5/2))/(99*b^2*x^14) + (4*c*(11*b*B - 6*A*c)*(b*x^2 + c*x^4)^(5/2))/(693*b^3*x^12) - (8*c^2*(11*b*B - 6*A*c)*(b*x^2 + c*x^4)^(5/2))/(3465*b^4*x^10)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^17, x, 6, -((A*(b*x^2 + c*x^4)^(5/2))/(13*b*x^18)) - ((13*b*B - 8*A*c)*(b*x^2 + c*x^4)^(5/2))/(143*b^2*x^16) + (2*c*(13*b*B - 8*A*c)*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^14) - (8*c^2*(13*b*B - 8*A*c)*(b*x^2 + c*x^4)^(5/2))/(3003*b^4*x^12) + (16*c^3*(13*b*B - 8*A*c)*(b*x^2 + c*x^4)^(5/2))/(15015*b^5*x^10)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^19, x, 7, -((A*(b*x^2 + c*x^4)^(5/2))/(15*b*x^20)) - ((3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(39*b^2*x^18) + (8*c*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(429*b^3*x^16) - (16*c^2*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(1287*b^4*x^14) + (64*c^3*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(9009*b^5*x^12) - (128*c^4*(3*b*B - 2*A*c)*(b*x^2 + c*x^4)^(5/2))/(45045*b^6*x^10)]

    @test_int [x^4*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 5, (16*b^3*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2))/(15015*c^5*x^5) - (8*b^2*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2))/(3003*c^4*x^3) + (2*b*(8*b*B - 13*A*c)*(b*x^2 + c*x^4)^(5/2))/(429*c^3*x) - ((8*b*B - 13*A*c)*x*(b*x^2 + c*x^4)^(5/2))/(143*c^2) + (B*x^3*(b*x^2 + c*x^4)^(5/2))/(13*c)]
    @test_int [x^2*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 4, -((8*b^2*(6*b*B - 11*A*c)*(b*x^2 + c*x^4)^(5/2))/(3465*c^4*x^5)) + (4*b*(6*b*B - 11*A*c)*(b*x^2 + c*x^4)^(5/2))/(693*c^3*x^3) - ((6*b*B - 11*A*c)*(b*x^2 + c*x^4)^(5/2))/(99*c^2*x) + (B*x*(b*x^2 + c*x^4)^(5/2))/(11*c)]
    @test_int [(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 3, (2*b*(4*b*B - 9*A*c)*(b*x^2 + c*x^4)^(5/2))/(315*c^3*x^5) - ((4*b*B - 9*A*c)*(b*x^2 + c*x^4)^(5/2))/(63*c^2*x^3) + (B*(b*x^2 + c*x^4)^(5/2))/(9*c*x)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^2, x, 2, -((2*b*B - 7*A*c)*(b*x^2 + c*x^4)^(5/2))/(35*c^2*x^5) + (B*(b*x^2 + c*x^4)^(5/2))/(7*c*x^3)]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^4, x, 5, (A*b*sqrt(b*x^2 + c*x^4))/x + (A*(b*x^2 + c*x^4)^(3/2))/(3*x^3) + (B*(b*x^2 + c*x^4)^(5/2))/(5*c*x^5) - A*b^(3/2)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^6, x, 5, ((2*b*B + 3*A*c)*sqrt(b*x^2 + c*x^4))/(2*x) + ((2*b*B + 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(6*b*x^3) - (A*(b*x^2 + c*x^4)^(5/2))/(2*b*x^7) - (1/2)*sqrt(b)*(2*b*B + 3*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^8, x, 5, (3*c*(4*b*B + A*c)*sqrt(b*x^2 + c*x^4))/(8*b*x) - ((4*b*B + A*c)*(b*x^2 + c*x^4)^(3/2))/(8*b*x^5) - (A*(b*x^2 + c*x^4)^(5/2))/(4*b*x^9) - (3*c*(4*b*B + A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*sqrt(b))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^10, x, 5, -((c*(6*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(16*b*x^3)) - ((6*b*B - A*c)*(b*x^2 + c*x^4)^(3/2))/(24*b*x^7) - (A*(b*x^2 + c*x^4)^(5/2))/(6*b*x^11) - (c^2*(6*b*B - A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(16*b^(3/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^12, x, 6, -((c*(8*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(64*b*x^5)) - (c^2*(8*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(128*b^2*x^3) - ((8*b*B - 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(48*b*x^9) - (A*(b*x^2 + c*x^4)^(5/2))/(8*b*x^13) + (c^3*(8*b*B - 3*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(128*b^(5/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^14, x, 7, -((c*(2*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(32*b*x^7)) - (c^2*(2*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(128*b^2*x^5) + (3*c^3*(2*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(256*b^3*x^3) - ((2*b*B - A*c)*(b*x^2 + c*x^4)^(3/2))/(16*b*x^11) - (A*(b*x^2 + c*x^4)^(5/2))/(10*b*x^15) - (3*c^4*(2*b*B - A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(256*b^(7/2))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^16, x, 8, -((c*(12*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(320*b*x^9)) - (c^2*(12*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(1920*b^2*x^7) + (c^3*(12*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(1536*b^3*x^5) - (c^4*(12*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(1024*b^4*x^3) - ((12*b*B - 7*A*c)*(b*x^2 + c*x^4)^(3/2))/(120*b*x^13) - (A*(b*x^2 + c*x^4)^(5/2))/(12*b*x^17) + (c^5*(12*b*B - 7*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(1024*b^(9/2))]

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

    @test_int [(x^7*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 7, (-5*b^2*(7*b*B - 8*A*c)*sqrt(b*x^2 + c*x^4))/(128*c^4) + (5*b*(7*b*B - 8*A*c)*x^2*sqrt(b*x^2 + c*x^4))/(192*c^3) - ((7*b*B - 8*A*c)*x^4*sqrt(b*x^2 + c*x^4))/(48*c^2) + (B*x^6*sqrt(b*x^2 + c*x^4))/(8*c) + (5*b^3*(7*b*B - 8*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(128*c^(9/2))]
    @test_int [(x^5*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 6, (b*(5*b*B - 6*A*c)*sqrt(b*x^2 + c*x^4))/(16*c^3) - ((5*b*B - 6*A*c)*x^2*sqrt(b*x^2 + c*x^4))/(24*c^2) + (B*x^4*sqrt(b*x^2 + c*x^4))/(6*c) - (b^2*(5*b*B - 6*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(7/2))]
    @test_int [(x^3*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 4, -(((3*b*B - 4*A*c - 2*B*c*x^2)*sqrt(b*x^2 + c*x^4))/(8*c^2)) + (b*(3*b*B - 4*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(5/2))]
    @test_int [(x*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 4, (B*sqrt(b*x^2 + c*x^4))/(2*c) - ((b*B - 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*c^(3/2))]
    @test_int [(A + B*x^2)/(x*sqrt(b*x^2 + c*x^4)), x, 4, -((A*sqrt(b*x^2 + c*x^4))/(b*x^2)) + (B*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/sqrt(c)]
    @test_int [(A + B*x^2)/(x^3*sqrt(b*x^2 + c*x^4)), x, 3, -((A*sqrt(b*x^2 + c*x^4))/(3*b*x^4)) - ((3*b*B - 2*A*c)*sqrt(b*x^2 + c*x^4))/(3*b^2*x^2)]
    @test_int [(A + B*x^2)/(x^5*sqrt(b*x^2 + c*x^4)), x, 4, -((A*sqrt(b*x^2 + c*x^4))/(5*b*x^6)) - ((5*b*B - 4*A*c)*sqrt(b*x^2 + c*x^4))/(15*b^2*x^4) + (2*c*(5*b*B - 4*A*c)*sqrt(b*x^2 + c*x^4))/(15*b^3*x^2)]
    @test_int [(A + B*x^2)/(x^7*sqrt(b*x^2 + c*x^4)), x, 5, -((A*sqrt(b*x^2 + c*x^4))/(7*b*x^8)) - ((7*b*B - 6*A*c)*sqrt(b*x^2 + c*x^4))/(35*b^2*x^6) + (4*c*(7*b*B - 6*A*c)*sqrt(b*x^2 + c*x^4))/(105*b^3*x^4) - (8*c^2*(7*b*B - 6*A*c)*sqrt(b*x^2 + c*x^4))/(105*b^4*x^2)]
    @test_int [(A + B*x^2)/(x^9*sqrt(b*x^2 + c*x^4)), x, 6, -((A*sqrt(b*x^2 + c*x^4))/(9*b*x^10)) - ((9*b*B - 8*A*c)*sqrt(b*x^2 + c*x^4))/(63*b^2*x^8) + (2*c*(9*b*B - 8*A*c)*sqrt(b*x^2 + c*x^4))/(105*b^3*x^6) - (8*c^2*(9*b*B - 8*A*c)*sqrt(b*x^2 + c*x^4))/(315*b^4*x^4) + (16*c^3*(9*b*B - 8*A*c)*sqrt(b*x^2 + c*x^4))/(315*b^5*x^2)]

    @test_int [(x^6*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 4, -((8*b^2*(6*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(105*c^4*x)) + (4*b*(6*b*B - 7*A*c)*x*sqrt(b*x^2 + c*x^4))/(105*c^3) - ((6*b*B - 7*A*c)*x^3*sqrt(b*x^2 + c*x^4))/(35*c^2) + (B*x^5*sqrt(b*x^2 + c*x^4))/(7*c)]
    @test_int [(x^4*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 3, (2*b*(4*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(15*c^3*x) - ((4*b*B - 5*A*c)*x*sqrt(b*x^2 + c*x^4))/(15*c^2) + (B*x^3*sqrt(b*x^2 + c*x^4))/(5*c)]
    @test_int [(x^2*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 2, -((2*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(3*c^2*x) + (B*x*sqrt(b*x^2 + c*x^4))/(3*c)]
    @test_int [(A + B*x^2)/sqrt(b*x^2 + c*x^4), x, 3, (B*sqrt(b*x^2 + c*x^4))/(c*x) - (A*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/sqrt(b)]
    @test_int [(A + B*x^2)/(x^2*sqrt(b*x^2 + c*x^4)), x, 3, -((A*sqrt(b*x^2 + c*x^4))/(2*b*x^3)) - ((2*b*B - A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*b^(3/2))]
    @test_int [(A + B*x^2)/(x^4*sqrt(b*x^2 + c*x^4)), x, 4, -((A*sqrt(b*x^2 + c*x^4))/(4*b*x^5)) - ((4*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(8*b^2*x^3) + (c*(4*b*B - 3*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(5/2))]

    @test_int [x^9*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 7, -(((b*B - A*c)*x^8)/(b*c*sqrt(b*x^2 + c*x^4))) + (5*b*(7*b*B - 6*A*c)*sqrt(b*x^2 + c*x^4))/(16*c^4) - (5*(7*b*B - 6*A*c)*x^2*sqrt(b*x^2 + c*x^4))/(24*c^3) + ((7*b*B - 6*A*c)*x^4*sqrt(b*x^2 + c*x^4))/(6*b*c^2) - (5*b^2*(7*b*B - 6*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(16*c^(9/2))]
    @test_int [x^7*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 6, -(((b*B - A*c)*x^6)/(b*c*sqrt(b*x^2 + c*x^4))) - (3*(5*b*B - 4*A*c)*sqrt(b*x^2 + c*x^4))/(8*c^3) + ((5*b*B - 4*A*c)*x^2*sqrt(b*x^2 + c*x^4))/(4*b*c^2) + (3*b*(5*b*B - 4*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(8*c^(7/2))]
    @test_int [x^5*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 5, -(((b*B - A*c)*x^4)/(b*c*sqrt(b*x^2 + c*x^4))) + ((3*b*B - 2*A*c)*sqrt(b*x^2 + c*x^4))/(2*b*c^2) - ((3*b*B - 2*A*c)*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/(2*c^(5/2))]
    @test_int [x^3*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 4, -(((b*B - A*c)*x^2)/(b*c*sqrt(b*x^2 + c*x^4))) + (B*atanh((sqrt(c)*x^2)/sqrt(b*x^2 + c*x^4)))/c^(3/2)]
    @test_int [x^1*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 2, -((A*b - (b*B - 2*A*c)*x^2)/(b^2*sqrt(b*x^2 + c*x^4)))]
    @test_int [(A + B*x^2)/(x^1*(b*x^2 + c*x^4)^(3/2)), x, 3, -(A/(3*b*x^2*sqrt(b*x^2 + c*x^4))) - ((3*b*B - 4*A*c)*(b + 2*c*x^2))/(3*b^3*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^3*(b*x^2 + c*x^4)^(3/2)), x, 4, -(A/(5*b*x^4*sqrt(b*x^2 + c*x^4))) - (5*b*B - 6*A*c)/(15*b^2*x^2*sqrt(b*x^2 + c*x^4)) + (4*c*(5*b*B - 6*A*c)*(b + 2*c*x^2))/(15*b^4*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^5*(b*x^2 + c*x^4)^(3/2)), x, 5, -(A/(7*b*x^6*sqrt(b*x^2 + c*x^4))) - (7*b*B - 8*A*c)/(35*b^2*x^4*sqrt(b*x^2 + c*x^4)) + (2*c*(7*b*B - 8*A*c))/(35*b^3*x^2*sqrt(b*x^2 + c*x^4)) - (8*c^2*(7*b*B - 8*A*c)*(b + 2*c*x^2))/(35*b^5*sqrt(b*x^2 + c*x^4))]

    @test_int [x^8*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 4, -(((b*B - A*c)*x^7)/(b*c*sqrt(b*x^2 + c*x^4))) + (8*b*(6*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(15*c^4*x) - (4*(6*b*B - 5*A*c)*x*sqrt(b*x^2 + c*x^4))/(15*c^3) + ((6*b*B - 5*A*c)*x^3*sqrt(b*x^2 + c*x^4))/(5*b*c^2)]
    @test_int [x^6*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 3, -(((b*B - A*c)*x^5)/(b*c*sqrt(b*x^2 + c*x^4))) - (2*(4*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(3*c^3*x) + ((4*b*B - 3*A*c)*x*sqrt(b*x^2 + c*x^4))/(3*b*c^2)]
    @test_int [x^4*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 2, -(((b*B - A*c)*x^3)/(b*c*sqrt(b*x^2 + c*x^4))) + ((2*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(b*c^2*x)]
    @test_int [x^2*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 3, -(((b*B - A*c)*x)/(b*c*sqrt(b*x^2 + c*x^4))) - (A*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/b^(3/2)]
    @test_int [x^0*(A + B*x^2)/(b*x^2 + c*x^4)^(3/2), x, 5, -(B/(3*c*x*sqrt(b*x^2 + c*x^4))) - (2*b*B - 3*A*c)/(3*b*c*x*sqrt(b*x^2 + c*x^4)) + ((2*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(2*b^2*c*x^3) - ((2*b*B - 3*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(2*b^(5/2))]
    @test_int [(A + B*x^2)/(x^2*(b*x^2 + c*x^4)^(3/2)), x, 5, -(A/(4*b*x^3*sqrt(b*x^2 + c*x^4))) + (4*b*B - 5*A*c)/(4*b^2*x*sqrt(b*x^2 + c*x^4)) - (3*(4*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(8*b^3*x^3) + (3*c*(4*b*B - 5*A*c)*atanh((sqrt(b)*x)/sqrt(b*x^2 + c*x^4)))/(8*b^(7/2))]

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

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

    @test_int [x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4), x, 3, (2*A*b*x^(13/2))/13 + (2*(b*B + A*c)*x^(17/2))/17 + (2*B*c*x^(21/2))/21]
    @test_int [x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4), x, 3, (2*A*b*x^(11/2))/11 + (2*(b*B + A*c)*x^(15/2))/15 + (2*B*c*x^(19/2))/19]
    @test_int [x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4), x, 3, (2*A*b*x^(9/2))/9 + (2*(b*B + A*c)*x^(13/2))/13 + (2*B*c*x^(17/2))/17]
    @test_int [sqrt(x)*(A + B*x^2)*(b*x^2 + c*x^4), x, 3, (2*A*b*x^(7/2))/7 + (2*(b*B + A*c)*x^(11/2))/11 + (2*B*c*x^(15/2))/15]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4))/sqrt(x), x, 3, (2*A*b*x^(5/2))/5 + (2*(b*B + A*c)*x^(9/2))/9 + (2*B*c*x^(13/2))/13]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4))/x^(3/2), x, 3, (2*A*b*x^(3/2))/3 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B*c*x^(11/2))/11]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4))/x^(5/2), x, 3, 2*A*b*sqrt(x) + (2*(b*B + A*c)*x^(5/2))/5 + (2*B*c*x^(9/2))/9]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4))/x^(7/2), x, 3, (-2*A*b)/sqrt(x) + (2*(b*B + A*c)*x^(3/2))/3 + (2*B*c*x^(7/2))/7]

    @test_int [x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2, x, 3, (2*A*b^2*x^(17/2))/17 + (2*b*(b*B + 2*A*c)*x^(21/2))/21 + (2*c*(2*b*B + A*c)*x^(25/2))/25 + (2*B*c^2*x^(29/2))/29]
    @test_int [x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2, x, 3, (2*A*b^2*x^(15/2))/15 + (2*b*(b*B + 2*A*c)*x^(19/2))/19 + (2*c*(2*b*B + A*c)*x^(23/2))/23 + (2*B*c^2*x^(27/2))/27]
    @test_int [x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^2, x, 3, (2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(17/2))/17 + (2*c*(2*b*B + A*c)*x^(21/2))/21 + (2*B*c^2*x^(25/2))/25]
    @test_int [sqrt(x)*(A + B*x^2)*(b*x^2 + c*x^4)^2, x, 3, (2*A*b^2*x^(11/2))/11 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(19/2))/19 + (2*B*c^2*x^(23/2))/23]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^2)/sqrt(x), x, 3, (2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(13/2))/13 + (2*c*(2*b*B + A*c)*x^(17/2))/17 + (2*B*c^2*x^(21/2))/21]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(3/2), x, 3, (2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(15/2))/15 + (2*B*c^2*x^(19/2))/19]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(5/2), x, 3, (2*A*b^2*x^(5/2))/5 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(13/2))/13 + (2*B*c^2*x^(17/2))/17]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^(7/2), x, 3, (2*A*b^2*x^(3/2))/3 + (2*b*(b*B + 2*A*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(11/2))/11 + (2*B*c^2*x^(15/2))/15]

    @test_int [x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3, x, 3, (2*A*b^3*x^(21/2))/21 + (2*b^2*(b*B + 3*A*c)*x^(25/2))/25 + (6*b*c*(b*B + A*c)*x^(29/2))/29 + (2*c^2*(3*b*B + A*c)*x^(33/2))/33 + (2*B*c^3*x^(37/2))/37]
    @test_int [x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3, x, 3, (2*A*b^3*x^(19/2))/19 + (2*b^2*(b*B + 3*A*c)*x^(23/2))/23 + (2*b*c*(b*B + A*c)*x^(27/2))/9 + (2*c^2*(3*b*B + A*c)*x^(31/2))/31 + (2*B*c^3*x^(35/2))/35]
    @test_int [x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^3, x, 3, (2*A*b^3*x^(17/2))/17 + (2*b^2*(b*B + 3*A*c)*x^(21/2))/21 + (6*b*c*(b*B + A*c)*x^(25/2))/25 + (2*c^2*(3*b*B + A*c)*x^(29/2))/29 + (2*B*c^3*x^(33/2))/33]
    @test_int [sqrt(x)*(A + B*x^2)*(b*x^2 + c*x^4)^3, x, 3, (2*A*b^3*x^(15/2))/15 + (2*b^2*(b*B + 3*A*c)*x^(19/2))/19 + (6*b*c*(b*B + A*c)*x^(23/2))/23 + (2*c^2*(3*b*B + A*c)*x^(27/2))/27 + (2*B*c^3*x^(31/2))/31]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/sqrt(x), x, 3, (2*A*b^3*x^(13/2))/13 + (2*b^2*(b*B + 3*A*c)*x^(17/2))/17 + (2*b*c*(b*B + A*c)*x^(21/2))/7 + (2*c^2*(3*b*B + A*c)*x^(25/2))/25 + (2*B*c^3*x^(29/2))/29]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(3/2), x, 3, (2*A*b^3*x^(11/2))/11 + (2*b^2*(b*B + 3*A*c)*x^(15/2))/15 + (6*b*c*(b*B + A*c)*x^(19/2))/19 + (2*c^2*(3*b*B + A*c)*x^(23/2))/23 + (2*B*c^3*x^(27/2))/27]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(5/2), x, 3, (2*A*b^3*x^(9/2))/9 + (2*b^2*(b*B + 3*A*c)*x^(13/2))/13 + (6*b*c*(b*B + A*c)*x^(17/2))/17 + (2*c^2*(3*b*B + A*c)*x^(21/2))/21 + (2*B*c^3*x^(25/2))/25]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^3)/x^(7/2), x, 3, (2*A*b^3*x^(7/2))/7 + (2*b^2*(b*B + 3*A*c)*x^(11/2))/11 + (2*b*c*(b*B + A*c)*x^(15/2))/5 + (2*c^2*(3*b*B + A*c)*x^(19/2))/19 + (2*B*c^3*x^(23/2))/23]

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

    @test_int [(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 14, (2*b*(b*B - A*c)*x^(3/2))/(3*c^3) - (2*(b*B - A*c)*x^(7/2))/(7*c^2) + (2*B*x^(11/2))/(11*c) + (b^(7/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(15/4)) - (b^(7/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(15/4)) - (b^(7/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(15/4)) + (b^(7/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(15/4))]
    @test_int [(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 14, (2*b*(b*B - A*c)*sqrt(x))/c^3 - (2*(b*B - A*c)*x^(5/2))/(5*c^2) + (2*B*x^(9/2))/(9*c) + (b^(5/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(13/4)) - (b^(5/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(13/4)) + (b^(5/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(13/4)) - (b^(5/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(13/4))]
    @test_int [(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 13, (-2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(7/2))/(7*c) - (b^(3/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(11/4)) + (b^(3/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(11/4)) + (b^(3/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(11/4)) - (b^(3/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(11/4))]
    @test_int [(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 13, (-2*(b*B - A*c)*sqrt(x))/c^2 + (2*B*x^(5/2))/(5*c) - (b^(1/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(9/4)) + (b^(1/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*c^(9/4)) - (b^(1/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(9/4)) + (b^(1/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*c^(9/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 12, (2*B*x^(3/2))/(3*c) + ((b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(1/4)*c^(7/4)) - ((b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(1/4)*c^(7/4)) - ((b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(1/4)*c^(7/4)) + ((b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(1/4)*c^(7/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4), x, 12, (2*B*sqrt(x))/c + ((b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(3/4)*c^(5/4)) - ((b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(3/4)*c^(5/4)) + ((b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(3/4)*c^(5/4)) - ((b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(3/4)*c^(5/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(b*x^2 + c*x^4), x, 12, (-2*A)/(b*sqrt(x)) - ((b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(5/4)*c^(3/4)) + ((b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(5/4)*c^(3/4)) + ((b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(5/4)*c^(3/4)) - ((b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(5/4)*c^(3/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(b*x^2 + c*x^4)), x, 12, (-2*A)/(3*b*x^(3/2)) - ((b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(7/4)*c^(1/4)) + ((b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(7/4)*c^(1/4)) - ((b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(7/4)*c^(1/4)) + ((b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(7/4)*c^(1/4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)), x, 13, (-2*A)/(5*b*x^(5/2)) - (2*(b*B - A*c))/(b^2*sqrt(x)) + (c^(1/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(9/4)) - (c^(1/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(9/4)) - (c^(1/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(9/4)) + (c^(1/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(9/4))]
    @test_int [(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)), x, 13, (-2*A)/(7*b*x^(7/2)) - (2*(b*B - A*c))/(3*b^2*x^(3/2)) + (c^(3/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(11/4)) - (c^(3/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(11/4)) + (c^(3/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(11/4)) - (c^(3/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(11/4))]
    @test_int [(A + B*x^2)/(x^(7/2)*(b*x^2 + c*x^4)), x, 14, (-2*A)/(9*b*x^(9/2)) - (2*(b*B - A*c))/(5*b^2*x^(5/2)) + (2*c*(b*B - A*c))/(b^3*sqrt(x)) - (c^(5/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(13/4)) + (c^(5/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(13/4)) + (c^(5/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(13/4)) - (c^(5/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(13/4))]
    @test_int [(A + B*x^2)/(x^(9/2)*(b*x^2 + c*x^4)), x, 14, (-2*A)/(11*b*x^(11/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(3*b^3*x^(3/2)) - (c^(7/4)*(b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(15/4)) + (c^(7/4)*(b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(sqrt(2)*b^(15/4)) - (c^(7/4)*(b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(15/4)) + (c^(7/4)*(b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(2*sqrt(2)*b^(15/4))]

    @test_int [(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 15, (b*(13*b*B - 9*A*c)*sqrt(x))/(2*c^4) - ((13*b*B - 9*A*c)*x^(5/2))/(10*c^3) + ((13*b*B - 9*A*c)*x^(9/2))/(18*b*c^2) - ((b*B - A*c)*x^(13/2))/(2*b*c*(b + c*x^2)) + (b^(5/4)*(13*b*B - 9*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(17/4)) - (b^(5/4)*(13*b*B - 9*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(17/4)) + (b^(5/4)*(13*b*B - 9*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(17/4)) - (b^(5/4)*(13*b*B - 9*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(17/4))]
    @test_int [(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 14, -((11*b*B - 7*A*c)*x^(3/2))/(6*c^3) + ((11*b*B - 7*A*c)*x^(7/2))/(14*b*c^2) - ((b*B - A*c)*x^(11/2))/(2*b*c*(b + c*x^2)) - (b^(3/4)*(11*b*B - 7*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(15/4)) + (b^(3/4)*(11*b*B - 7*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(15/4)) + (b^(3/4)*(11*b*B - 7*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(15/4)) - (b^(3/4)*(11*b*B - 7*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(15/4))]
    @test_int [(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 14, -((9*b*B - 5*A*c)*sqrt(x))/(2*c^3) + ((9*b*B - 5*A*c)*x^(5/2))/(10*b*c^2) - ((b*B - A*c)*x^(9/2))/(2*b*c*(b + c*x^2)) - (b^(1/4)*(9*b*B - 5*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*c^(13/4)) - (b^(1/4)*(9*b*B - 5*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(13/4)) + (b^(1/4)*(9*b*B - 5*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*c^(13/4))]
    @test_int [(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 13, ((7*b*B - 3*A*c)*x^(3/2))/(6*b*c^2) - ((b*B - A*c)*x^(7/2))/(2*b*c*(b + c*x^2)) + ((7*b*B - 3*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(1/4)*c^(11/4)) - ((7*b*B - 3*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(1/4)*c^(11/4)) - ((7*b*B - 3*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(1/4)*c^(11/4)) + ((7*b*B - 3*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(1/4)*c^(11/4))]
    @test_int [(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 13, ((5*b*B - A*c)*sqrt(x))/(2*b*c^2) - ((b*B - A*c)*x^(5/2))/(2*b*c*(b + c*x^2)) + ((5*b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(3/4)*c^(9/4)) - ((5*b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(3/4)*c^(9/4)) + ((5*b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(3/4)*c^(9/4)) - ((5*b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(3/4)*c^(9/4))]
    @test_int [(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 12, -((b*B - A*c)*x^(3/2))/(2*b*c*(b + c*x^2)) - ((3*b*B + A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(5/4)*c^(7/4)) + ((3*b*B + A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(5/4)*c^(7/4)) + ((3*b*B + A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(5/4)*c^(7/4)) - ((3*b*B + A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(5/4)*c^(7/4))]
    @test_int [(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 12, -((b*B - A*c)*sqrt(x))/(2*b*c*(b + c*x^2)) - ((b*B + 3*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(7/4)*c^(5/4)) + ((b*B + 3*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(7/4)*c^(5/4)) - ((b*B + 3*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(7/4)*c^(5/4)) + ((b*B + 3*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(7/4)*c^(5/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 13, (b*B - 5*A*c)/(2*b^2*c*sqrt(x)) - (b*B - A*c)/(2*b*c*sqrt(x)*(b + c*x^2)) - ((b*B - 5*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(9/4)*c^(3/4)) + ((b*B - 5*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(9/4)*c^(3/4)) + ((b*B - 5*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(9/4)*c^(3/4)) - ((b*B - 5*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(9/4)*c^(3/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 13, (3*b*B - 7*A*c)/(6*b^2*c*x^(3/2)) - (b*B - A*c)/(2*b*c*x^(3/2)*(b + c*x^2)) - ((3*b*B - 7*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(11/4)*c^(1/4)) + ((3*b*B - 7*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(11/4)*c^(1/4)) - ((3*b*B - 7*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(11/4)*c^(1/4)) + ((3*b*B - 7*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(11/4)*c^(1/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(b*x^2 + c*x^4)^2, x, 14, (5*b*B - 9*A*c)/(10*b^2*c*x^(5/2)) - (5*b*B - 9*A*c)/(2*b^3*sqrt(x)) - (b*B - A*c)/(2*b*c*x^(5/2)*(b + c*x^2)) + (c^(1/4)*(5*b*B - 9*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(13/4)) - (c^(1/4)*(5*b*B - 9*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(13/4)) - (c^(1/4)*(5*b*B - 9*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(13/4)) + (c^(1/4)*(5*b*B - 9*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(13/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(b*x^2 + c*x^4)^2), x, 14, (7*b*B - 11*A*c)/(14*b^2*c*x^(7/2)) - (7*b*B - 11*A*c)/(6*b^3*x^(3/2)) - (b*B - A*c)/(2*b*c*x^(7/2)*(b + c*x^2)) + (c^(3/4)*(7*b*B - 11*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(15/4)) + (c^(3/4)*(7*b*B - 11*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(15/4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^2), x, 15, (9*b*B - 13*A*c)/(18*b^2*c*x^(9/2)) - (9*b*B - 13*A*c)/(10*b^3*x^(5/2)) + (c*(9*b*B - 13*A*c))/(2*b^4*sqrt(x)) - (b*B - A*c)/(2*b*c*x^(9/2)*(b + c*x^2)) - (c^(5/4)*(9*b*B - 13*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(4*sqrt(2)*b^(17/4)) + (c^(5/4)*(9*b*B - 13*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(17/4)) - (c^(5/4)*(9*b*B - 13*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(8*sqrt(2)*b^(17/4))]

    @test_int [(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 15, (-9*(13*b*B - 5*A*c)*sqrt(x))/(16*c^4) + (9*(13*b*B - 5*A*c)*x^(5/2))/(80*b*c^3) - ((b*B - A*c)*x^(13/2))/(4*b*c*(b + c*x^2)^2) - ((13*b*B - 5*A*c)*x^(9/2))/(16*b*c^2*(b + c*x^2)) - (9*b^(1/4)*(13*b*B - 5*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*c^(17/4)) - (9*b^(1/4)*(13*b*B - 5*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*c^(17/4))]
    @test_int [(x^(21/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 14, (7*(11*b*B - 3*A*c)*x^(3/2))/(48*b*c^3) - ((b*B - A*c)*x^(11/2))/(4*b*c*(b + c*x^2)^2) - ((11*b*B - 3*A*c)*x^(7/2))/(16*b*c^2*(b + c*x^2)) + (7*(11*b*B - 3*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(1/4)*c^(15/4)) - (7*(11*b*B - 3*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(1/4)*c^(15/4)) - (7*(11*b*B - 3*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(1/4)*c^(15/4)) + (7*(11*b*B - 3*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(1/4)*c^(15/4))]
    @test_int [(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 14, (5*(9*b*B - A*c)*sqrt(x))/(16*b*c^3) - ((b*B - A*c)*x^(9/2))/(4*b*c*(b + c*x^2)^2) - ((9*b*B - A*c)*x^(5/2))/(16*b*c^2*(b + c*x^2)) + (5*(9*b*B - A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(3/4)*c^(13/4)) - (5*(9*b*B - A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(3/4)*c^(13/4)) + (5*(9*b*B - A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(3/4)*c^(13/4)) - (5*(9*b*B - A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(3/4)*c^(13/4))]
    @test_int [(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 13, -((b*B - A*c)*x^(7/2))/(4*b*c*(b + c*x^2)^2) - ((7*b*B + A*c)*x^(3/2))/(16*b*c^2*(b + c*x^2)) - (3*(7*b*B + A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(5/4)*c^(11/4)) + (3*(7*b*B + A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(5/4)*c^(11/4)) + (3*(7*b*B + A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(5/4)*c^(11/4)) - (3*(7*b*B + A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(5/4)*c^(11/4))]
    @test_int [(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 13, -((b*B - A*c)*x^(5/2))/(4*b*c*(b + c*x^2)^2) - ((5*b*B + 3*A*c)*sqrt(x))/(16*b*c^2*(b + c*x^2)) - ((5*b*B + 3*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(7/4)*c^(9/4)) + ((5*b*B + 3*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(7/4)*c^(9/4)) - ((5*b*B + 3*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(7/4)*c^(9/4)) + ((5*b*B + 3*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(7/4)*c^(9/4))]
    @test_int [(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 13, -((b*B - A*c)*x^(3/2))/(4*b*c*(b + c*x^2)^2) + ((3*b*B + 5*A*c)*x^(3/2))/(16*b^2*c*(b + c*x^2)) - ((3*b*B + 5*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(9/4)*c^(7/4)) + ((3*b*B + 5*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(9/4)*c^(7/4)) - ((3*b*B + 5*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(9/4)*c^(7/4))]
    @test_int [(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 13, -((b*B - A*c)*sqrt(x))/(4*b*c*(b + c*x^2)^2) + ((b*B + 7*A*c)*sqrt(x))/(16*b^2*c*(b + c*x^2)) - (3*(b*B + 7*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(11/4)*c^(5/4)) + (3*(b*B + 7*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(11/4)*c^(5/4)) - (3*(b*B + 7*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(11/4)*c^(5/4)) + (3*(b*B + 7*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(11/4)*c^(5/4))]
    @test_int [(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 14, (5*(b*B - 9*A*c))/(16*b^3*c*sqrt(x)) - (b*B - A*c)/(4*b*c*sqrt(x)*(b + c*x^2)^2) - (b*B - 9*A*c)/(16*b^2*c*sqrt(x)*(b + c*x^2)) - (5*(b*B - 9*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(13/4)*c^(3/4)) + (5*(b*B - 9*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(13/4)*c^(3/4)) + (5*(b*B - 9*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(13/4)*c^(3/4)) - (5*(b*B - 9*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(13/4)*c^(3/4))]
    @test_int [(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 14, (7*(3*b*B - 11*A*c))/(48*b^3*c*x^(3/2)) - (b*B - A*c)/(4*b*c*x^(3/2)*(b + c*x^2)^2) - (3*b*B - 11*A*c)/(16*b^2*c*x^(3/2)*(b + c*x^2)) - (7*(3*b*B - 11*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(15/4)*c^(1/4)) + (7*(3*b*B - 11*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(15/4)*c^(1/4)) - (7*(3*b*B - 11*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(15/4)*c^(1/4)) + (7*(3*b*B - 11*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(15/4)*c^(1/4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 15, (9*(5*b*B - 13*A*c))/(80*b^3*c*x^(5/2)) - (9*(5*b*B - 13*A*c))/(16*b^4*sqrt(x)) - (b*B - A*c)/(4*b*c*x^(5/2)*(b + c*x^2)^2) - (5*b*B - 13*A*c)/(16*b^2*c*x^(5/2)*(b + c*x^2)) + (9*c^(1/4)*(5*b*B - 13*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(17/4)) + (9*c^(1/4)*(5*b*B - 13*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(17/4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 15, (11*(7*b*B - 15*A*c))/(112*b^3*c*x^(7/2)) - (11*(7*b*B - 15*A*c))/(48*b^4*x^(3/2)) - (b*B - A*c)/(4*b*c*x^(7/2)*(b + c*x^2)^2) - (7*b*B - 15*A*c)/(16*b^2*c*x^(7/2)*(b + c*x^2)) + (11*c^(3/4)*(7*b*B - 15*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(19/4)) + (11*c^(3/4)*(7*b*B - 15*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(19/4)) - (11*c^(3/4)*(7*b*B - 15*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(19/4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(b*x^2 + c*x^4)^3, x, 16, (13*(9*b*B - 17*A*c))/(144*b^3*c*x^(9/2)) - (13*(9*b*B - 17*A*c))/(80*b^4*x^(5/2)) + (13*c*(9*b*B - 17*A*c))/(16*b^5*sqrt(x)) - (b*B - A*c)/(4*b*c*x^(9/2)*(b + c*x^2)^2) - (9*b*B - 17*A*c)/(16*b^2*c*x^(9/2)*(b + c*x^2)) - (13*c^(5/4)*(9*b*B - 17*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(21/4)) + (13*c^(5/4)*(9*b*B - 17*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(21/4)) + (13*c^(5/4)*(9*b*B - 17*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(21/4)) - (13*c^(5/4)*(9*b*B - 17*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(21/4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(b*x^2 + c*x^4)^3), x, 16, (15*(11*b*B - 19*A*c))/(176*b^3*c*x^(11/2)) - (15*(11*b*B - 19*A*c))/(112*b^4*x^(7/2)) + (5*c*(11*b*B - 19*A*c))/(16*b^5*x^(3/2)) - (b*B - A*c)/(4*b*c*x^(11/2)*(b + c*x^2)^2) - (11*b*B - 19*A*c)/(16*b^2*c*x^(11/2)*(b + c*x^2)) - (15*c^(7/4)*(11*b*B - 19*A*c)*atan(1 - (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(23/4)) + (15*c^(7/4)*(11*b*B - 19*A*c)*atan(1 + (sqrt(2)*c^(1/4)*sqrt(x))/b^(1/4)))/(32*sqrt(2)*b^(23/4)) - (15*c^(7/4)*(11*b*B - 19*A*c)*log(sqrt(b) - sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(23/4)) + (15*c^(7/4)*(11*b*B - 19*A*c)*log(sqrt(b) + sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x))/(64*sqrt(2)*b^(23/4))]

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

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

    @test_int [x^(5/2)*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 7, (4*b^2*(3*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(231*c^3*sqrt(x)) - (4*b*(3*b*B - 5*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(385*c^2) - (2*(3*b*B - 5*A*c)*x^(7/2)*sqrt(b*x^2 + c*x^4))/(55*c) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(3/2))/(15*c) - (2*b^(11/4)*(3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(3/2)*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 8, (4*b^2*(7*b*B - 13*A*c)*x^(3/2)*(b + c*x^2))/(195*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (4*b*(7*b*B - 13*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(585*c^2) - (2*(7*b*B - 13*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(117*c) + (2*B*sqrt(x)*(b*x^2 + c*x^4)^(3/2))/(13*c) - (4*b^(9/4)*(7*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (2*b^(9/4)*(7*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)*(A + B*x^2)*sqrt(b*x^2 + c*x^4), x, 6, -((4*b*(5*b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(231*c^2*sqrt(x))) - (2*(5*b*B - 11*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c) + (2*B*(b*x^2 + c*x^4)^(3/2))/(11*c*sqrt(x)) + (2*b^(7/4)*(5*b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/sqrt(x), x, 7, -((4*b*(b*B - 3*A*c)*x^(3/2)*(b + c*x^2))/(15*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (2*(b*B - 3*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(15*c) + (2*B*(b*x^2 + c*x^4)^(3/2))/(9*c*x^(3/2)) + (4*b^(5/4)*(b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (2*b^(5/4)*(b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(3/2), x, 5, -((2*(b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(21*c*sqrt(x))) + (2*B*(b*x^2 + c*x^4)^(3/2))/(7*c*x^(5/2)) - (2*b^(3/4)*(b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(5/2), x, 7, (4*(b*B + 5*A*c)*x^(3/2)*(b + c*x^2))/(5*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (2*(b*B + 5*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*b) - (2*A*(b*x^2 + c*x^4)^(3/2))/(b*x^(7/2)) - (4*b^(1/4)*(b*B + 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(3/4)*sqrt(b*x^2 + c*x^4)) + (2*b^(1/4)*(b*B + 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(7/2), x, 5, (2*(b*B + A*c)*sqrt(b*x^2 + c*x^4))/(3*b*sqrt(x)) - (2*A*(b*x^2 + c*x^4)^(3/2))/(3*b*x^(9/2)) + (2*(b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*b^(1/4)*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(9/2), x, 7, (4*sqrt(c)*(5*b*B + A*c)*x^(3/2)*(b + c*x^2))/(5*b*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*(5*b*B + A*c)*sqrt(b*x^2 + c*x^4))/(5*b*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(3/2))/(5*b*x^(11/2)) - (4*c^(1/4)*(5*b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^2 + c*x^4)) + (2*c^(1/4)*(5*b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(11/2), x, 5, -((2*(7*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(21*b*x^(5/2))) - (2*A*(b*x^2 + c*x^4)^(3/2))/(7*b*x^(13/2)) + (2*c^(3/4)*(7*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*b^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(13/2), x, 8, (4*c^(3/2)*(3*b*B - A*c)*x^(3/2)*(b + c*x^2))/(15*b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*(3*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(15*b*x^(7/2)) - (4*c*(3*b*B - A*c)*sqrt(b*x^2 + c*x^4))/(15*b^2*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(3/2))/(9*b*x^(15/2)) - (4*c^(5/4)*(3*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(7/4)*sqrt(b*x^2 + c*x^4)) + (2*c^(5/4)*(3*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*sqrt(b*x^2 + c*x^4))/x^(15/2), x, 6, -((2*(11*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(77*b*x^(9/2))) - (4*c*(11*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(231*b^2*x^(5/2)) - (2*A*(b*x^2 + c*x^4)^(3/2))/(11*b*x^(17/2)) - (2*c^(7/4)*(11*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*b^(9/4)*sqrt(b*x^2 + c*x^4))]

    @test_int [x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 11, (88*b^5*(3*b*B - 5*A*c)*x^(3/2)*(b + c*x^2))/(16575*c^(9/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (88*b^4*(3*b*B - 5*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(49725*c^4) + (88*b^3*(3*b*B - 5*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(69615*c^3) - (8*b^2*(3*b*B - 5*A*c)*x^(9/2)*sqrt(b*x^2 + c*x^4))/(7735*c^2) - (4*b*(3*b*B - 5*A*c)*x^(13/2)*sqrt(b*x^2 + c*x^4))/(595*c) - (2*(3*b*B - 5*A*c)*x^(9/2)*(b*x^2 + c*x^4)^(3/2))/(105*c) + (2*B*x^(5/2)*(b*x^2 + c*x^4)^(5/2))/(25*c) - (88*b^(21/4)*(3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(16575*c^(19/4)*sqrt(b*x^2 + c*x^4)) + (44*b^(21/4)*(3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(16575*c^(19/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(5/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 9, -((24*b^4*(13*b*B - 23*A*c)*sqrt(b*x^2 + c*x^4))/(33649*c^4*sqrt(x))) + (72*b^3*(13*b*B - 23*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(168245*c^3) - (8*b^2*(13*b*B - 23*A*c)*x^(7/2)*sqrt(b*x^2 + c*x^4))/(24035*c^2) - (4*b*(13*b*B - 23*A*c)*x^(11/2)*sqrt(b*x^2 + c*x^4))/(2185*c) - (2*(13*b*B - 23*A*c)*x^(7/2)*(b*x^2 + c*x^4)^(3/2))/(437*c) + (2*B*x^(3/2)*(b*x^2 + c*x^4)^(5/2))/(23*c) + (12*b^(19/4)*(13*b*B - 23*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(33649*c^(17/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 10, -((8*b^4*(11*b*B - 21*A*c)*x^(3/2)*(b + c*x^2))/(3315*c^(7/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (8*b^3*(11*b*B - 21*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(9945*c^3) - (8*b^2*(11*b*B - 21*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(13923*c^2) - (4*b*(11*b*B - 21*A*c)*x^(9/2)*sqrt(b*x^2 + c*x^4))/(1547*c) - (2*(11*b*B - 21*A*c)*x^(5/2)*(b*x^2 + c*x^4)^(3/2))/(357*c) + (2*B*sqrt(x)*(b*x^2 + c*x^4)^(5/2))/(21*c) + (8*b^(17/4)*(11*b*B - 21*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3315*c^(15/4)*sqrt(b*x^2 + c*x^4)) - (4*b^(17/4)*(11*b*B - 21*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3315*c^(15/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [sqrt(x)*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x, 8, (8*b^3*(9*b*B - 19*A*c)*sqrt(b*x^2 + c*x^4))/(4389*c^3*sqrt(x)) - (8*b^2*(9*b*B - 19*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(7315*c^2) - (4*b*(9*b*B - 19*A*c)*x^(7/2)*sqrt(b*x^2 + c*x^4))/(1045*c) - (2*(9*b*B - 19*A*c)*x^(3/2)*(b*x^2 + c*x^4)^(3/2))/(285*c) + (2*B*(b*x^2 + c*x^4)^(5/2))/(19*c*sqrt(x)) - (4*b^(15/4)*(9*b*B - 19*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(4389*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/sqrt(x), x, 9, (8*b^3*(7*b*B - 17*A*c)*x^(3/2)*(b + c*x^2))/(1105*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (8*b^2*(7*b*B - 17*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(3315*c^2) - (4*b*(7*b*B - 17*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(663*c) - (2*(7*b*B - 17*A*c)*sqrt(x)*(b*x^2 + c*x^4)^(3/2))/(221*c) + (2*B*(b*x^2 + c*x^4)^(5/2))/(17*c*x^(3/2)) - (8*b^(13/4)*(7*b*B - 17*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(1105*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (4*b^(13/4)*(7*b*B - 17*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(1105*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(3/2), x, 7, -((8*b^2*(b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(231*c^2*sqrt(x))) - (4*b*(b*B - 3*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c) - (2*(b*B - 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(33*c*sqrt(x)) + (2*B*(b*x^2 + c*x^4)^(5/2))/(15*c*x^(5/2)) + (4*b^(11/4)*(b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(5/2), x, 8, -((8*b^2*(3*b*B - 13*A*c)*x^(3/2)*(b + c*x^2))/(195*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (4*b*(3*b*B - 13*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(195*c) - (2*(3*b*B - 13*A*c)*(b*x^2 + c*x^4)^(3/2))/(117*c*x^(3/2)) + (2*B*(b*x^2 + c*x^4)^(5/2))/(13*c*x^(7/2)) + (8*b^(9/4)*(3*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (4*b^(9/4)*(3*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(7/2), x, 6, -((4*b*(b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(77*c*sqrt(x))) - (2*(b*B - 11*A*c)*(b*x^2 + c*x^4)^(3/2))/(77*c*x^(5/2)) + (2*B*(b*x^2 + c*x^4)^(5/2))/(11*c*x^(9/2)) - (4*b^(7/4)*(b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(9/2), x, 8, (8*b*(b*B + 9*A*c)*x^(3/2)*(b + c*x^2))/(15*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (4/15)*(b*B + 9*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4) + (2*(b*B + 9*A*c)*(b*x^2 + c*x^4)^(3/2))/(9*b*x^(3/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(b*x^(11/2)) - (8*b^(5/4)*(b*B + 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(3/4)*sqrt(b*x^2 + c*x^4)) + (4*b^(5/4)*(b*B + 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(11/2), x, 6, (4*(3*b*B + 7*A*c)*sqrt(b*x^2 + c*x^4))/(21*sqrt(x)) + (2*(3*b*B + 7*A*c)*(b*x^2 + c*x^4)^(3/2))/(21*b*x^(5/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(3*b*x^(13/2)) + (4*b^(3/4)*(3*b*B + 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(13/2), x, 8, (24*sqrt(c)*(b*B + A*c)*x^(3/2)*(b + c*x^2))/(5*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + (12*c*(b*B + A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*b) - (2*(b*B + A*c)*(b*x^2 + c*x^4)^(3/2))/(b*x^(7/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(5*b*x^(15/2)) - (24*b^(1/4)*c^(1/4)*(b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4)) + (12*b^(1/4)*c^(1/4)*(b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(15/2), x, 6, (4*c*(7*b*B + 3*A*c)*sqrt(b*x^2 + c*x^4))/(21*b*sqrt(x)) - (2*(7*b*B + 3*A*c)*(b*x^2 + c*x^4)^(3/2))/(21*b*x^(9/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(7*b*x^(17/2)) + (4*c^(3/4)*(7*b*B + 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*b^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [((A + B*x^2)*(b*x^2 + c*x^4)^(3/2))/x^(17/2), x, 8, (8*c^(3/2)*(9*b*B + A*c)*x^(3/2)*(b + c*x^2))/(15*b*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (4*c*(9*b*B + A*c)*sqrt(b*x^2 + c*x^4))/(15*b*x^(3/2)) - (2*(9*b*B + A*c)*(b*x^2 + c*x^4)^(3/2))/(45*b*x^(11/2)) - (2*A*(b*x^2 + c*x^4)^(5/2))/(9*b*x^(19/2)) - (8*c^(5/4)*(9*b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(3/4)*sqrt(b*x^2 + c*x^4)) + (4*c^(5/4)*(9*b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(3/4)*sqrt(b*x^2 + c*x^4))]

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

    @test_int [(x^(13/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 7, -((2*b^2*(13*b*B - 15*A*c)*sqrt(b*x^2 + c*x^4))/(77*c^4*sqrt(x))) + (6*b*(13*b*B - 15*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(385*c^3) - (2*(13*b*B - 15*A*c)*x^(7/2)*sqrt(b*x^2 + c*x^4))/(165*c^2) + (2*B*x^(11/2)*sqrt(b*x^2 + c*x^4))/(15*c) + (b^(11/4)*(13*b*B - 15*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(77*c^(17/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(11/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 8, -((14*b^2*(11*b*B - 13*A*c)*x^(3/2)*(b + c*x^2))/(195*c^(7/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (14*b*(11*b*B - 13*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(585*c^3) - (2*(11*b*B - 13*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(117*c^2) + (2*B*x^(9/2)*sqrt(b*x^2 + c*x^4))/(13*c) + (14*b^(9/4)*(11*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(15/4)*sqrt(b*x^2 + c*x^4)) - (7*b^(9/4)*(11*b*B - 13*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(195*c^(15/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(9/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 6, (10*b*(9*b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(231*c^3*sqrt(x)) - (2*(9*b*B - 11*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c^2) + (2*B*x^(7/2)*sqrt(b*x^2 + c*x^4))/(11*c) - (5*b^(7/4)*(9*b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(7/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 7, (2*b*(7*b*B - 9*A*c)*x^(3/2)*(b + c*x^2))/(15*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*(7*b*B - 9*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(45*c^2) + (2*B*x^(5/2)*sqrt(b*x^2 + c*x^4))/(9*c) - (2*b^(5/4)*(7*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(11/4)*sqrt(b*x^2 + c*x^4)) + (b^(5/4)*(7*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(5/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 5, -((2*(5*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(21*c^2*sqrt(x))) + (2*B*x^(3/2)*sqrt(b*x^2 + c*x^4))/(7*c) + (b^(3/4)*(5*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(3/2)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 6, -((2*(3*b*B - 5*A*c)*x^(3/2)*(b + c*x^2))/(5*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) + (2*B*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*c) + (2*b^(1/4)*(3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(7/4)*sqrt(b*x^2 + c*x^4)) - (b^(1/4)*(3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(sqrt(x)*(A + B*x^2))/sqrt(b*x^2 + c*x^4), x, 4, (2*B*sqrt(b*x^2 + c*x^4))/(3*c*sqrt(x)) - ((b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*b^(1/4)*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(sqrt(x)*sqrt(b*x^2 + c*x^4)), x, 6, (2*(b*B + A*c)*x^(3/2)*(b + c*x^2))/(b*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*A*sqrt(b*x^2 + c*x^4))/(b*x^(3/2)) - (2*(b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*c^(3/4)*sqrt(b*x^2 + c*x^4)) + ((b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(3/2)*sqrt(b*x^2 + c*x^4)), x, 4, -((2*A*sqrt(b*x^2 + c*x^4))/(3*b*x^(5/2))) + ((3*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(3*b^(5/4)*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(5/2)*sqrt(b*x^2 + c*x^4)), x, 7, (2*sqrt(c)*(5*b*B - 3*A*c)*x^(3/2)*(b + c*x^2))/(5*b^2*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (2*A*sqrt(b*x^2 + c*x^4))/(5*b*x^(7/2)) - (2*(5*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(5*b^2*x^(3/2)) - (2*c^(1/4)*(5*b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(7/4)*sqrt(b*x^2 + c*x^4)) + (c^(1/4)*(5*b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(7/2)*sqrt(b*x^2 + c*x^4)), x, 5, -((2*A*sqrt(b*x^2 + c*x^4))/(7*b*x^(9/2))) - (2*(7*b*B - 5*A*c)*sqrt(b*x^2 + c*x^4))/(21*b^2*x^(5/2)) - (c^(3/4)*(7*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(21*b^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(9/2)*sqrt(b*x^2 + c*x^4)), x, 8, -((2*c^(3/2)*(9*b*B - 7*A*c)*x^(3/2)*(b + c*x^2))/(15*b^3*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4))) - (2*A*sqrt(b*x^2 + c*x^4))/(9*b*x^(11/2)) - (2*(9*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(45*b^2*x^(7/2)) + (2*c*(9*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(15*b^3*x^(3/2)) + (2*c^(5/4)*(9*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(11/4)*sqrt(b*x^2 + c*x^4)) - (c^(5/4)*(9*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(11/2)*sqrt(b*x^2 + c*x^4)), x, 6, -((2*A*sqrt(b*x^2 + c*x^4))/(11*b*x^(13/2))) - (2*(11*b*B - 9*A*c)*sqrt(b*x^2 + c*x^4))/(77*b^2*x^(9/2)) + (10*c*(11*b*B - 9*A*c)*sqrt(b*x^2 + c*x^4))/(231*b^3*x^(5/2)) + (5*c^(7/4)*(11*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(231*b^(13/4)*sqrt(b*x^2 + c*x^4))]

    @test_int [(x^(17/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 7, -(((b*B - A*c)*x^(15/2))/(b*c*sqrt(b*x^2 + c*x^4))) + (15*b*(13*b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(77*c^4*sqrt(x)) - (9*(13*b*B - 11*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(77*c^3) + ((13*b*B - 11*A*c)*x^(7/2)*sqrt(b*x^2 + c*x^4))/(11*b*c^2) - (15*b^(7/4)*(13*b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(154*c^(17/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(15/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 8, -(((b*B - A*c)*x^(13/2))/(b*c*sqrt(b*x^2 + c*x^4))) + (7*b*(11*b*B - 9*A*c)*x^(3/2)*(b + c*x^2))/(15*c^(7/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (7*(11*b*B - 9*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(45*c^3) + ((11*b*B - 9*A*c)*x^(5/2)*sqrt(b*x^2 + c*x^4))/(9*b*c^2) - (7*b^(5/4)*(11*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*c^(15/4)*sqrt(b*x^2 + c*x^4)) + (7*b^(5/4)*(11*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(30*c^(15/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(13/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 6, -(((b*B - A*c)*x^(11/2))/(b*c*sqrt(b*x^2 + c*x^4))) - (5*(9*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(21*c^3*sqrt(x)) + ((9*b*B - 7*A*c)*x^(3/2)*sqrt(b*x^2 + c*x^4))/(7*b*c^2) + (5*b^(3/4)*(9*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(42*c^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 7, -(((b*B - A*c)*x^(9/2))/(b*c*sqrt(b*x^2 + c*x^4))) - (3*(7*b*B - 5*A*c)*x^(3/2)*(b + c*x^2))/(5*c^(5/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + ((7*b*B - 5*A*c)*sqrt(x)*sqrt(b*x^2 + c*x^4))/(5*b*c^2) + (3*b^(1/4)*(7*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*c^(11/4)*sqrt(b*x^2 + c*x^4)) - (3*b^(1/4)*(7*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(10*c^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 5, -(((b*B - A*c)*x^(7/2))/(b*c*sqrt(b*x^2 + c*x^4))) + ((5*b*B - 3*A*c)*sqrt(b*x^2 + c*x^4))/(3*b*c^2*sqrt(x)) - ((5*b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(6*b^(1/4)*c^(9/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(7/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 6, -(((b*B - A*c)*x^(5/2))/(b*c*sqrt(b*x^2 + c*x^4))) + ((3*b*B - A*c)*x^(3/2)*(b + c*x^2))/(b*c^(3/2)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - ((3*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(3/4)*c^(7/4)*sqrt(b*x^2 + c*x^4)) + ((3*b*B - A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(3/4)*c^(7/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(5/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 4, -(((b*B - A*c)*x^(3/2))/(b*c*sqrt(b*x^2 + c*x^4))) + ((b*B + A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(5/4)*c^(5/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 7, -((2*A*sqrt(x))/(b*sqrt(b*x^2 + c*x^4))) + ((b*B - 3*A*c)*x^(5/2))/(b^2*sqrt(b*x^2 + c*x^4)) - ((b*B - 3*A*c)*x^(3/2)*(b + c*x^2))/(b^2*sqrt(c)*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) + ((b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(b^(7/4)*c^(3/4)*sqrt(b*x^2 + c*x^4)) - ((b*B - 3*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(2*b^(7/4)*c^(3/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(sqrt(x)*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2), x, 5, -((2*A)/(3*b*sqrt(x)*sqrt(b*x^2 + c*x^4))) + ((3*b*B - 5*A*c)*x^(3/2))/(3*b^2*sqrt(b*x^2 + c*x^4)) + ((3*b*B - 5*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(6*b^(9/4)*c^(1/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(sqrt(x)*(b*x^2 + c*x^4)^(3/2)), x, 8, -((2*A)/(5*b*x^(3/2)*sqrt(b*x^2 + c*x^4))) + ((5*b*B - 7*A*c)*sqrt(x))/(5*b^2*sqrt(b*x^2 + c*x^4)) + (3*sqrt(c)*(5*b*B - 7*A*c)*x^(3/2)*(b + c*x^2))/(5*b^3*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (3*(5*b*B - 7*A*c)*sqrt(b*x^2 + c*x^4))/(5*b^3*x^(3/2)) - (3*c^(1/4)*(5*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(5*b^(11/4)*sqrt(b*x^2 + c*x^4)) + (3*c^(1/4)*(5*b*B - 7*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(10*b^(11/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(3/2)*(b*x^2 + c*x^4)^(3/2)), x, 6, -((2*A)/(7*b*x^(5/2)*sqrt(b*x^2 + c*x^4))) + (7*b*B - 9*A*c)/(7*b^2*sqrt(x)*sqrt(b*x^2 + c*x^4)) - (5*(7*b*B - 9*A*c)*sqrt(b*x^2 + c*x^4))/(21*b^3*x^(5/2)) - (5*c^(3/4)*(7*b*B - 9*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(42*b^(13/4)*sqrt(b*x^2 + c*x^4))]
    @test_int [(A + B*x^2)/(x^(5/2)*(b*x^2 + c*x^4)^(3/2)), x, 9, -((2*A)/(9*b*x^(7/2)*sqrt(b*x^2 + c*x^4))) + (9*b*B - 11*A*c)/(9*b^2*x^(3/2)*sqrt(b*x^2 + c*x^4)) - (7*c^(3/2)*(9*b*B - 11*A*c)*x^(3/2)*(b + c*x^2))/(15*b^4*(sqrt(b) + sqrt(c)*x)*sqrt(b*x^2 + c*x^4)) - (7*(9*b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(45*b^3*x^(7/2)) + (7*c*(9*b*B - 11*A*c)*sqrt(b*x^2 + c*x^4))/(15*b^4*x^(3/2)) + (7*c^(5/4)*(9*b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.E(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(15*b^(15/4)*sqrt(b*x^2 + c*x^4)) - (7*c^(5/4)*(9*b*B - 11*A*c)*x*(sqrt(b) + sqrt(c)*x)*sqrt((b + c*x^2)/(sqrt(b) + sqrt(c)*x)^2)*Elliptic.F(2*atan((c^(1/4)*sqrt(x))/b^(1/4)), 1/2))/(30*b^(15/4)*sqrt(b*x^2 + c*x^4))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x^2)*(a*x^2+b*x^4)^p*with*m*symbolic=#

    @test_int [x^m*(A + B*x^2)*(b*x^2 + c*x^4)^3, x, 3, (A*b^3*x^(7 + m))/(7 + m) + (b^2*(b*B + 3*A*c)*x^(9 + m))/(9 + m) + (3*b*c*(b*B + A*c)*x^(11 + m))/(11 + m) + (c^2*(3*b*B + A*c)*x^(13 + m))/(13 + m) + (B*c^3*x^(15 + m))/(15 + m)]
    @test_int [x^m*(A + B*x^2)*(b*x^2 + c*x^4)^2, x, 3, (A*b^2*x^(5 + m))/(5 + m) + (b*(b*B + 2*A*c)*x^(7 + m))/(7 + m) + (c*(2*b*B + A*c)*x^(9 + m))/(9 + m) + (B*c^2*x^(11 + m))/(11 + m)]
    @test_int [x^m*(A + B*x^2)*(b*x^2 + c*x^4)^1, x, 3, (A*b*x^(3 + m))/(3 + m) + ((b*B + A*c)*x^(5 + m))/(5 + m) + (B*c*x^(7 + m))/(7 + m)]
    @test_int [x^m*(A + B*x^2)/(b*x^2 + c*x^4)^1, x, 3, -((B*x^(-1 + m))/(c*(1 - m))) + ((b*B - A*c)*x^(-1 + m)*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + m), (1 + m)/2, -((c*x^2)/b)))/(b*c*(1 - m))]
    @test_int [x^m*(A + B*x^2)/(b*x^2 + c*x^4)^2, x, 3, -(((b*B - A*c)*x^(-3 + m))/(2*b*c*(b + c*x^2))) + ((b*B*(3 - m) - A*c*(5 - m))*x^(-3 + m)*HypergeometricFunctions._₂F₁(1, (1/2)*(-3 + m), (1/2)*(-1 + m), -((c*x^2)/b)))/(2*b^2*c*(3 - m))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*(d+e*x^2)*(a*x^2+b*x^4)^p*with*p*symbolic=#

    @test_int [x^m*(A + B*x^2)*(b*x^2 + c*x^4)^p, x, 4, (B*x^(-1 + m)*(b*x^2 + c*x^4)^(1 + p))/(c*(3 + m + 4*p)) - ((b*B*(1 + m + 2*p) - A*c*(3 + m + 4*p))*x^(1 + m)*(b*x^2 + c*x^4)^p*HypergeometricFunctions._₂F₁(-p, (1/2)*(1 + m + 2*p), (1/2)*(3 + m + 2*p), -((c*x^2)/b)))/((1 + (c*x^2)/b)^p*(c*(1 + m + 2*p)*(3 + m + 4*p)))]

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

    @test_int [x^(n - j*p - 1)*(a*x^j + b*x^(j + n))^p*(c + d*x^n), x, 2, If(13>=8, -(((a*d - b*c*(2 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(x^(j*(1 + p))*(b^2*n*(1 + p)*(2 + p)))) + (d*x^(n - j*(1 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(b*n*(2 + p)), -(((a*d - b*c*(2 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(x^(j*(1 + p))*(b^2*n*(2 + 3*p + p^2)))) + (d*x^(n - j*(1 + p))*(a*x^j + b*x^(j + n))^(1 + p))/(b*n*(2 + p)))]

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

    @test_int [(e*x)^(7/4)*(a*x^j + b*x^(j + n))^(5/3)*(c + d*x^n)^q, x, 4, If(13<11, (12*a*e*x^(2 + j)*(e*x)^(3/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(2/3)*AppellF1((33 + 20*j)/(12*n), -(5/3), -q, 1 + (11/4 + (5*j)/3)/n, -((b*x^n)/a), -((d*x^n)/c)))/((1 + (d*x^n)/c)^q*((33 + 20*j)*(1 + (b*x^n)/a)^(2/3))), (12*a*e*x^(2 + j)*(e*x)^(3/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(2/3)*AppellF1((33 + 20*j)/(12*n), -(5/3), -q, (33 + 20*j + 12*n)/(12*n), -((b*x^n)/a), -((d*x^n)/c)))/((1 + (d*x^n)/c)^q*((33 + 20*j)*(1 + (b*x^n)/a)^(2/3))))]

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

    @test_int [(4 + 3*x^4)/(5*x + 2*x^5), x, 4, (4*log(x))/5 + (7*log(5 + 2*x^4))/40]
    @test_int [(1 + x^6)/(x - x^7), x, 4, log(x) - log(1 - x^6)/3]
    @test_int [(8 + 5*x^10)/(2*x - x^11), x, 4, 4*log(x) - (9/10)*log(2 - x^10)]
    @test_int [(-3 + 2*x)/(-x^2 + x^3), x, 3, -(3/x) - log(1 - x) + log(x)]

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(e*x)^m*(a*x^j+b*x^k)^p*(c*x^r+d*x^s)^q=#

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

    @test_int [x^m*(a*(m + q + 1)*x^q + b*(m + q + n*(p + 1) + 1)*x^(n + q))*(a + b*x^n)^p, x, 2, x^(1 + m + q)*(a + b*x^n)^(1 + p)]

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

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

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

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

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

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

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

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

max_stars_repo_path: test/1 Algebraic functions/1.1 Binomial products/1.1.4 Improper/1.1.4.3 (e x)^m (a x^j+b x^k)^p (c+d x^n)^q.jl


text:
module TestDay17

import AOC2021.Day17.part1, AOC2021.Day17.part2
using Test

test_input = "target area: x=20..30, y=-10..-5"

@testset "Day 17" begin
    @testset "part 1" begin
        @test part1(test_input) == 45
    end
    @testset "part 2" begin
        @test part2(test_input) == 112
    end
end

end # module

ext: jl