Integrand size = 34, antiderivative size = 889 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left ((A c+a C) d-a B e+(B c d-(A c+a C) e) x^2\right )}{6 a \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}}-\frac {e \left (B c d^3-2 A c d^2 e-5 a C d^2 e+4 a B d e^2-3 a A e^3\right ) x}{6 a d \left (c d^2-a e^2\right )^2 \left (d+e x^2\right ) \sqrt {a-c x^4}}+\frac {x \left (d \left (A c \left (5 c^2 d^4-18 a c d^2 e^2-17 a^2 e^4\right )-a \left (5 a^2 C e^4+4 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+3 c \left (B \left (c^2 d^5-6 a c d^3 e^2-5 a^2 d e^4\right )+2 \left (a C d^2 e \left (c d^2+4 a e^2\right )-A \left (c^2 d^4 e-5 a c d^2 e^3-a^2 e^5\right )\right )\right ) x^2\right )}{12 a^2 d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} \left (B \left (c^2 d^5-6 a c d^3 e^2-5 a^2 d e^4\right )+2 \left (a C d^2 e \left (c d^2+4 a e^2\right )-A \left (c^2 d^4 e-5 a c d^2 e^3-a^2 e^5\right )\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 )}{4 a^{5/4} d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {\left (5 A c^{5/2} d^4+5 a^{5/2} C d e^3+\sqrt {a} c^2 d^3 (3 B d-A e)-a c^{3/2} d^2 \left (C d^2-e (5 B d-19 A e)\right )+a^{3/2} c d e \left (5 C d^2-e (13 B d-11 A e)\right )-a^2 \sqrt {c} e^2 \left (19 C d^2-3 e (5 B d-2 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 )}{12 a^{7/4} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} e^2 \left (c d^2 \left (7 C d^2-e (9 B d-11 A e)\right )+a e^2 \left (3 C d^2-e (B d+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 )}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}} \] Output:
1/6*x*((A*c+C*a)*d-B*a*e+(B*c*d-(A*c+C*a)*e)*x^2)/a/(-a*e^2+c*d^2)/(e*x^2+ d)/(-c*x^4+a)^(3/2)-1/6*e*(-3*A*a*e^3-2*A*c*d^2*e+4*B*a*d*e^2+B*c*d^3-5*C* a*d^2*e)*x/a/d/(-a*e^2+c*d^2)^2/(e*x^2+d)/(-c*x^4+a)^(1/2)+1/12*x*(d*(A*c* (-17*a^2*e^4-18*a*c*d^2*e^2+5*c^2*d^4)-a*(5*a^2*C*e^4+4*a*c*d*e^2*(-7*B*e+ 6*C*d)+c^2*d^3*(-2*B*e+C*d)))+3*c*(B*(-5*a^2*d*e^4-6*a*c*d^3*e^2+c^2*d^5)+ 2*a*C*d^2*e*(4*a*e^2+c*d^2)-2*A*(-a^2*e^5-5*a*c*d^2*e^3+c^2*d^4*e))*x^2)/a ^2/d/(-a*e^2+c*d^2)^3/(-c*x^4+a)^(1/2)-1/4*c^(1/4)*(B*(-5*a^2*d*e^4-6*a*c* d^3*e^2+c^2*d^5)+2*a*C*d^2*e*(4*a*e^2+c*d^2)-2*A*(-a^2*e^5-5*a*c*d^2*e^3+c ^2*d^4*e))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(5/4)/d/(-a* e^2+c*d^2)^3/(-c*x^4+a)^(1/2)+1/12*(5*A*c^(5/2)*d^4+5*a^(5/2)*C*d*e^3+a^(1 /2)*c^2*d^3*(-A*e+3*B*d)-a*c^(3/2)*d^2*(C*d^2-e*(-19*A*e+5*B*d))+a^(3/2)*c *d*e*(5*C*d^2-e*(-11*A*e+13*B*d))-a^2*c^(1/2)*e^2*(19*C*d^2-3*e*(-2*A*e+5* B*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(7/4)/c^(1/4)/d/ (c^(1/2)*d-a^(1/2)*e)^2/(c^(1/2)*d+a^(1/2)*e)^3/(-c*x^4+a)^(1/2)+1/2*a^(1/ 4)*e^2*(c*d^2*(7*C*d^2-e*(-11*A*e+9*B*d))+a*e^2*(3*C*d^2-e*(A*e+B*d)))*(1- c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/c^(1/4 )/d^2/(-a*e^2+c*d^2)^3/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 13.69 (sec) , antiderivative size = 782, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d x \left (-6 a^2 e^4 \left (C d^2+e (-B d+A e)\right ) \left (a-c x^4\right )^2+2 a d \left (c d^2-a e^2\right ) \left (d+e x^2\right ) \left (a^2 C e^2+B c^2 d^2 x^2+A c \left (a e^2+c d \left (d-2 e x^2\right )\right )+a c \left (C d \left (d-2 e x^2\right )+B e \left (-2 d+e x^2\right )\right )\right )-d \left (d+e x^2\right ) \left (a-c x^4\right ) \left (5 a^3 C e^4-3 B c^3 d^4 x^2+A c \left (11 a^2 e^4+6 a c d e^2 \left (3 d-5 e x^2\right )+c^2 d^3 \left (-5 d+6 e x^2\right )\right )+a^2 c e^2 \left (18 C d \left (d-e x^2\right )+B e \left (-22 d+9 e x^2\right )\right )+a c^2 d^2 \left (C d \left (d-6 e x^2\right )+2 B e \left (-d+9 e x^2\right )\right )\right )\right )+i \left (d+e x^2\right ) \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d \left (2 a C d^2 e \left (c d^2+4 a e^2\right )+B \left (c^2 d^5-6 a c d^3 e^2-5 a^2 d e^4\right )+2 A \left (-c^2 d^4 e+5 a c d^2 e^3+a^2 e^5\right )\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+d \left (-\sqrt {c} d+\sqrt {a} e\right ) \left (5 A c^{5/2} d^4+5 a^{5/2} C d e^3+\sqrt {a} c^2 d^3 (3 B d-A e)+a^2 \sqrt {c} e^2 \left (-19 C d^2+3 e (5 B d-2 A e)\right )+a^{3/2} c d e \left (5 C d^2+e (-13 B d+11 A e)\right )-a c^{3/2} d^2 \left (C d^2+e (-5 B d+19 A e)\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+6 a^2 e^2 \left (c \left (-7 C d^4+d^2 e (9 B d-11 A e)\right )+a e^2 \left (-3 C d^2+e (B d+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 )}{12 a^2 \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} d^2 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((d + e*x^2)^2*(a - c*x^4)^(5/2)),x]
Output:
(Sqrt[-(Sqrt[c]/Sqrt[a])]*d*x*(-6*a^2*e^4*(C*d^2 + e*(-(B*d) + A*e))*(a - c*x^4)^2 + 2*a*d*(c*d^2 - a*e^2)*(d + e*x^2)*(a^2*C*e^2 + B*c^2*d^2*x^2 + A*c*(a*e^2 + c*d*(d - 2*e*x^2)) + a*c*(C*d*(d - 2*e*x^2) + B*e*(-2*d + e*x ^2))) - d*(d + e*x^2)*(a - c*x^4)*(5*a^3*C*e^4 - 3*B*c^3*d^4*x^2 + A*c*(11 *a^2*e^4 + 6*a*c*d*e^2*(3*d - 5*e*x^2) + c^2*d^3*(-5*d + 6*e*x^2)) + a^2*c *e^2*(18*C*d*(d - e*x^2) + B*e*(-22*d + 9*e*x^2)) + a*c^2*d^2*(C*d*(d - 6* e*x^2) + 2*B*e*(-d + 9*e*x^2)))) + I*(d + e*x^2)*(a - c*x^4)*Sqrt[1 - (c*x ^4)/a]*(3*Sqrt[a]*Sqrt[c]*d*(2*a*C*d^2*e*(c*d^2 + 4*a*e^2) + B*(c^2*d^5 - 6*a*c*d^3*e^2 - 5*a^2*d*e^4) + 2*A*(-(c^2*d^4*e) + 5*a*c*d^2*e^3 + a^2*e^5 ))*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + d*(-(Sqrt[c]*d) + Sqrt[a]*e)*(5*A*c^(5/2)*d^4 + 5*a^(5/2)*C*d*e^3 + Sqrt[a]*c^2*d^3*(3*B*d - A*e) + a^2*Sqrt[c]*e^2*(-19*C*d^2 + 3*e*(5*B*d - 2*A*e)) + a^(3/2)*c*d* e*(5*C*d^2 + e*(-13*B*d + 11*A*e)) - a*c^(3/2)*d^2*(C*d^2 + e*(-5*B*d + 19 *A*e)))*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 6*a^2*e^2*( c*(-7*C*d^4 + d^2*e*(9*B*d - 11*A*e)) + a*e^2*(-3*C*d^2 + e*(B*d + A*e)))* EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]* x], -1]))/(12*a^2*Sqrt[-(Sqrt[c]/Sqrt[a])]*d^2*(c*d^2 - a*e^2)^3*(d + e*x^ 2)*(a - c*x^4)^(3/2))
Time = 2.58 (sec) , antiderivative size = 1340, normalized size of antiderivative = 1.51, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a-c x^4\right )^{5/2} \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {e^2 \left (A e^2-B d e+C d^2\right )}{\sqrt {a-c x^4} \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )^2}+\frac {-c x^2 \left (2 d e (a C+A c)-B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (C d-2 B e)\right )}{\left (a-c x^4\right )^{5/2} \left (c d^2-a e^2\right )^2}+\frac {e^2 \left (a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{\sqrt {a-c x^4} \left (d+e x^2\right ) \left (c d^2-a e^2\right )^3}+\frac {c \left (e x^2 \left (a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )-a e^2 \left (3 C d^2-e (2 B d-A e)\right )-c \left (C d^4-d^2 e (2 B d-3 A e)\right )\right )}{\left (a-c x^4\right )^{3/2} \left (c d^2-a e^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (C d^2-B e d+A e^2\right ) x \sqrt {a-c x^4} e^4}{2 d \left (c d^2-a e^2\right )^3 \left (e x^2+d\right )}-\frac {a^{3/4} \sqrt [4]{c} \left (C d^2-B e d+A e^2\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 ) e^3}{2 d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (C d^2-B e d+A e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) e^2}{2 d \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]{a} \left (3 c d^2-a e^2\right ) \left (C d^2-B e d+A e^2\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 ) e^2}{2 \sqrt [4]{c} d^2 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (2 c C d^3-c e (3 B d-4 A e) d+a e^2 (2 C d-B e)\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 ) e^2}{\sqrt [4]{c} d \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} \left (2 c C d^3-c e (3 B d-4 A e) d+a e^2 (2 C d-B e)\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 ) e}{2 \sqrt [4]{a} \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{c} \left (2 (A c+a C) d e-B \left (c d^2+a e^2\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 )}{4 a^{5/4} \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {\left (\frac {(A c+a C) \left (5 c d^2-6 \sqrt {a} \sqrt {c} e d+5 a e^2\right )}{\sqrt {a}}+B \left (3 c^{3/2} d^2-10 \sqrt {a} c e d+3 a \sqrt {c} e^2\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 )}{12 a^{5/4} \sqrt [4]{c} \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a (2 C d-B e) e^2-\sqrt {a} \sqrt {c} \left (C d^2-e (B d-A e)\right ) e+c \left (C d^3-d e (2 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 )}{2 a^{3/4} \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {c x \left (a \left (3 C d^2-e (2 B d-A e)\right ) e^2-\left (2 c C d^3-c e (3 B d-4 A e) d+a e^2 (2 C d-B e)\right ) x^2 e+c \left (C d^4-d^2 e (2 B d-3 A e)\right )\right )}{2 a \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}+\frac {x \left (5 \left (A c \left (c d^2+a e^2\right )+a \left (a C e^2+c d (C d-2 B e)\right )\right )-3 c \left (2 (A c+a C) d e-B \left (c d^2+a e^2\right )\right ) x^2\right )}{12 a^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}+\frac {x \left (-c \left (2 (A c+a C) d e-B \left (c d^2+a e^2\right )\right ) x^2+A c \left (c d^2+a e^2\right )+a \left (a C e^2+c d (C d-2 B e)\right )\right )}{6 a \left (c d^2-a e^2\right )^2 \left (a-c x^4\right )^{3/2}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((d + e*x^2)^2*(a - c*x^4)^(5/2)),x]
Output:
(x*(A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + c*d*(C*d - 2*B*e)) - c*(2*(A*c + a* C)*d*e - B*(c*d^2 + a*e^2))*x^2))/(6*a*(c*d^2 - a*e^2)^2*(a - c*x^4)^(3/2) ) - (c*x*(c*(C*d^4 - d^2*e*(2*B*d - 3*A*e)) + a*e^2*(3*C*d^2 - e*(2*B*d - A*e)) - e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) + a*e^2*(2*C*d - B*e))*x^2))/ (2*a*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) + (x*(5*(A*c*(c*d^2 + a*e^2) + a*( a*C*e^2 + c*d*(C*d - 2*B*e))) - 3*c*(2*(A*c + a*C)*d*e - B*(c*d^2 + a*e^2) )*x^2))/(12*a^2*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (e^4*(C*d^2 - B*d*e + A*e^2)*x*Sqrt[a - c*x^4])/(2*d*(c*d^2 - a*e^2)^3*(d + e*x^2)) - (a^(3/4)* c^(1/4)*e^3*(C*d^2 - B*d*e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[( c^(1/4)*x)/a^(1/4)], -1])/(2*d*(c*d^2 - a*e^2)^3*Sqrt[a - c*x^4]) - (c^(1/ 4)*e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) + a*e^2*(2*C*d - B*e))*Sqrt[1 - (c *x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(1/4)*(c*d^2 - a *e^2)^3*Sqrt[a - c*x^4]) + (c^(1/4)*(2*(A*c + a*C)*d*e - B*(c*d^2 + a*e^2) )*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(4*a^(5/ 4)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) - (a^(1/4)*c^(1/4)*e^2*(C*d^2 - B*d* e + A*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) /(2*d*(Sqrt[c]*d + Sqrt[a]*e)*(c*d^2 - a*e^2)^2*Sqrt[a - c*x^4]) + ((((A*c + a*C)*(5*c*d^2 - 6*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2))/Sqrt[a] + B*(3*c^(3/2 )*d^2 - 10*Sqrt[a]*c*d*e + 3*a*Sqrt[c]*e^2))*Sqrt[1 - (c*x^4)/a]*EllipticF [ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(12*a^(5/4)*c^(1/4)*(c*d^2 - a*e^2)^...
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2455 vs. \(2 (807 ) = 1614\).
Time = 1.00 (sec) , antiderivative size = 2456, normalized size of antiderivative = 2.76
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2456\) |
| elliptic | \(\text {Expression too large to display}\) | \(4267\) |
Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
C/e^2*(1/6/a*x/c^2*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+5/12/a^2*x/(-(x^4-a/c)*c)^ (1/2)+5/12/a^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/e^2*(B*e-2*C*d)*((1/6/c/a*e/(a*e^2-c*d^2)*x^3-1/6/c/a*d/(a*e^2-c *d^2)*x)*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+2*c*(1/8*e*(3*a*e^2-c*d^2)/a^2/(a*e^ 2-c*d^2)^2*x^3-1/24*d*(11*a*e^2-5*c*d^2)/a^2/(a*e^2-c*d^2)^2*x)/(-(x^4-a/c )*c)^(1/2)-11/12*c*d/a/(a*e^2-c*d^2)^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)*Elliptic F(x*(c^(1/2)/a^(1/2))^(1/2),I)*e^2+5/12*c^2*d^3/a^2/(a*e^2-c*d^2)^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)+3/4*c^(1/2)*e^ 3/a^(1/2)/(a*e^2-c*d^2)^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)-1/4*c^(3/2)*e/a^(3/2)/(a*e^2-c*d^2)^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)*d^2-3/4*c^(1/2)*e^3/a^(1/2 )/(a*e^2-c*d^2)^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)+1/4*c^(3/2)*e/a^(3/2)/(a*e^2-c*d^2)^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...
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(5/2),x, algorithm="frica s")
Output:
Timed out
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((C*x**4+B*x**2+A)/(e*x**2+d)**2/(-c*x**4+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(5/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(5/2)*(e*x^2 + d)^2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}} {\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(5/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)/((-c*x^4 + a)^(5/2)*(e*x^2 + d)^2), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (a-c\,x^4\right )}^{5/2}\,{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(5/2)*(d + e*x^2)^2),x)
Output:
int((A + B*x^2 + C*x^4)/((a - c*x^4)^(5/2)*(d + e*x^2)^2), x)
\[ \int \frac {A+B x^2+C x^4}{\left (d+e x^2\right )^2 \left (a-c x^4\right )^{5/2}} \, dx=\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} e^{2} x^{16}-2 c^{3} d e \,x^{14}+3 a \,c^{2} e^{2} x^{12}-c^{3} d^{2} x^{12}+6 a \,c^{2} d e \,x^{10}-3 a^{2} c \,e^{2} x^{8}+3 a \,c^{2} d^{2} x^{8}-6 a^{2} c d e \,x^{6}+a^{3} e^{2} x^{4}-3 a^{2} c \,d^{2} x^{4}+2 a^{3} d e \,x^{2}+a^{3} d^{2}}d x \right ) a +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{4}}{-c^{3} e^{2} x^{16}-2 c^{3} d e \,x^{14}+3 a \,c^{2} e^{2} x^{12}-c^{3} d^{2} x^{12}+6 a \,c^{2} d e \,x^{10}-3 a^{2} c \,e^{2} x^{8}+3 a \,c^{2} d^{2} x^{8}-6 a^{2} c d e \,x^{6}+a^{3} e^{2} x^{4}-3 a^{2} c \,d^{2} x^{4}+2 a^{3} d e \,x^{2}+a^{3} d^{2}}d x \right ) c +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} e^{2} x^{16}-2 c^{3} d e \,x^{14}+3 a \,c^{2} e^{2} x^{12}-c^{3} d^{2} x^{12}+6 a \,c^{2} d e \,x^{10}-3 a^{2} c \,e^{2} x^{8}+3 a \,c^{2} d^{2} x^{8}-6 a^{2} c d e \,x^{6}+a^{3} e^{2} x^{4}-3 a^{2} c \,d^{2} x^{4}+2 a^{3} d e \,x^{2}+a^{3} d^{2}}d x \right ) b \] Input:
int((C*x^4+B*x^2+A)/(e*x^2+d)^2/(-c*x^4+a)^(5/2),x)
Output:
int(sqrt(a - c*x**4)/(a**3*d**2 + 2*a**3*d*e*x**2 + a**3*e**2*x**4 - 3*a** 2*c*d**2*x**4 - 6*a**2*c*d*e*x**6 - 3*a**2*c*e**2*x**8 + 3*a*c**2*d**2*x** 8 + 6*a*c**2*d*e*x**10 + 3*a*c**2*e**2*x**12 - c**3*d**2*x**12 - 2*c**3*d* e*x**14 - c**3*e**2*x**16),x)*a + int((sqrt(a - c*x**4)*x**4)/(a**3*d**2 + 2*a**3*d*e*x**2 + a**3*e**2*x**4 - 3*a**2*c*d**2*x**4 - 6*a**2*c*d*e*x**6 - 3*a**2*c*e**2*x**8 + 3*a*c**2*d**2*x**8 + 6*a*c**2*d*e*x**10 + 3*a*c**2 *e**2*x**12 - c**3*d**2*x**12 - 2*c**3*d*e*x**14 - c**3*e**2*x**16),x)*c + int((sqrt(a - c*x**4)*x**2)/(a**3*d**2 + 2*a**3*d*e*x**2 + a**3*e**2*x**4 - 3*a**2*c*d**2*x**4 - 6*a**2*c*d*e*x**6 - 3*a**2*c*e**2*x**8 + 3*a*c**2* d**2*x**8 + 6*a*c**2*d*e*x**10 + 3*a*c**2*e**2*x**12 - c**3*d**2*x**12 - 2 *c**3*d*e*x**14 - c**3*e**2*x**16),x)*b