\(\int \frac {(A+B x^2) (d+e x^2)^2}{(a+c x^4)^{3/2}} \, dx\) [9]

Optimal result

Integrand size = 28, antiderivative size = 694 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {B e^2 x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (B c d^2+2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {a+c x^4}}+\frac {\left (B c d^2+2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c^{7/4} \sqrt {a+c x^4}}+\frac {e (2 B d+A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {\left (B c d^2+2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2-2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{3/4} c^{7/4} \sqrt {a+c x^4}} \]

[Out]

1/2*x*(A*c*d^2-2*a*B*d*e-a*A*e^2+(2*A*c*d*e-B*a*e^2+B*c*d^2)*x^2)/a/c/(c*x^4+a)^(1/2)+B*e^2*x*(c*x^4+a)^(1/2)/
c^(3/2)/(a^(1/2)+x^2*c^(1/2))-1/2*(2*A*c*d*e-B*a*e^2+B*c*d^2)*x*(c*x^4+a)^(1/2)/a/c^(3/2)/(a^(1/2)+x^2*c^(1/2)
)-a^(1/4)*B*e^2*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*ar
ctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(7/4)/
(c*x^4+a)^(1/2)+1/2*(2*A*c*d*e-B*a*e^2+B*c*d^2)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4
)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2
)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/c^(7/4)/(c*x^4+a)^(1/2)+1/2*a^(1/4)*B*e^2*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)
^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c
^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(7/4)/(c*x^4+a)^(1/2)+1/2*e*(A*e+2*B*d)*(cos(2*arctan(c^(1
/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/
2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/c^(5/4)/(c*x^4+a)^(1/2)-1/4*(cos(2
*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4)
)),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*(B*c*d^2+2*A*c*d*e-B*a*e^2-(-A*a*e^2+A*c*d^2-2*B*a*d*e)*c^(1/2)/a^(1/2))
*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/c^(7/4)/(c*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1735, 1193, 1212, 226, 1210, 311} \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\frac {\sqrt {c} \left (-a A e^2-2 a B d e+A c d^2\right )}{\sqrt {a}}-a B e^2+2 A c d e+B c d^2\right )}{4 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) \left (-a B e^2+2 A c d e+B c d^2\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} (A e+2 B d) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4} \left (-a B e^2+2 A c d e+B c d^2\right )}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (x^2 \left (-a B e^2+2 A c d e+B c d^2\right )-a A e^2-2 a B d e+A c d^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c^{7/4} \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {a+c x^4}}+\frac {B e^2 x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )} \]

[In]

Int[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]

[Out]

(x*(A*c*d^2 - 2*a*B*d*e - a*A*e^2 + (B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x^2))/(2*a*c*Sqrt[a + c*x^4]) + (B*e^2*x*S
qrt[a + c*x^4])/(c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*c
^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x
^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(7/4)*Sqrt[a + c*x^4]) + ((B*c*d^2 + 2*A*c*d*e - a*B*
e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(2*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4]) + (a^(1/4)*B*e^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[
a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(7/4)*Sqrt[a + c*x^4]) + (e*(2*B*d +
A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(2*a^(1/4)*c^(5/4)*Sqrt[a + c*x^4]) - ((B*c*d^2 + 2*A*c*d*e - a*B*e^2 - (Sqrt[c]*(A*c*d^2 - 2*a*B*d*
e - a*A*e^2))/Sqrt[a])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[
(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1193

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)*((a + c*x^4)^(p + 1)/
(4*a*(p + 1))), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x],
x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 1212

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(e + d*q)/q, Int
[1/Sqrt[a + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a
, c, d, e}, x] && PosQ[c/a]

Rule 1735

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a +
c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2] && NeQ[c*d^
2 + a*e^2, 0] && IntegerQ[p + 1/2] && IntegerQ[q]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2}{c \left (a+c x^4\right )^{3/2}}+\frac {e (2 B d+A e)}{c \sqrt {a+c x^4}}+\frac {B e^2 x^2}{c \sqrt {a+c x^4}}\right ) \, dx \\ & = \frac {\int \frac {A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{c}+\frac {\left (B e^2\right ) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx}{c}+\frac {(e (2 B d+A e)) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{c} \\ & = \frac {x \left (A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {e (2 B d+A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {\int \frac {-A c d^2+2 a B d e+a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2}{\sqrt {a+c x^4}} \, dx}{2 a c}+\frac {\left (\sqrt {a} B e^2\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{c^{3/2}}-\frac {\left (\sqrt {a} B e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{c^{3/2}} \\ & = \frac {x \left (A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {B e^2 x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{7/4} \sqrt {a+c x^4}}+\frac {e (2 B d+A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}+\frac {\left (B c d^2+2 A c d e-a B e^2\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} c^{3/2}}-\frac {\left (B c d^2+2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2-2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} c^{3/2}} \\ & = \frac {x \left (A c d^2-2 a B d e-a A e^2+\left (B c d^2+2 A c d e-a B e^2\right ) x^2\right )}{2 a c \sqrt {a+c x^4}}+\frac {B e^2 x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (B c d^2+2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{c^{7/4} \sqrt {a+c x^4}}+\frac {\left (B c d^2+2 A c d e-a B e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} B e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 c^{7/4} \sqrt {a+c x^4}}+\frac {e (2 B d+A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{5/4} \sqrt {a+c x^4}}-\frac {\left (B c d^2+2 A c d e-a B e^2-\frac {\sqrt {c} \left (A c d^2-2 a B d e-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{3/4} c^{7/4} \sqrt {a+c x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.14 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.24 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 A \left (c d^2-a e^2\right ) x+6 a B e x \left (-d+e x^2\right )+3 \left (A c d^2+2 a B d e+a A e^2\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 \left (B c d^2+2 A c d e-3 a B e^2\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a c \sqrt {a+c x^4}} \]

[In]

Integrate[((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x]

[Out]

(3*A*(c*d^2 - a*e^2)*x + 6*a*B*e*x*(-d + e*x^2) + 3*(A*c*d^2 + 2*a*B*d*e + a*A*e^2)*x*Sqrt[1 + (c*x^4)/a]*Hype
rgeometric2F1[1/4, 1/2, 5/4, -((c*x^4)/a)] + 2*(B*c*d^2 + 2*A*c*d*e - 3*a*B*e^2)*x^3*Sqrt[1 + (c*x^4)/a]*Hyper
geometric2F1[3/4, 3/2, 7/4, -((c*x^4)/a)])/(6*a*c*Sqrt[a + c*x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.47

method	result	size
elliptic	\(-\frac {2 c \left (-\frac {\left (2 A c d e -B a \,e^{2}+B c \,d^{2}\right ) x^{3}}{4 a \,c^{2}}+\frac {\left (a A \,e^{2}-A c \,d^{2}+2 a B d e \right ) x}{4 c^{2} a}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {e \left (A e +2 B d \right )}{c}-\frac {a A \,e^{2}-A c \,d^{2}+2 a B d e}{2 a c}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (\frac {B \,e^{2}}{c}-\frac {2 A c d e -B a \,e^{2}+B c \,d^{2}}{2 a c}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\)	\(323\)
default	\(d^{2} A \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+B \,e^{2} \left (-\frac {x^{3}}{2 c \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\left (A \,e^{2}+2 B e d \right ) \left (-\frac {x}{2 c \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\left (2 A e d +B \,d^{2}\right ) \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )\)	\(458\)

[In]

int((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2*c*(-1/4*(2*A*c*d*e-B*a*e^2+B*c*d^2)/a/c^2*x^3+1/4*(A*a*e^2-A*c*d^2+2*B*a*d*e)/c^2/a*x)/((x^4+a/c)*c)^(1/2)+
(e*(A*e+2*B*d)/c-1/2*(A*a*e^2-A*c*d^2+2*B*a*d*e)/a/c)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2
)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+I*(1/c*B*e^2-1/2*(2
*A*c*d*e-B*a*e^2+B*c*d^2)/a/c)*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*
c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^
(1/2))^(1/2),I))

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.48 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {{\left ({\left (B a c^{2} d^{2} + 2 \, A a c^{2} d e - 3 \, B a^{2} c e^{2}\right )} x^{5} + {\left (B a^{2} c d^{2} + 2 \, A a^{2} c d e - 3 \, B a^{3} e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (2 \, {\left (A + B\right )} a c^{2} d e + {\left (B a c^{2} + A c^{3}\right )} d^{2} - {\left (3 \, B a^{2} c - A a c^{2}\right )} e^{2}\right )} x^{5} + {\left (2 \, {\left (A + B\right )} a^{2} c d e + {\left (B a^{2} c + A a c^{2}\right )} d^{2} - {\left (3 \, B a^{3} - A a^{2} c\right )} e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (2 \, B a^{2} c e^{2} x^{4} - B a^{2} c d^{2} - 2 \, A a^{2} c d e + 3 \, B a^{3} e^{2} + {\left (A a c^{2} d^{2} - 2 \, B a^{2} c d e - A a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{2 \, {\left (a^{2} c^{3} x^{5} + a^{3} c^{2} x\right )}} \]

[In]

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(((B*a*c^2*d^2 + 2*A*a*c^2*d*e - 3*B*a^2*c*e^2)*x^5 + (B*a^2*c*d^2 + 2*A*a^2*c*d*e - 3*B*a^3*e^2)*x)*sqrt
(c)*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4)/x), -1) - ((2*(A + B)*a*c^2*d*e + (B*a*c^2 + A*c^3)*d^2 - (3*B
*a^2*c - A*a*c^2)*e^2)*x^5 + (2*(A + B)*a^2*c*d*e + (B*a^2*c + A*a*c^2)*d^2 - (3*B*a^3 - A*a^2*c)*e^2)*x)*sqrt
(c)*(-a/c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) - (2*B*a^2*c*e^2*x^4 - B*a^2*c*d^2 - 2*A*a^2*c*d*e + 3
*B*a^3*e^2 + (A*a*c^2*d^2 - 2*B*a^2*c*d*e - A*a^2*c*e^2)*x^2)*sqrt(c*x^4 + a))/(a^2*c^3*x^5 + a^3*c^2*x)

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((B*x**2+A)*(e*x**2+d)**2/(c*x**4+a)**(3/2),x)

[Out]

Integral((A + B*x**2)*(d + e*x**2)**2/(a + c*x**4)**(3/2), x)

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + a)^(3/2), x)

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^2}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \]

[In]

int(((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2),x)

[Out]

int(((A + B*x^2)*(d + e*x^2)^2)/(a + c*x^4)^(3/2), x)