\(\int \frac {A+B x^2+C x^4}{(d+e x^2)^3 \sqrt {a-c x^4}} \, dx\) [46]

Optimal result

Integrand size = 34, antiderivative size = 615 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {\left (c d^2 \left (C d^2-e (5 B d-9 A e)\right )+a e^2 \left (5 C d^2-e (B d+3 A e)\right )\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} \left (c d^2 \left (C d^2-e (5 B d-9 A e)\right )+a e^2 \left (5 C d^2-e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 e \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (c d^2 \left (C d^2+e (3 B d-7 A e)\right )+2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 e^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (c^2 d^4 \left (C d^2+3 e (B d-5 A e)\right )-a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )-2 a c d^2 e^2 \left (5 C d^2-e (5 B d+3 A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 e^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \] Output:

-1/4*(A*e^2-B*d*e+C*d^2)*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)^2-1 
/8*(c*d^2*(C*d^2-e*(-9*A*e+5*B*d))+a*e^2*(5*C*d^2-e*(3*A*e+B*d)))*x*(-c*x^ 
4+a)^(1/2)/d^2/(-a*e^2+c*d^2)^2/(e*x^2+d)-1/8*a^(3/4)*c^(1/4)*(c*d^2*(C*d^ 
2-e*(-9*A*e+5*B*d))+a*e^2*(5*C*d^2-e*(3*A*e+B*d)))*(1-c*x^4/a)^(1/2)*Ellip 
ticE(c^(1/4)*x/a^(1/4),I)/d^2/e/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)+1/8*a^(1 
/4)*c^(1/4)*(c*d^2*(C*d^2+e*(-7*A*e+3*B*d))+2*a^(1/2)*c^(1/2)*d*e*(C*d^2-e 
*(-A*e+B*d))-a*e^2*(5*C*d^2-e*(3*A*e+B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^ 
(1/4)*x/a^(1/4),I)/d^2/e^2/(c^(1/2)*d+a^(1/2)*e)/(-a*e^2+c*d^2)/(-c*x^4+a) 
^(1/2)-1/8*a^(1/4)*(c^2*d^4*(C*d^2+3*e*(-5*A*e+B*d))-a^2*e^4*(3*C*d^2+e*(3 
*A*e+B*d))-2*a*c*d^2*e^2*(5*C*d^2-e*(3*A*e+5*B*d)))*(1-c*x^4/a)^(1/2)*Elli 
pticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4)/d^3/e^2/(-a*e^2+c 
*d^2)^2/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.59 (sec) , antiderivative size = 521, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\frac {-\frac {d x \left (a-c x^4\right ) \left (2 d \left (c d^2-a e^2\right ) \left (C d^2+e (-B d+A e)\right )+\left (c C d^4+5 a C d^2 e^2-a e^3 (B d+3 A e)+c d^2 e (-5 B d+9 A e)\right ) \left (d+e x^2\right )\right )}{\left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (-\sqrt {a} \sqrt {c} d e \left (c C d^4+5 a C d^2 e^2-a e^3 (B d+3 A e)+c d^2 e (-5 B d+9 A e)\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c \left (C d^4+d^2 e (3 B d-7 A e)\right )+2 \sqrt {a} \sqrt {c} d e \left (C d^2+e (-B d+A e)\right )+a e^2 \left (-5 C d^2+e (B d+3 A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (-c^2 \left (C d^6+3 d^4 e (B d-5 A e)\right )+a^2 e^4 \left (3 C d^2+e (B d+3 A e)\right )+2 a c d^2 e^2 \left (5 C d^2-e (5 B d+3 A e)\right )\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (c d^2 e-a e^3\right )^2}}{8 d^3 \sqrt {a-c x^4}} \] Input:

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

Output:

(-((d*x*(a - c*x^4)*(2*d*(c*d^2 - a*e^2)*(C*d^2 + e*(-(B*d) + A*e)) + (c*C 
*d^4 + 5*a*C*d^2*e^2 - a*e^3*(B*d + 3*A*e) + c*d^2*e*(-5*B*d + 9*A*e))*(d 
+ e*x^2)))/((c*d^2 - a*e^2)^2*(d + e*x^2)^2)) - (I*Sqrt[1 - (c*x^4)/a]*(-( 
Sqrt[a]*Sqrt[c]*d*e*(c*C*d^4 + 5*a*C*d^2*e^2 - a*e^3*(B*d + 3*A*e) + c*d^2 
*e*(-5*B*d + 9*A*e))*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]) 
 + Sqrt[c]*d*(Sqrt[c]*d - Sqrt[a]*e)*(c*(C*d^4 + d^2*e*(3*B*d - 7*A*e)) + 
2*Sqrt[a]*Sqrt[c]*d*e*(C*d^2 + e*(-(B*d) + A*e)) + a*e^2*(-5*C*d^2 + e*(B* 
d + 3*A*e)))*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-(c^2 
*(C*d^6 + 3*d^4*e*(B*d - 5*A*e))) + a^2*e^4*(3*C*d^2 + e*(B*d + 3*A*e)) + 
2*a*c*d^2*e^2*(5*C*d^2 - e*(5*B*d + 3*A*e)))*EllipticPi[-((Sqrt[a]*e)/(Sqr 
t[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqr 
t[a])]*(c*d^2*e - a*e^3)^2))/(8*d^3*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 604, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2211, 2211, 2235, 25, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}

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.

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

\(\Big \downarrow \) 2211

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

\(\Big \downarrow \) 2211

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

\(\Big \downarrow \) 2235

\(\displaystyle \frac {\frac {-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}-\frac {\int -\frac {c \left (d \left (c \left (C d^4+e (3 B d-7 A e) d^2\right )-a e^2 \left (7 C d^2-e (3 B d+A e)\right )\right )-e \left (c C d^4+5 a C e^2 d^2-c e (5 B d-9 A e) d^2-a e^3 (B d+3 A e)\right ) x^2\right )}{\sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {c \left (d \left (c \left (C d^4+e (3 B d-7 A e) d^2\right )-a e^2 \left (7 C d^2-e (3 B d+A e)\right )\right )-e \left (c C d^4+5 a C e^2 d^2-c e (5 B d-9 A e) d^2-a e^3 (B d+3 A e)\right ) x^2\right )}{\sqrt {a-c x^4}}dx}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {c \int \frac {d \left (c \left (C d^4+e (3 B d-7 A e) d^2\right )-a e^2 \left (7 C d^2-e (3 B d+A e)\right )\right )-e \left (c C d^4+5 a C e^2 d^2-c e (5 B d-9 A e) d^2-a e^3 (B d+3 A e)\right ) x^2}{\sqrt {a-c x^4}}dx}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1513

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-\frac {\sqrt {a} e \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}-\frac {e \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}-\frac {e \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {e \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {e \sqrt {1-\frac {c x^4}{a}} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {\sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{e^2}-\frac {\left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1543

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{e^2}-\frac {\sqrt {1-\frac {c x^4}{a}} \left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{e^2 \sqrt {a-c x^4}}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (2 \sqrt {a} \sqrt {c} d e \left (C d^2-e (B d-A e)\right )-a e^2 \left (5 C d^2-e (3 A e+B d)\right )+c \left (d^2 e (3 B d-7 A e)+C d^4\right )\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{e^2}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-a^2 e^4 \left (e (3 A e+B d)+3 C d^2\right )-2 a c d^2 e^2 \left (5 C d^2-e (3 A e+5 B d)\right )+c^2 \left (3 d^4 e (B d-5 A e)+C d^6\right )\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d e^2 \sqrt {a-c x^4}}}{2 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (-a e^3 (3 A e+B d)+5 a C d^2 e^2-c d^2 e (5 B d-9 A e)+c C d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {x \sqrt {a-c x^4} \left (A e^2-B d e+C d^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\)

Input:

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

Output:

-1/4*((C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e 
*x^2)^2) + (-1/2*((c*C*d^4 + 5*a*C*d^2*e^2 - c*d^2*e*(5*B*d - 9*A*e) - a*e 
^3*(B*d + 3*A*e))*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e*x^2)) + ((c 
*(-((a^(3/4)*e*(c*C*d^4 + 5*a*C*d^2*e^2 - c*d^2*e*(5*B*d - 9*A*e) - a*e^3* 
(B*d + 3*A*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], 
-1])/(c^(3/4)*Sqrt[a - c*x^4])) + (a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(c*(C*d 
^4 + d^2*e*(3*B*d - 7*A*e)) + 2*Sqrt[a]*Sqrt[c]*d*e*(C*d^2 - e*(B*d - A*e) 
) - a*e^2*(5*C*d^2 - e*(B*d + 3*A*e)))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSi 
n[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4])))/e^2 - (a^(1/4)*(c 
^2*(C*d^6 + 3*d^4*e*(B*d - 5*A*e)) - a^2*e^4*(3*C*d^2 + e*(B*d + 3*A*e)) - 
 2*a*c*d^2*e^2*(5*C*d^2 - e*(5*B*d + 3*A*e)))*Sqrt[1 - (c*x^4)/a]*Elliptic 
Pi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)* 
d*e^2*Sqrt[a - c*x^4]))/(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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

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

rule 1542

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

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

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 2235

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si 
mp[-(e^2)^(-1)   Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( 
C*d^2 - B*d*e + A*e^2)/e^2   Int[1/((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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1615 vs. \(2 (553 ) = 1106\).

Time = 2.16 (sec) , antiderivative size = 1616, normalized size of antiderivative = 2.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1616\)
elliptic	\(\text {Expression too large to display}\)	\(2762\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((C*x^4+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**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)*c + 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