\(\int \frac {A+B x^2}{(d+e x^2)^3 (a+c x^4)^{3/2}} \, dx\) [80]

Optimal result

Integrand size = 28, antiderivative size = 1142 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \left (a+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Output:

-1/4*e*(-A*e+B*d)*x/d/(a*e^2+c*d^2)/(e*x^2+d)^2/(c*x^4+a)^(1/2)-1/8*e*(-3* 
A*a*e^3-13*A*c*d^2*e-B*a*d*e^2+9*B*c*d^3)*x/d^2/(a*e^2+c*d^2)^2/(e*x^2+d)/ 
(c*x^4+a)^(1/2)+1/8*c*x*(d*(a*B*d*e*(-7*a*e^2+23*c*d^2)+A*(-a^2*e^4-27*a*c 
*d^2*e^2+4*c^2*d^4))-(3*A*e*(-a^2*e^4-7*a*c*d^2*e^2+4*c^2*d^4)-B*(a^2*d*e^ 
4-25*a*c*d^3*e^2+4*c^2*d^5))*x^2)/a/d^2/(a*e^2+c*d^2)^3/(c*x^4+a)^(1/2)+1/ 
8*c^(1/2)*(3*A*e*(-a^2*e^4-7*a*c*d^2*e^2+4*c^2*d^4)-B*(a^2*d*e^4-25*a*c*d^ 
3*e^2+4*c^2*d^5))*x*(c*x^4+a)^(1/2)/a/d^2/(a*e^2+c*d^2)^3/(a^(1/2)+c^(1/2) 
*x^2)+1/16*e^(3/2)*(3*A*e*(a^2*e^4+2*a*c*d^2*e^2+21*c^2*d^4)-B*(-a^2*d*e^4 
-26*a*c*d^3*e^2+35*c^2*d^5))*arctan((a*e^2+c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/ 
(c*x^4+a)^(1/2))/d^(5/2)/(a*e^2+c*d^2)^(7/2)-1/8*c^(1/4)*(3*A*e*(-a^2*e^4- 
7*a*c*d^2*e^2+4*c^2*d^4)-B*(a^2*d*e^4-25*a*c*d^3*e^2+4*c^2*d^5))*(a^(1/2)+ 
c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arc 
tan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/d^2/(a*e^2+c*d^2)^3/(c*x^4+a) 
^(1/2)+1/8*c^(1/4)*(2*A*c^2*d^4-2*a*c*d^2*e*(-5*A*e+2*B*d)-2*a^(1/2)*c^(3/ 
2)*d^3*(-2*A*e+B*d)+a^(3/2)*c^(1/2)*d*e^2*(-A*e+3*B*d)+a^2*e^3*(3*A*e+B*d) 
)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJ 
acobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(5/4)/d^2/(c^(1/2)*d-a^ 
(1/2)*e)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)-1/32*e*(c^(1/2)*d+a^(1/2)*e)*(3*A 
*e*(a^2*e^4+2*a*c*d^2*e^2+21*c^2*d^4)-B*(-a^2*d*e^4-26*a*c*d^3*e^2+35*c^2* 
d^5))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*E...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 12.91 (sec) , antiderivative size = 630, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \left (a+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d x \left (-2 a d e^3 (B d-A e) \left (c d^2+a e^2\right ) \left (a+c x^4\right )+a e^3 \left (-13 B c d^3+17 A c d^2 e+a B d e^2+3 a A e^3\right ) \left (d+e x^2\right ) \left (a+c x^4\right )+4 c d^2 \left (d+e x^2\right )^2 \left (B \left (-a^2 e^3+c^2 d^3 x^2+3 a c d e \left (d-e x^2\right )\right )+A c \left (c d^2 \left (d-3 e x^2\right )+a e^2 \left (-3 d+e x^2\right )\right )\right )\right )-\left (d+e x^2\right )^2 \sqrt {1+\frac {c x^4}{a}} \left (\sqrt {a} \sqrt {c} d \left (3 A e \left (-4 c^2 d^4+7 a c d^2 e^2+a^2 e^4\right )+B \left (4 c^2 d^5-25 a c d^3 e^2+a^2 d e^4\right )\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+i \left (\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (4 A c^2 d^4+4 i \sqrt {a} c^{3/2} d^3 (B d-2 A e)+19 a c d^2 e (B d-A e)-2 i a^{3/2} \sqrt {c} d e^2 (3 B d-A e)-a^2 e^3 (B d+3 A e)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+a e \left (3 A e \left (21 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )+B \left (-35 c^2 d^5+26 a c d^3 e^2+a^2 d e^4\right )\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )\right )}{8 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (c d^3+a d e^2\right )^3 \left (d+e x^2\right )^2 \sqrt {a+c x^4}} \] Input:

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

Output:

(Sqrt[(I*Sqrt[c])/Sqrt[a]]*d*x*(-2*a*d*e^3*(B*d - A*e)*(c*d^2 + a*e^2)*(a 
+ c*x^4) + a*e^3*(-13*B*c*d^3 + 17*A*c*d^2*e + a*B*d*e^2 + 3*a*A*e^3)*(d + 
 e*x^2)*(a + c*x^4) + 4*c*d^2*(d + e*x^2)^2*(B*(-(a^2*e^3) + c^2*d^3*x^2 + 
 3*a*c*d*e*(d - e*x^2)) + A*c*(c*d^2*(d - 3*e*x^2) + a*e^2*(-3*d + e*x^2)) 
)) - (d + e*x^2)^2*Sqrt[1 + (c*x^4)/a]*(Sqrt[a]*Sqrt[c]*d*(3*A*e*(-4*c^2*d 
^4 + 7*a*c*d^2*e^2 + a^2*e^4) + B*(4*c^2*d^5 - 25*a*c*d^3*e^2 + a^2*d*e^4) 
)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + I*(Sqrt[c]*d*(Sq 
rt[c]*d - I*Sqrt[a]*e)*(4*A*c^2*d^4 + (4*I)*Sqrt[a]*c^(3/2)*d^3*(B*d - 2*A 
*e) + 19*a*c*d^2*e*(B*d - A*e) - (2*I)*a^(3/2)*Sqrt[c]*d*e^2*(3*B*d - A*e) 
 - a^2*e^3*(B*d + 3*A*e))*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x] 
, -1] + a*e*(3*A*e*(21*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + B*(-35*c^2*d^5 
 + 26*a*c*d^3*e^2 + a^2*d*e^4))*EllipticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I 
*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])))/(8*a*Sqrt[(I*Sqrt[c])/Sqrt[a 
]]*(c*d^3 + a*d*e^2)^3*(d + e*x^2)^2*Sqrt[a + c*x^4])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(2452\) vs. \(2(1142)=2284\).

Time = 4.05 (sec) , antiderivative size = 2452, normalized size of antiderivative = 2.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2259, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2259

\(\displaystyle \int \left (\frac {e (A e-B d)}{\sqrt {a+c x^4} \left (d+e x^2\right )^3 \left (a e^2+c d^2\right )}+\frac {e \left (a B e^2+2 A c d e-B c d^2\right )}{\sqrt {a+c x^4} \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {c e \left (-a A e^3+3 a B d e^2+3 A c d^2 e-B c d^3\right )}{\sqrt {a+c x^4} \left (d+e x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {c \left (c x^2 \left (a A e^3-3 a B d e^2-3 A c d^2 e+B c d^3\right )+A c d \left (c d^2-3 a e^2\right )+a B e \left (3 c d^2-a e^2\right )\right )}{\left (a+c x^4\right )^{3/2} \left (a e^2+c d^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {c x^4+a} e^3}{8 d^2 \left (c d^2+a e^2\right )^3 \left (e x^2+d\right )}-\frac {\left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a} e^3}{2 d \left (c d^2+a e^2\right )^3 \left (e x^2+d\right )}-\frac {(B d-A e) x \sqrt {c x^4+a} e^3}{4 d \left (c d^2+a e^2\right )^2 \left (e x^2+d\right )^2}-\frac {3 \sqrt [4]{a} \sqrt [4]{c} (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}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right ) e^2}{8 d^2 \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}-\frac {\sqrt [4]{a} \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 \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 )^3 \sqrt {c x^4+a}}+\frac {3 \sqrt {c} (B d-A e) \left (3 c d^2+a e^2\right ) x \sqrt {c x^4+a} e^2}{8 d^2 \left (c d^2+a e^2\right )^3 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {\sqrt {c} \left (B c d^2-2 A c e d-a B e^2\right ) x \sqrt {c x^4+a} e^2}{2 d \left (c d^2+a e^2\right )^3 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {\left (3 c d^2+a e^2\right ) \left (B c d^2-2 A c e d-a B e^2\right ) \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 )^{7/2}}-\frac {c \left (B c d^3-3 A c e d^2-3 a B e^2 d+a A e^3\right ) \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 )^{7/2}}-\frac {3 (B d-A e) \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {c x^4+a}}\right ) e^{3/2}}{16 d^{5/2} \left (c d^2+a e^2\right )^{7/2}}-\frac {\sqrt [4]{c} (B d-A e) \left (4 c d^2-\sqrt {a} \sqrt {c} e d+3 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}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{8 \sqrt [4]{a} 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 {\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}} \operatorname {EllipticF}\left (2 \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 )^2 \sqrt {c x^4+a}}-\frac {c^{5/4} \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}} \operatorname {EllipticF}\left (2 \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 )^3 \sqrt {c x^4+a}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c d^2+a e^2\right ) \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}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \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 )^3 \sqrt {c x^4+a}}+\frac {a^{3/4} c^{3/4} \left (\frac {\sqrt {c} d}{\sqrt {a}}+e\right )^2 \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}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{4 d \left (c d^2-a e^2\right ) \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \left (5 c^2 d^4+2 a c e^2 d^2+a^2 e^4\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 {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) e}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}+\frac {c^{5/4} \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} \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}+\frac {c^{3/4} \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} \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}-\frac {c^{3/2} \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 \left (c d^2+a e^2\right )^3 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {c 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 \left (c d^2+a e^2\right )^3 \sqrt {c x^4+a}}\)

Input:

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

Output:

(c*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*d^2 + a*e^2)^3*Sqrt[a + c* 
x^4]) + (3*Sqrt[c]*e^2*(B*d - A*e)*(3*c*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(8 
*d^2*(c*d^2 + a*e^2)^3*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*e^2*(B*c*d^2 - 
2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/(2*d*(c*d^2 + a*e^2)^3*(Sqrt[a] + 
Sqrt[c]*x^2)) - (c^(3/2)*(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*d^2 + a*e^2)^3*(Sqrt[a] + Sqrt[c]*x^2)) - (e^3*( 
B*d - A*e)*x*Sqrt[a + c*x^4])/(4*d*(c*d^2 + a*e^2)^2*(d + e*x^2)^2) - (3*e 
^3*(B*d - A*e)*(3*c*d^2 + a*e^2)*x*Sqrt[a + c*x^4])/(8*d^2*(c*d^2 + a*e^2) 
^3*(d + e*x^2)) - (e^3*(B*c*d^2 - 2*A*c*d*e - a*B*e^2)*x*Sqrt[a + c*x^4])/ 
(2*d*(c*d^2 + a*e^2)^3*(d + e*x^2)) - (e^(3/2)*(3*c*d^2 + a*e^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]*Sqr 
t[a + c*x^4])])/(4*d^(3/2)*(c*d^2 + a*e^2)^(7/2)) - (c*e^(3/2)*(B*c*d^3 - 
3*A*c*d^2*e - 3*a*B*d*e^2 + a*A*e^3)*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)^(7/2)) - (3*e^(3/ 
2)*(B*d - A*e)*(5*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c*d^2 + 
a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + c*x^4])])/(16*d^(5/2)*(c*d^2 + a*e^2)^ 
(7/2)) - (3*a^(1/4)*c^(1/4)*e^2*(B*d - A*e)*(3*c*d^2 + a*e^2)*(Sqrt[a] + S 
qrt[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])/(8*d^2*(c*d^2 + a*e^2)^3*Sqrt[a + c*x^4]) ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2259

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] && IntegerQ[p + 1/ 
2] && IntegerQ[q]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.03 (sec) , antiderivative size = 2326, normalized size of antiderivative = 2.04

method	result	size
default	\(\text {Expression too large to display}\)	\(2326\)
elliptic	\(\text {Expression too large to display}\)	\(2596\)

Input:

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

Output:

B/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*d*e* 
c/(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)/(c*(a/c+x^4)) 
^(1/2)-1/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^ 
(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^( 
1/2),I)*e^2*c/(a*e^2+c*d^2)^2+1/2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x 
^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*Elliptic 
F(x*(I*c^(1/2)/a^(1/2))^(1/2),I)*c^2/a/(a*e^2+c*d^2)^2*d^2-1/2*I*a^(1/2)/( 
I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/ 
a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*c^(1/2)*e^3/(a*e^2+c*d^2)^2/d*EllipticF(x*( 
I*c^(1/2)/a^(1/2))^(1/2),I)+1/2*I*a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c 
^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)* 
c^(1/2)*e^3/(a*e^2+c*d^2)^2/d*EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+I/a 
^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1 
/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*c^(3/2)*d*e/(a*e^2+c*d^2)^2*Ellipti 
cF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-I/a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I 
*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2 
)*c^(3/2)*d*e/(a*e^2+c*d^2)^2*EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/2 
*e^4/(a*e^2+c*d^2)^2/d^2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2 
))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I*c 
^(1/2)/a^(1/2))^(1/2),I/c^(1/2)*a^(1/2)/d*e,(-I/a^(1/2)*c^(1/2))^(1/2)/...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \left (a+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} e^{3} x^{14}+3 c^{2} d \,e^{2} x^{12}+2 a c \,e^{3} x^{10}+3 c^{2} d^{2} e \,x^{10}+6 a c d \,e^{2} x^{8}+c^{2} d^{3} x^{8}+a^{2} e^{3} x^{6}+6 a c \,d^{2} e \,x^{6}+3 a^{2} d \,e^{2} x^{4}+2 a c \,d^{3} x^{4}+3 a^{2} d^{2} e \,x^{2}+a^{2} d^{3}}d x \right ) a +\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c^{2} e^{3} x^{14}+3 c^{2} d \,e^{2} x^{12}+2 a c \,e^{3} x^{10}+3 c^{2} d^{2} e \,x^{10}+6 a c d \,e^{2} x^{8}+c^{2} d^{3} x^{8}+a^{2} e^{3} x^{6}+6 a c \,d^{2} e \,x^{6}+3 a^{2} d \,e^{2} x^{4}+2 a c \,d^{3} x^{4}+3 a^{2} d^{2} e \,x^{2}+a^{2} d^{3}}d x \right ) b \] Input:

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

Output:

int(sqrt(a + c*x**4)/(a**2*d**3 + 3*a**2*d**2*e*x**2 + 3*a**2*d*e**2*x**4 
+ a**2*e**3*x**6 + 2*a*c*d**3*x**4 + 6*a*c*d**2*e*x**6 + 6*a*c*d*e**2*x**8 
 + 2*a*c*e**3*x**10 + c**2*d**3*x**8 + 3*c**2*d**2*e*x**10 + 3*c**2*d*e**2 
*x**12 + c**2*e**3*x**14),x)*a + int((sqrt(a + c*x**4)*x**2)/(a**2*d**3 + 
3*a**2*d**2*e*x**2 + 3*a**2*d*e**2*x**4 + a**2*e**3*x**6 + 2*a*c*d**3*x**4 
 + 6*a*c*d**2*e*x**6 + 6*a*c*d*e**2*x**8 + 2*a*c*e**3*x**10 + c**2*d**3*x* 
*8 + 3*c**2*d**2*e*x**10 + 3*c**2*d*e**2*x**12 + c**2*e**3*x**14),x)*b