\(\int \frac {A+B x^2}{(d+e x^2)^3 \sqrt {a+c x^4}} \, dx\) [73]

Optimal result

Integrand size = 28, antiderivative size = 875 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\frac {\sqrt {c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+c x^4}}{4 d \left (c d^2+a e^2\right ) \left (d+e x^2\right )^2}-\frac {e \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) x \sqrt {a+c x^4}}{8 d^2 \left (c d^2+a e^2\right )^2 \left (d+e x^2\right )}+\frac {\left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\right ) \arctan \left (\frac {\sqrt {c d^2+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{16 d^{5/2} \sqrt {e} \left (c d^2+a e^2\right )^{5/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 B c d^3-9 A c d^2 e-a B d e^2-3 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 )}{8 d^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (4 A c d^2+\sqrt {a} \sqrt {c} d (B d-A e)+a e (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 )}{8 \sqrt [4]{a} d^2 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )-B \left (3 c^2 d^5-10 a c d^3 e^2-a^2 d e^4\right )\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 {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 )}{32 \sqrt [4]{a} \sqrt [4]{c} d^3 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \] Output:

1/8*c^(1/2)*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+5*B*c*d^3)*x*(c*x^4+a)^(1/2) 
/d^2/(a*e^2+c*d^2)^2/(a^(1/2)+c^(1/2)*x^2)-1/4*e*(-A*e+B*d)*x*(c*x^4+a)^(1 
/2)/d/(a*e^2+c*d^2)/(e*x^2+d)^2-1/8*e*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+5* 
B*c*d^3)*x*(c*x^4+a)^(1/2)/d^2/(a*e^2+c*d^2)^2/(e*x^2+d)+1/16*(3*A*e*(a^2* 
e^4+2*a*c*d^2*e^2+5*c^2*d^4)-B*(-a^2*d*e^4-10*a*c*d^3*e^2+3*c^2*d^5))*arct 
an((a*e^2+c*d^2)^(1/2)*x/d^(1/2)/e^(1/2)/(c*x^4+a)^(1/2))/d^(5/2)/e^(1/2)/ 
(a*e^2+c*d^2)^(5/2)-1/8*a^(1/4)*c^(1/4)*(-3*A*a*e^3-9*A*c*d^2*e-B*a*d*e^2+ 
5*B*c*d^3)*(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*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/d^2/(a*e^2+c*d^2) 
^2/(c*x^4+a)^(1/2)+1/8*c^(1/4)*(4*A*c*d^2+a^(1/2)*c^(1/2)*d*(-A*e+B*d)+a*e 
*(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)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/d^2/( 
c^(1/2)*d-a^(1/2)*e)/(a*e^2+c*d^2)/(c*x^4+a)^(1/2)-1/32*(c^(1/2)*d+a^(1/2) 
*e)*(3*A*e*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)-B*(-a^2*d*e^4-10*a*c*d^3*e^2+ 
3*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 
)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(1/2)*d-a^(1/2)*e)^2 
/a^(1/2)/c^(1/2)/d/e,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d^3/e/(c^(1/2)*d-a^(1/2) 
*e)/(a*e^2+c*d^2)^2/(c*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 12.27 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.52 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+c x^4}} \, dx=\frac {-\frac {d e^2 x \left (a+c x^4\right ) \left (2 d (B d-A e) \left (c d^2+a e^2\right )+\left (5 B c d^3-9 A c d^2 e-a B d e^2-3 a A e^3\right ) \left (d+e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1+\frac {c x^4}{a}} \left (-i \sqrt {a} \sqrt {c} d e \left (-5 B c d^3+9 A c d^2 e+a B d e^2+3 a A e^3\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (\sqrt {c} d-i \sqrt {a} e\right ) \left (A e \left (-7 c d^2+2 i \sqrt {a} \sqrt {c} d e-3 a e^2\right )+B d \left (3 c d^2-2 i \sqrt {a} \sqrt {c} d e-a e^2\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (3 A e \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right )+B \left (-3 c^2 d^5+10 a c d^3 e^2+a^2 d e^4\right )\right ) \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}}{8 d^3 e \left (c d^2+a e^2\right )^2 \sqrt {a+c x^4}} \] Input:

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

Output:

(-((d*e^2*x*(a + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + a*e^2) + (5*B*c*d^3 - 9* 
A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*(d + e*x^2)))/(d + e*x^2)^2) - (I*Sqrt[ 
1 + (c*x^4)/a]*((-I)*Sqrt[a]*Sqrt[c]*d*e*(-5*B*c*d^3 + 9*A*c*d^2*e + a*B*d 
*e^2 + 3*a*A*e^3)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + 
Sqrt[c]*d*(Sqrt[c]*d - I*Sqrt[a]*e)*(A*e*(-7*c*d^2 + (2*I)*Sqrt[a]*Sqrt[c] 
*d*e - 3*a*e^2) + B*d*(3*c*d^2 - (2*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2))*Ellip 
ticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (3*A*e*(5*c^2*d^4 + 2*a 
*c*d^2*e^2 + a^2*e^4) + B*(-3*c^2*d^5 + 10*a*c*d^3*e^2 + a^2*d*e^4))*Ellip 
ticPi[((-I)*Sqrt[a]*e)/(Sqrt[c]*d), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x] 
, -1]))/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(8*d^3*e*(c*d^2 + a*e^2)^2*Sqrt[a + c*x 
^4])
 

Rubi [A] (verified)

Time = 2.23 (sec) , antiderivative size = 788, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {2211, 25, 2211, 25, 2233, 27, 1510, 2227, 27, 761, 2221}

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.

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

\(\Big \downarrow \) 2211

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2211

\(\displaystyle \frac {-\frac {\int -\frac {c e \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) x^4+4 c d \left (2 B c d^3-4 A c e d^2-a B e^2 d-a A e^3\right ) x^2+a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c e^2 d^2+3 a^2 e^4\right )}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {c e \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right ) x^4+4 c d \left (2 B c d^3-4 A c e d^2-a B e^2 d-a A e^3\right ) x^2+a B d e \left (7 c d^2+a e^2\right )+A \left (8 c^2 d^4+5 a c e^2 d^2+3 a^2 e^4\right )}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 2233

\(\displaystyle \frac {\frac {\frac {\int \frac {c e \left (8 A c^2 d^4+\sqrt {a} c^{3/2} (5 B d-9 A e) d^3+a c e (7 B d+5 A e) d^2-a^{3/2} \sqrt {c} e^2 (B d+3 A e) d-\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right )-4 c d \left (2 B c d^3-4 A c e d^2-a B e^2 d-a A e^3\right )\right ) x^2+a^2 e^3 (B d+3 A e)\right )}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{c e}-\sqrt {a} \sqrt {c} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {8 A c^2 d^4+\sqrt {a} c^{3/2} (5 B d-9 A e) d^3+a c e (7 B d+5 A e) d^2-a^{3/2} \sqrt {c} e^2 (B d+3 A e) d-\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right )-4 c d \left (2 B c d^3-4 A c e d^2-a B e^2 d-a A e^3\right )\right ) x^2+a^2 e^3 (B d+3 A e)}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx-\sqrt {c} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {\int \frac {8 A c^2 d^4+\sqrt {a} c^{3/2} (5 B d-9 A e) d^3+a c e (7 B d+5 A e) d^2-a^{3/2} \sqrt {c} e^2 (B d+3 A e) d-\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (5 B c d^3-9 A c e d^2-a B e^2 d-3 a A e^3\right )-4 c d \left (2 B c d^3-4 A c e d^2-a B e^2 d-a A e^3\right )\right ) x^2+a^2 e^3 (B d+3 A e)}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx-\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 2227

\(\displaystyle \frac {\frac {-\frac {\sqrt {a} \left (3 A e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right )-B \left (-a^2 d e^4-10 a c d^3 e^2+3 c^2 d^5\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2+c d^2\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {\left (3 A e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right )-B \left (-a^2 d e^4-10 a c d^3 e^2+3 c^2 d^5\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2+c d^2\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A c d^2\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}-\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {-\frac {\left (3 A e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right )-B \left (-a^2 d e^4-10 a c d^3 e^2+3 c^2 d^5\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A c d^2\right )}{\sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}-\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\frac {-\frac {\left (3 A e \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right )-B \left (-a^2 d e^4-10 a c d^3 e^2+3 c^2 d^5\right )\right ) \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\sqrt {a} e+\sqrt {c} d\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {c} d}{\sqrt {a}}-e\right )^2}{4 \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {x \sqrt {a e^2+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+c x^4}}\right )}{2 \sqrt {d} \sqrt {e} \sqrt {a e^2+c d^2}}\right )}{\sqrt {c} d-\sqrt {a} e}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2+c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A c d^2\right )}{\sqrt [4]{a} \sqrt {a+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}-\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} \left (-3 a A e^3-a B d e^2-9 A c d^2 e+5 B c d^3\right )}{2 d \left (d+e x^2\right ) \left (a e^2+c d^2\right )}}{4 d \left (a e^2+c d^2\right )}-\frac {e x \sqrt {a+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2+c d^2\right )}\)

Input:

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

Output:

-1/4*(e*(B*d - A*e)*x*Sqrt[a + c*x^4])/(d*(c*d^2 + a*e^2)*(d + e*x^2)^2) + 
 (-1/2*(e*(5*B*c*d^3 - 9*A*c*d^2*e - a*B*d*e^2 - 3*a*A*e^3)*x*Sqrt[a + c*x 
^4])/(d*(c*d^2 + a*e^2)*(d + e*x^2)) + (-(Sqrt[c]*(5*B*c*d^3 - 9*A*c*d^2*e 
 - a*B*d*e^2 - 3*a*A*e^3)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) 
+ (a^(1/4)*(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^(1/4)*Sqrt[a + c*x^ 
4]))) + (c^(1/4)*(c*d^2 + a*e^2)*(4*A*c*d^2 + Sqrt[a]*Sqrt[c]*d*(B*d - A*e 
) + a*e*(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])/(a^(1/4)*( 
Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + c*x^4]) - ((3*A*e*(5*c^2*d^4 + 2*a*c*d^2*e 
^2 + a^2*e^4) - B*(3*c^2*d^5 - 10*a*c*d^3*e^2 - a^2*d*e^4))*(-1/2*((Sqrt[c 
]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + 
c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 + a*e^2]) + ((Sqrt[c]*d + Sqrt[a]*e) 
*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip 
ticPi[-1/4*(Sqrt[a]*((Sqrt[c]*d)/Sqrt[a] - e)^2)/(Sqrt[c]*d*e), 2*ArcTan[( 
c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + c*x^4])))/(Sqrt 
[c]*d - Sqrt[a]*e))/(2*d*(c*d^2 + a*e^2)))/(4*d*(c*d^2 + a*e^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761

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 1510

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]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 2211

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol 
] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] 
}, Simp[(-(C*d^2 - B*d*e + A*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] + Simp[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*d*(C*d - B*e) + A*(a*e^2*( 
2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*( 
C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1]
 

rule 2221

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)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

rule 2227

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

rule 2233

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]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim 
p[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] && PosQ[c/a]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.02 (sec) , antiderivative size = 1591, normalized size of antiderivative = 1.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1591\)
elliptic	\(\text {Expression too large to display}\)	\(1982\)

Input:

int((B*x^2+A)/(e*x^2+d)^3/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

B/e*(1/2*e^2/(a*e^2+c*d^2)/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)-1/2*c/(a*e^2+c*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)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I 
)-1/2*I*e*c^(1/2)/(a*e^2+c*d^2)/d*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)* 
EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+1/2*I*e*c^(1/2)/(a*e^2+c*d^2)/d*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)*EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/ 
2),I)+1/2/(a*e^2+c*d^2)/d^2*e^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)/(I*c^(1/2)/a^(1/2))^(1/2))*a+3/2/(a*e^2+c*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)/(I*c^(1/2)/a^(1/2))^(1/2))*c)+(A*e-B*d)/e*(1/4*e^2/ 
(a*e^2+c*d^2)/d*x*(c*x^4+a)^(1/2)/(e*x^2+d)^2+3/8*e^2*(a*e^2+3*c*d^2)/(a*e 
^2+c*d^2)^2/d^2*x*(c*x^4+a)^(1/2)/(e*x^2+d)-1/8*c/d/(a*e^2+c*d^2)^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)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)*a*e^2-7 
/8*c^2*d/(a*e^2+c*d^2)^2/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(...
 

Fricas [F(-1)]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((B*x**2+A)/(e*x**2+d)**3/(c*x**4+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x^2)/((a + c*x^4)^(1/2)*(d + e*x^2)^3),x)
 

Output:

int((A + B*x^2)/((a + c*x^4)^(1/2)*(d + e*x^2)^3), x)
 

Reduce [F]

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

int((B*x^2+A)/(e*x^2+d)^3/(c*x^4+a)^(1/2),x)
 

Output:

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