3.1.13 \(\int \frac {A+B x^2}{(d+e x^2)^2 (a+c x^4)^{3/2}} \, dx\) [13]

Optimal. Leaf size=1494 \[ \frac {c 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 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} (B d-A e) \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} 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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (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} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a 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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {a^{3/4} e \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \]

[Out]

-1/4*e^(3/2)*(-A*e+B*d)*(a*e^2+3*c*d^2)*arctan(x*(a*e^2+c*d^2)^(1/2)/d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(3/2)/
(a*e^2+c*d^2)^(5/2)-1/2*e^(3/2)*(-2*A*c*d*e-B*a*e^2+B*c*d^2)*arctan(x*(a*e^2+c*d^2)^(1/2)/d^(1/2)/e^(1/2)/(c*x
^4+a)^(1/2))/(a*e^2+c*d^2)^(5/2)/d^(1/2)+1/2*c*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/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/2*e^3*(-A*e+B*d)*x*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)^2/(e*x^2+d)+1/2*e^2*(-A
*e+B*d)*x*c^(1/2)*(c*x^4+a)^(1/2)/d/(a*e^2+c*d^2)^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
^(1/2)*(c*x^4+a)^(1/2)/a/(a*e^2+c*d^2)^2/(a^(1/2)+x^2*c^(1/2))-1/2*a^(1/4)*c^(1/4)*e^2*(-A*e+B*d)*(cos(2*arcta
n(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)/d/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)+1/
2*c^(1/4)*(-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)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/2*c^(1/4)*e*(-A*e+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)/d/(a*e^2+c*d^2)/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+
a)^(1/2)-1/2*c^(1/4)*e*(-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)))*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)/(a*e^2+c*d^2)^2/(-e*a^(1/2)+d*c^(1/2))/(c*x^4+a)^(1/2)+1/8*e*(-A*e+B*d)*(a
*e^2+3*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)))*EllipticPi(sin(2*arc
tan(c^(1/4)*x/a^(1/4))),-1/4*(-e*a^(1/2)+d*c^(1/2))^2/d/e/a^(1/2)/c^(1/2),1/2*2^(1/2))*(e*a^(1/2)+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^(1/4)/c^(1/4)/d^2/(a*e^2+c*d^2)^2/(-e*a^(1/2)
+d*c^(1/2))/(c*x^4+a)^(1/2)+1/4*a^(3/4)*e*(-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)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(-e*a^(1/2)+d*c^(1/2))^2
/d/e/a^(1/2)/c^(1/2),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*(e+d*c^(1/2)/a^(1/2))^2*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2
))^2)^(1/2)/c^(1/4)/d/(-a*e^2+c*d^2)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/4*c^(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-a*B*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)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.36, antiderivative size = 1494, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1735, 1193, 1212, 226, 1210, 1238, 1729, 1723, 1721, 1231} \begin {gather*} -\frac {(B d-A e) x \sqrt {c x^4+a} e^3}{2 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )}-\frac {\sqrt [4]{a} \sqrt [4]{c} (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 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{2 d \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt {c} (B d-A e) x \sqrt {c x^4+a} e^2}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {(B d-A e) \left (3 c d^2+a e^2\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (B c d^2-2 A c e d-a B e^2\right ) \text {ArcTan}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} \left (B c d^2-2 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}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} (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}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {c x^4+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a 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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {a^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \left (B c d^2-2 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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e}{4 \sqrt [4]{c} d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (B c d^2-2 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}} E\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt [4]{c} \left (B c d^2-2 A c e d-a B e^2-\frac {\sqrt {c} \left (A c d^2+2 a B e d-a A e^2\right )}{\sqrt {a}}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 a^{3/4} \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt {c} \left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {c x \left (A c d^2+2 a B e d-a A e^2+\left (B c d^2-2 A c e d-a B e^2\right ) x^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {c x^4+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*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*d^2 + a*e^2)^2*Sqrt[a + c*
x^4]) + (Sqrt[c]*e^2*(B*d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[c]
*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*a*(c*d^2 + a*e^2)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*(B*
d - A*e)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^2*(d + e*x^2)) - (e^(3/2)*(B*d - A*e)*(3*c*d^2 + a*e^2)*ArcTa
n[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(5/2)) - (e^(3/2)*(B*
c*d^2 - 2*A*c*d*e - a*B*e^2)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(2*Sqrt[d]*(c*
d^2 + a*e^2)^(5/2)) - (a^(1/4)*c^(1/4)*e^2*(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])/(2*d*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (c^(1/4)
*(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*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*e*(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)*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)*Sqrt[a + c*x^4]) - (c^(1/4)*e*(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]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^
(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) - (c^(1/4)*(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*d^2 + a*e^2)^2*Sqrt[a
+ c*x^4]) + (e*(Sqrt[c]*d + Sqrt[a]*e)*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/
(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sqrt[c]*d - Sqrt[a]*e)^2/(Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*
x)/a^(1/4)], 1/2])/(8*a^(1/4)*c^(1/4)*d^2*(Sqrt[c]*d - Sqrt[a]*e)*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^4]) + (a^(3/4
)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(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]*EllipticPi[-1/4*(Sqrt[c]*d - Sqrt[a]*e)^2/(Sqrt[a]*Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^
(1/4)], 1/2])/(4*c^(1/4)*d*(c*d^2 - a*e^2)*(c*d^2 + a*e^2)^2*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 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 1231

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

Rule 1238

Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt
[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*d*(q + 1)*(c*d^2 + a*e^2)), Int[((d + e*x^2)^(q + 1
)/Sqrt[a + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x], x],
x] /; FreeQ[{a, c, d, e}, x] && ILtQ[q, -1]

Rule 1721

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

Rule 1723

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

Rule 1729

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x
, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Dist[-C/(e*q), Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] +
Dist[1/(c*e), Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /;
FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && 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*} \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \left (a+c x^4\right )^{3/2}} \, dx &=\int \left (\frac {c \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 )}{\left (c d^2+a e^2\right )^2 \left (a+c x^4\right )^{3/2}}+\frac {e (-B d+A e)}{\left (c d^2+a e^2\right ) \left (d+e x^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (-B c d^2+2 A c d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}}\right ) \, dx\\ &=\frac {c \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}{\left (c d^2+a e^2\right )^2}-\frac {(e (B d-A e)) \int \frac {1}{\left (d+e x^2\right )^2 \sqrt {a+c x^4}} \, dx}{c d^2+a e^2}-\frac {\left (e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=\frac {c 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 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {c \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 \left (c d^2+a e^2\right )^2}+\frac {(e (B d-A e)) \int \frac {-2 c d^2-a e^2+2 c d e x^2+c e^2 x^4}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} e \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {a} e^2 \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{\left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac {c 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 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{c} e \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {(B d-A e) \int \frac {\sqrt {a} c^{3/2} d e^2+c e \left (-2 c d^2-a e^2\right )+\left (2 c^2 d e^2-c e^2 \left (c d-\sqrt {a} \sqrt {c} e\right )\right ) x^2}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 c d \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {a} \sqrt {c} e^2 (B d-A e)\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 d \left (c d^2+a e^2\right )^2}+\frac {\left (\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}-\frac {\left (\sqrt {c} \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 )\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{2 \sqrt {a} \left (c d^2+a e^2\right )^2}\\ &=\frac {c 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 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} 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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} e (B d-A e)\right ) \int \frac {1}{\sqrt {a+c x^4}} \, dx}{d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )}+\frac {\left (\sqrt {a} e^2 (B d-A e) \left (3 c d^2+a e^2\right )\right ) \int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a+c x^4}} \, dx}{2 d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2}\\ &=\frac {c 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 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt {c} e^2 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt {c} \left (B c d^2-2 A c d e-a B e^2\right ) x \sqrt {a+c x^4}}{2 a \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^3 (B d-A e) x \sqrt {a+c x^4}}{2 d \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}-\frac {e^{3/2} (B d-A e) \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{4 d^{3/2} \left (c d^2+a e^2\right )^{5/2}}-\frac {e^{3/2} \left (B c d^2-2 A c d e-a B e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} 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}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 d \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e (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} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} e \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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{c} \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} \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (3 c d^2+a 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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) \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}} \Pi \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 10.85, size = 427, normalized size = 0.29 \begin {gather*} \frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d \left (a e^3 (-B d+A e) x \left (a+c x^4\right )+c d x \left (d+e x^2\right ) \left (-a A e^2+B c d^2 x^2+A c d \left (d-2 e x^2\right )+a B e \left (2 d-e x^2\right )\right )\right )-\left (d+e x^2\right ) \sqrt {1+\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d \left (-B c d^3+2 A c d^2 e+2 a B d e^2-a A e^3\right ) E\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A c d^2+i \sqrt {a} \sqrt {c} d (B d-A e)+a e (2 B d-A e)\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+a e \left (-5 B c d^3+7 A c d^2 e+a B d e^2+a A e^3\right ) \Pi \left (-\frac {i \sqrt {a} e}{\sqrt {c} d};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^3+a d e^2\right )^2 \left (d+e x^2\right ) \sqrt {a+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.17, size = 1384, normalized size = 0.93

method	result	size
default	\(\text {Expression too large to display}\)	\(1384\)
elliptic	\(\text {Expression too large to display}\)	\(1664\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B/e*(-2*c*(1/4/a*e/(a*e^2+c*d^2)*x^3-1/4*d/a/(a*e^2+c*d^2)*x)/((x^4+a/c)*c)^(1/2)+1/2*c*d/a/(a*e^2+c*d^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)*Elliptic
F(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I/a^(1/2)*c^(1/2)*e/(a*e^2+c*d^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)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-1/
2*I/a^(1/2)*c^(1/2)*e/(a*e^2+c*d^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)*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/(a*e^2+c*d^2)*e^2/d/(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)*EllipticPi(x*(I/a^(
1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))+(A*e-B*d)/e*(
1/2*e^4/(a*e^2+c*d^2)^2/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)-2*c*(1/2/a*c*d*e/(a*e^2+c*d^2)^2*x^3+1/4/a*(a*e^2-c*d^2)
/(a*e^2+c*d^2)^2*x)/((x^4+a/c)*c)^(1/2)-1/(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)*e^2*c/(a*e^2+c*d^2)^2+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)*EllipticF
(x*(I/a^(1/2)*c^(1/2))^(1/2),I)*c^2/a/(a*e^2+c*d^2)^2*d^2-1/2*I*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)*e^3/d/(a*e^2+c*d^2)^2*EllipticF(x*
(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*I*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)*e^3/d/(a*e^2+c*d^2)^2*EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+
I/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^(3/2)*d*e/(a*e^2+c*d^2)^2*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-I/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^(3/2)*d*e/(a*e^2+c*d^2)^2*E
llipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+1/2*e^4/d^2/(a*e^2+c*d^2)^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)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),I*a^(1/
2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*a+7/2*e^2/(a*e^2+c*d^2)^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)*EllipticPi(x*(I/a^(
1/2)*c^(1/2))^(1/2),I*a^(1/2)/c^(1/2)*e/d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2))*c)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x^{2}}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {B\,x^2+A}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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