\(\int \frac {(A+B x^2) (d+e x^2)^3}{(a+c x^4)^{3/2}} \, dx\) [8]

Optimal result

Integrand size = 28, antiderivative size = 912 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt {a+c x^4}}+\frac {B e^3 x \sqrt {a+c x^4}}{3 c^2}+\frac {e^2 (3 B d+A e) x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} e^2 (3 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}} 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^3+3 A c d^2 e-3 a B d e^2-a A e^3\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 {a^{3/4} B e^3 \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 )}{6 c^{9/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} e^2 (3 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 c^{7/4} \sqrt {a+c x^4}}+\frac {e \left (3 B c d^2+3 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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}+\frac {\left (A c^2 d^3+a^2 B e^3-3 a c d e (B d+A e)+a^{3/2} \sqrt {c} e^2 (3 B d+A e)-\sqrt {a} c^{3/2} d^2 (B d+3 A e)\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^{5/4} c^{9/4} \sqrt {a+c x^4}} \]

[Out]

1/2*x*(A*c*d*(-3*a*e^2+c*d^2)-a*B*e*(-a*e^2+3*c*d^2)+c*(-A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2+B*c*d^3)*x^2)/a/c^2/(
c*x^4+a)^(1/2)+1/3*B*e^3*x*(c*x^4+a)^(1/2)/c^2+e^2*(A*e+3*B*d)*x*(c*x^4+a)^(1/2)/c^(3/2)/(a^(1/2)+x^2*c^(1/2))
-1/2*(-A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2+B*c*d^3)*x*(c*x^4+a)^(1/2)/a/c^(3/2)/(a^(1/2)+x^2*c^(1/2))-a^(1/4)*e^2*
(A*e+3*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)))*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)/c^(7/4)/(c*x
^4+a)^(1/2)+1/2*(-A*a*e^3+3*A*c*d^2*e-3*B*a*d*e^2+B*c*d^3)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*ar
ctan(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)-1/6*a^(3/4)*B*e^3*(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^(9/4)/(c*x^4+a)^(1/2)+1/2*a^(1/4)*e^2*(A*e+3*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)/c^(7/4)/(c*x^4+a)^(1/
2)+1/2*e*(3*A*c*d*e-B*a*e^2+3*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)))*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^(9/4)/(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*c^2*d^3+a^2*B*e^3-3*a*c*d*e*(A*e+B
*d)-c^(3/2)*d^2*(3*A*e+B*d)*a^(1/2)+a^(3/2)*e^2*(A*e+3*B*d)*c^(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^(5/4)/c^(9/4)/(c*x^4+a)^(1/2)

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1735, 1193, 1212, 226, 1210, 311, 327} \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {a^{3/4} B \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e^3}{6 c^{9/4} \sqrt {c x^4+a}}+\frac {B x \sqrt {c x^4+a} e^3}{3 c^2}-\frac {\sqrt [4]{a} (3 B d+A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{c^{7/4} \sqrt {c x^4+a}}+\frac {\sqrt [4]{a} (3 B d+A e) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e^2}{2 c^{7/4} \sqrt {c x^4+a}}+\frac {(3 B d+A e) x \sqrt {c x^4+a} e^2}{c^{3/2} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {\left (3 B c d^2+3 A c e d-a B e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{2 \sqrt [4]{a} c^{9/4} \sqrt {c x^4+a}}+\frac {\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\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 {c x^4+a}}+\frac {\left (A c^2 d^3-\sqrt {a} c^{3/2} (B d+3 A e) d^2-3 a c e (B d+A e) d+a^2 B e^3+a^{3/2} \sqrt {c} e^2 (3 B d+A e)\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} c^{9/4} \sqrt {c x^4+a}}-\frac {\left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x \sqrt {c x^4+a}}{2 a c^{3/2} \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {x \left (c \left (B c d^3+3 A c e d^2-3 a B e^2 d-a A e^3\right ) x^2+A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )\right )}{2 a c^2 \sqrt {c x^4+a}} \]

[In]

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

[Out]

(x*(A*c*d*(c*d^2 - 3*a*e^2) - a*B*e*(3*c*d^2 - a*e^2) + c*(B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x^2)
)/(2*a*c^2*Sqrt[a + c*x^4]) + (B*e^3*x*Sqrt[a + c*x^4])/(3*c^2) + (e^2*(3*B*d + A*e)*x*Sqrt[a + c*x^4])/(c^(3/
2)*(Sqrt[a] + Sqrt[c]*x^2)) - ((B*c*d^3 + 3*A*c*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x*Sqrt[a + c*x^4])/(2*a*c^(3/2)
*(Sqrt[a] + Sqrt[c]*x^2)) - (a^(1/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqr
t[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^3 + 3*A*c*d^2*
e - 3*a*B*d*e^2 - a*A*e^3)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*Arc
Tan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(3/4)*c^(7/4)*Sqrt[a + c*x^4]) - (a^(3/4)*B*e^3*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(6*c^(9/4)*Sqrt[a +
c*x^4]) + (a^(1/4)*e^2*(3*B*d + A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*c^(7/4)*Sqrt[a + c*x^4]) + (e*(3*B*c*d^2 + 3*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]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/
2])/(2*a^(1/4)*c^(9/4)*Sqrt[a + c*x^4]) + ((A*c^2*d^3 + a^2*B*e^3 - 3*a*c*d*e*(B*d + A*e) + a^(3/2)*Sqrt[c]*e^
2*(3*B*d + A*e) - Sqrt[a]*c^(3/2)*d^2*(B*d + 3*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])/(4*a^(5/4)*c^(9/4)*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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{c^2 \left (a+c x^4\right )^{3/2}}+\frac {e \left (3 B c d^2+3 A c d e-a B e^2\right )}{c^2 \sqrt {a+c x^4}}+\frac {e^2 (3 B d+A e) x^2}{c \sqrt {a+c x^4}}+\frac {B e^3 x^4}{c \sqrt {a+c x^4}}\right ) \, dx \\ & = \frac {\int \frac {A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{\left (a+c x^4\right )^{3/2}} \, dx}{c^2}+\frac {\left (B e^3\right ) \int \frac {x^4}{\sqrt {a+c x^4}} \, dx}{c}+\frac {\left (e^2 (3 B d+A e)\right ) \int \frac {x^2}{\sqrt {a+c x^4}} \, dx}{c}+\frac {\left (e \left (3 B c d^2+3 A c d e-a B e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{c^2} \\ & = \frac {x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt {a+c x^4}}+\frac {B e^3 x \sqrt {a+c x^4}}{3 c^2}+\frac {e \left (3 B c d^2+3 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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}-\frac {\int \frac {-A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2}{\sqrt {a+c x^4}} \, dx}{2 a c^2}-\frac {\left (a B e^3\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{3 c^2}+\frac {\left (\sqrt {a} e^2 (3 B d+A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{c^{3/2}}-\frac {\left (\sqrt {a} e^2 (3 B d+A e)\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 \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt {a+c x^4}}+\frac {B e^3 x \sqrt {a+c x^4}}{3 c^2}+\frac {e^2 (3 B d+A e) x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} e^2 (3 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}} 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 {a^{3/4} B e^3 \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 )}{6 c^{9/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} e^2 (3 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 c^{7/4} \sqrt {a+c x^4}}+\frac {e \left (3 B c d^2+3 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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}+\frac {\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\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 (A c^2 d^3+a^2 B e^3-3 a c d e (B d+A e)+a^{3/2} \sqrt {c} e^2 (3 B d+A e)-\sqrt {a} c^{3/2} d^2 (B d+3 A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 a c^2} \\ & = \frac {x \left (A c d \left (c d^2-3 a e^2\right )-a B e \left (3 c d^2-a e^2\right )+c \left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x^2\right )}{2 a c^2 \sqrt {a+c x^4}}+\frac {B e^3 x \sqrt {a+c x^4}}{3 c^2}+\frac {e^2 (3 B d+A e) x \sqrt {a+c x^4}}{c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (B c d^3+3 A c d^2 e-3 a B d e^2-a A e^3\right ) x \sqrt {a+c x^4}}{2 a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} e^2 (3 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}} 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^3+3 A c d^2 e-3 a B d e^2-a A e^3\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 {a^{3/4} B e^3 \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 )}{6 c^{9/4} \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} e^2 (3 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 c^{7/4} \sqrt {a+c x^4}}+\frac {e \left (3 B c d^2+3 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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} c^{9/4} \sqrt {a+c x^4}}+\frac {\left (A c^2 d^3+a^2 B e^3-3 a c d e (B d+A e)+a^{3/2} \sqrt {c} e^2 (3 B d+A e)-\sqrt {a} c^{3/2} d^2 (B d+3 A e)\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^{5/4} c^{9/4} \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 = 10.19 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.24 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 A c x \left (c d^3+a e^2 \left (-3 d+2 e x^2\right )\right )+a B e x \left (5 a e^2+c \left (-9 d^2+18 d e x^2+2 e^2 x^4\right )\right )+\left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right ) x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 c \left (B c d^3+3 A c d^2 e-9 a B d e^2-3 a A e^3\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a c^2 \sqrt {a+c x^4}} \]

[In]

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

[Out]

(3*A*c*x*(c*d^3 + a*e^2*(-3*d + 2*e*x^2)) + a*B*e*x*(5*a*e^2 + c*(-9*d^2 + 18*d*e*x^2 + 2*e^2*x^4)) + (a*B*e*(
9*c*d^2 - 5*a*e^2) + 3*A*c*d*(c*d^2 + 3*a*e^2))*x*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^
4)/a)] + 2*c*(B*c*d^3 + 3*A*c*d^2*e - 9*a*B*d*e^2 - 3*a*A*e^3)*x^3*Sqrt[1 + (c*x^4)/a]*Hypergeometric2F1[3/4,
3/2, 7/4, -((c*x^4)/a)])/(6*a*c^2*Sqrt[a + c*x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.29 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.47

method	result	size
elliptic	\(-\frac {2 c \left (\frac {\left (A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}\right ) x^{3}}{4 c^{2} a}+\frac {\left (3 A a c d \,e^{2}-A \,c^{2} d^{3}-a^{2} B \,e^{3}+3 B a c \,d^{2} e \right ) x}{4 a \,c^{3}}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {B \,e^{3} x \sqrt {c \,x^{4}+a}}{3 c^{2}}+\frac {\left (\frac {e \left (3 A c d e -B a \,e^{2}+3 B c \,d^{2}\right )}{c^{2}}-\frac {3 A a c d \,e^{2}-A \,c^{2} d^{3}-a^{2} B \,e^{3}+3 B a c \,d^{2} e}{2 c^{2} a}-\frac {B \,e^{3} a}{3 c^{2}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \left (\frac {e^{2} \left (A e +3 B d \right )}{c}+\frac {A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}}{2 c a}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\)	\(426\)
default	\(A \,d^{3} \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+B \,e^{3} \left (\frac {x a}{2 c^{2} \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {x \sqrt {c \,x^{4}+a}}{3 c^{2}}-\frac {5 a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{6 c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\left (e^{3} A +3 B \,e^{2} d \right ) \left (-\frac {x^{3}}{2 c \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+\left (3 A \,d^{2} e +d^{3} B \right ) \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (-\frac {x}{2 c \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )\)	\(588\)
risch	\(\frac {B \,e^{3} x \sqrt {c \,x^{4}+a}}{3 c^{2}}+\frac {3 A \,c^{2} d^{3} \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 )-a^{2} B \,e^{3} \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 )+\left (3 A \,c^{2} e^{3}+9 B \,c^{2} d \,e^{2}\right ) \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 (9 A \,c^{2} d^{2} e +3 B \,c^{2} d^{3}\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 )+\left (9 A \,c^{2} d \,e^{2}-4 B a c \,e^{3}+9 B \,c^{2} d^{2} e \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 )}{3 c^{2}}\)	\(632\)

[In]

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

[Out]

-2*c*(1/4/c^2*(A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2-B*c*d^3)/a*x^3+1/4/a/c^3*(3*A*a*c*d*e^2-A*c^2*d^3-B*a^2*e^3+3*B
*a*c*d^2*e)*x)/((x^4+a/c)*c)^(1/2)+1/3*B*e^3*x*(c*x^4+a)^(1/2)/c^2+(e*(3*A*c*d*e-B*a*e^2+3*B*c*d^2)/c^2-1/2/c^
2/a*(3*A*a*c*d*e^2-A*c^2*d^3-B*a^2*e^3+3*B*a*c*d^2*e)-1/3*B*e^3/c^2*a)/(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*e^2*(A*e+3*B*d)+1/2/c*(A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2-B*c*d^3)/a)*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 = 0.12 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.49 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+c x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left ({\left (B a c^{2} d^{3} + 3 \, A a c^{2} d^{2} e - 9 \, B a^{2} c d e^{2} - 3 \, A a^{2} c e^{3}\right )} x^{5} + {\left (B a^{2} c d^{3} + 3 \, A a^{2} c d^{2} e - 9 \, B a^{3} d e^{2} - 3 \, A a^{3} e^{3}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (9 \, {\left (A + B\right )} a c^{2} d^{2} e - {\left (9 \, A + 5 \, B\right )} a^{2} c e^{3} + 3 \, {\left (B a c^{2} + A c^{3}\right )} d^{3} - 9 \, {\left (3 \, B a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{5} + {\left (9 \, {\left (A + B\right )} a^{2} c d^{2} e - {\left (9 \, A + 5 \, B\right )} a^{3} e^{3} + 3 \, {\left (B a^{2} c + A a c^{2}\right )} d^{3} - 9 \, {\left (3 \, B a^{3} - A a^{2} c\right )} d e^{2}\right )} x\right )} \sqrt {c} \left (-\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (2 \, B a^{2} c e^{3} x^{6} - 3 \, B a^{2} c d^{3} - 9 \, A a^{2} c d^{2} e + 27 \, B a^{3} d e^{2} + 9 \, A a^{3} e^{3} + 6 \, {\left (3 \, B a^{2} c d e^{2} + A a^{2} c e^{3}\right )} x^{4} + {\left (3 \, A a c^{2} d^{3} - 9 \, B a^{2} c d^{2} e - 9 \, A a^{2} c d e^{2} + 5 \, B a^{3} e^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + a}}{6 \, {\left (a^{2} c^{3} x^{5} + a^{3} c^{2} x\right )}} \]

[In]

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

[Out]

-1/6*(3*((B*a*c^2*d^3 + 3*A*a*c^2*d^2*e - 9*B*a^2*c*d*e^2 - 3*A*a^2*c*e^3)*x^5 + (B*a^2*c*d^3 + 3*A*a^2*c*d^2*
e - 9*B*a^3*d*e^2 - 3*A*a^3*e^3)*x)*sqrt(c)*(-a/c)^(3/4)*elliptic_e(arcsin((-a/c)^(1/4)/x), -1) - ((9*(A + B)*
a*c^2*d^2*e - (9*A + 5*B)*a^2*c*e^3 + 3*(B*a*c^2 + A*c^3)*d^3 - 9*(3*B*a^2*c - A*a*c^2)*d*e^2)*x^5 + (9*(A + B
)*a^2*c*d^2*e - (9*A + 5*B)*a^3*e^3 + 3*(B*a^2*c + A*a*c^2)*d^3 - 9*(3*B*a^3 - A*a^2*c)*d*e^2)*x)*sqrt(c)*(-a/
c)^(3/4)*elliptic_f(arcsin((-a/c)^(1/4)/x), -1) - (2*B*a^2*c*e^3*x^6 - 3*B*a^2*c*d^3 - 9*A*a^2*c*d^2*e + 27*B*
a^3*d*e^2 + 9*A*a^3*e^3 + 6*(3*B*a^2*c*d*e^2 + A*a^2*c*e^3)*x^4 + (3*A*a*c^2*d^3 - 9*B*a^2*c*d^2*e - 9*A*a^2*c
*d*e^2 + 5*B*a^3*e^3)*x^2)*sqrt(c*x^4 + a))/(a^2*c^3*x^5 + a^3*c^2*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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