Integrand size = 34, antiderivative size = 402 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (A c \left (c d^2+a e^2\right )+a \left (a C e^2+c d (C d+2 B e)\right )+c \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right ) x^2\right )}{6 a c^2 \left (a-c x^4\right )^{3/2}}+\frac {x \left (A c \left (5 c d^2-a e^2\right )-a \left (7 a C e^2+c d (C d+2 B e)\right )+3 c \left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) x^2\right )}{12 a^2 c^2 \sqrt {a-c x^4}}-\frac {\left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{5/4} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (5 A c^2 d^2+5 a^2 C e^2+3 \sqrt {a} c^{3/2} d (B d+2 A e)-3 a^{3/2} \sqrt {c} e (2 C d+B e)-a c \left (C d^2+e (2 B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c^{9/4} \sqrt {a-c x^4}} \] Output:
1/6*x*(A*c*(a*e^2+c*d^2)+a*(C*a*e^2+c*d*(2*B*e+C*d))+c*(2*(A*c+C*a)*d*e+B* (a*e^2+c*d^2))*x^2)/a/c^2/(-c*x^4+a)^(3/2)+1/12*x*(A*c*(-a*e^2+5*c*d^2)-a* (7*C*a*e^2+c*d*(2*B*e+C*d))+3*c*(2*A*c*d*e-B*a*e^2+B*c*d^2-2*C*a*d*e)*x^2) /a^2/c^2/(-c*x^4+a)^(1/2)-1/4*(2*A*c*d*e-B*a*e^2+B*c*d^2-2*C*a*d*e)*(1-c*x ^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(5/4)/c^(7/4)/(-c*x^4+a)^(1/2 )+1/12*(5*A*c^2*d^2+5*a^2*C*e^2+3*a^(1/2)*c^(3/2)*d*(2*A*e+B*d)-3*a^(3/2)* c^(1/2)*e*(B*e+2*C*d)-a*c*(C*d^2+e*(A*e+2*B*d)))*(1-c*x^4/a)^(1/2)*Ellipti cF(c^(1/4)*x/a^(1/4),I)/a^(7/4)/c^(9/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.50 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (-5 a^3 C e^2-5 A c^3 d^2 x^4+a c^2 \left (d (C d+2 B e) x^4+A \left (7 d^2+e^2 x^4\right )\right )+a^2 c \left (e \left (2 B d+A e+4 B e x^2\right )+C \left (d^2+8 d e x^2+7 e^2 x^4\right )\right )\right )+\left (A c \left (5 c d^2-a e^2\right )+a \left (5 a C e^2-c d (C d+2 B e)\right )\right ) x \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )-4 c \left (-B c d^2-2 A c d e+2 a C d e+a B e^2\right ) x^3 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{12 a^2 c^2 \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x]
Output:
(x*(-5*a^3*C*e^2 - 5*A*c^3*d^2*x^4 + a*c^2*(d*(C*d + 2*B*e)*x^4 + A*(7*d^2 + e^2*x^4)) + a^2*c*(e*(2*B*d + A*e + 4*B*e*x^2) + C*(d^2 + 8*d*e*x^2 + 7 *e^2*x^4))) + (A*c*(5*c*d^2 - a*e^2) + a*(5*a*C*e^2 - c*d*(C*d + 2*B*e)))* x*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4) /a] - 4*c*(-(B*c*d^2) - 2*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x^3*(a - c*x^4)*S qrt[1 - (c*x^4)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, (c*x^4)/a])/(12*a^2*c^ 2*(a - c*x^4)^(3/2))
Time = 1.40 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.69, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {-2 a C e^2-c e (A e+2 B d)-c e x^2 (B e+2 C d)-c C d^2}{c^2 \left (a-c x^4\right )^{3/2}}+\frac {c x^2 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )}{c^2 \left (a-c x^4\right )^{5/2}}+\frac {C e^2}{c^2 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (\sqrt {a} \sqrt {c} e (B e+2 C d)+2 a C e^2+c e (A e+2 B d)+c C d^2\right )}{2 a^{3/4} c^{9/4} \sqrt {a-c x^4}}+\frac {\sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (3 \sqrt {c} \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+\frac {5 \left (A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )\right )}{\sqrt {a}}\right )}{12 a^{5/4} c^{9/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )}{4 a^{5/4} c^{7/4} \sqrt {a-c x^4}}+\frac {x \left (3 c x^2 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+5 \left (A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )\right )\right )}{12 a^2 c^2 \sqrt {a-c x^4}}-\frac {x \left (2 a C e^2+c e (A e+2 B d)+c e x^2 (B e+2 C d)+c C d^2\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {x \left (c x^2 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )\right )}{6 a c^2 \left (a-c x^4\right )^{3/2}}+\frac {e \sqrt {1-\frac {c x^4}{a}} (B e+2 C d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} C e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{9/4} \sqrt {a-c x^4}}\) |
Input:
Int[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(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^2*(a - c*x^4)^(3/2)) - (x*(c*C*d^ 2 + 2*a*C*e^2 + c*e*(2*B*d + A*e) + c*e*(2*C*d + B*e)*x^2))/(2*a*c^2*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^2*Sqrt[a - c*x^4]) + (e*(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^(7/4)*Sqrt[a - c*x^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^(7/4)*Sqrt[a - c*x^4]) + (a^(1/4)*C*e^2*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(9/4)*Sqrt[a - c*x^4]) - ((c*C*d^2 + 2*a*C*e^2 + c*e*(2*B*d + A*e) + Sqrt[a]*Sqrt[c]*e*(2 *C*d + B*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1 ])/(2*a^(3/4)*c^(9/4)*Sqrt[a - c*x^4]) + ((3*Sqrt[c]*(2*(A*c + a*C)*d*e + B*(c*d^2 + a*e^2)) + (5*(A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + c*d*(C*d + 2*B *e))))/Sqrt[a])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(12*a^(5/4)*c^(9/4)*Sqrt[a - c*x^4])
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]
Time = 1.99 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.11
| method | result | size |
| elliptic | \(\frac {\left (\frac {\left (2 A c d e +B a \,e^{2}+B c \,d^{2}+2 C a d e \right ) x^{3}}{6 c^{3} a}+\frac {\left (A a c \,e^{2}+A \,c^{2} d^{2}+2 B a c d e +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{6 a \,c^{4}}\right ) \sqrt {-c \,x^{4}+a}}{\left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {2 c \left (\frac {\left (2 A c d e -B a \,e^{2}+B c \,d^{2}-2 C a d e \right ) x^{3}}{8 c^{2} a^{2}}-\frac {\left (A a c \,e^{2}-5 A \,c^{2} d^{2}+2 B a c d e +7 a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{24 a^{2} c^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {C \,e^{2}}{c^{2}}-\frac {A a c \,e^{2}-5 A \,c^{2} d^{2}+2 B a c d e +7 a^{2} C \,e^{2}+C a c \,d^{2}}{12 c^{2} a^{2}}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (2 A c d e -B a \,e^{2}+B c \,d^{2}-2 C a d e \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 c^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(447\) |
| default | \(A \,d^{2} \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+d \left (2 A e +B d \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+e \left (B e +2 C d \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x^{3}}{4 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 c^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A \,e^{2}+2 B d e +C \,d^{2}\right ) \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x}{12 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 c a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C \,e^{2} \left (\frac {a x \sqrt {-c \,x^{4}+a}}{6 c^{4} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {7 x}{12 c^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(698\) |
Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/6/c^3*(2*A*c*d*e+B*a*e^2+B*c*d^2+2*C*a*d*e)/a*x^3+1/6/a/c^4*(A*a*c*e^2+ A*c^2*d^2+2*B*a*c*d*e+C*a^2*e^2+C*a*c*d^2)*x)*(-c*x^4+a)^(1/2)/(x^4-a/c)^2 +2*c*(1/8/c^2*(2*A*c*d*e-B*a*e^2+B*c*d^2-2*C*a*d*e)/a^2*x^3-1/24/a^2/c^3*( A*a*c*e^2-5*A*c^2*d^2+2*B*a*c*d*e+7*C*a^2*e^2+C*a*c*d^2)*x)/(-(x^4-a/c)*c) ^(1/2)+(C*e^2/c^2-1/12/c^2/a^2*(A*a*c*e^2-5*A*c^2*d^2+2*B*a*c*d*e+7*C*a^2* e^2+C*a*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)+1/4/c^(3/2)*(2*A*c*d*e-B*a*e^2+B*c*d^2-2*C*a*d*e)/a^(3/2)/(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)-EllipticE(x*(c^ (1/2)/a^(1/2))^(1/2),I))
Time = 0.11 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.62 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (B c^{4} d^{2} - B a c^{3} e^{2} - 2 \, {\left (C a c^{3} - A c^{4}\right )} d e\right )} x^{8} + B a^{2} c^{2} d^{2} - B a^{3} c e^{2} - 2 \, {\left (B a c^{3} d^{2} - B a^{2} c^{2} e^{2} - 2 \, {\left (C a^{2} c^{2} - A a c^{3}\right )} d e\right )} x^{4} - 2 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left ({\left ({\left (C a c^{3} - {\left (5 \, A + 3 \, B\right )} c^{4}\right )} d^{2} + 2 \, {\left ({\left (B + 3 \, C\right )} a c^{3} - 3 \, A c^{4}\right )} d e - {\left (5 \, C a^{2} c^{2} - {\left (A + 3 \, B\right )} a c^{3}\right )} e^{2}\right )} x^{8} - 2 \, {\left ({\left (C a^{2} c^{2} - {\left (5 \, A + 3 \, B\right )} a c^{3}\right )} d^{2} + 2 \, {\left ({\left (B + 3 \, C\right )} a^{2} c^{2} - 3 \, A a c^{3}\right )} d e - {\left (5 \, C a^{3} c - {\left (A + 3 \, B\right )} a^{2} c^{2}\right )} e^{2}\right )} x^{4} + {\left (C a^{3} c - {\left (5 \, A + 3 \, B\right )} a^{2} c^{2}\right )} d^{2} + 2 \, {\left ({\left (B + 3 \, C\right )} a^{3} c - 3 \, A a^{2} c^{2}\right )} d e - {\left (5 \, C a^{4} - {\left (A + 3 \, B\right )} a^{3} c\right )} e^{2}\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, {\left (B c^{4} d^{2} - B a c^{3} e^{2} - 2 \, {\left (C a c^{3} - A c^{4}\right )} d e\right )} x^{7} - {\left (2 \, B a c^{3} d e + {\left (C a c^{3} - 5 \, A c^{4}\right )} d^{2} + {\left (7 \, C a^{2} c^{2} + A a c^{3}\right )} e^{2}\right )} x^{5} - {\left (5 \, B a c^{3} d^{2} - B a^{2} c^{2} e^{2} - 2 \, {\left (C a^{2} c^{2} - 5 \, A a c^{3}\right )} d e\right )} x^{3} - {\left (2 \, B a^{2} c^{2} d e + {\left (C a^{2} c^{2} + 7 \, A a c^{3}\right )} d^{2} - {\left (5 \, C a^{3} c - A a^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {-c x^{4} + a}}{12 \, {\left (a^{2} c^{5} x^{8} - 2 \, a^{3} c^{4} x^{4} + a^{4} c^{3}\right )}} \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="frica s")
Output:
-1/12*(3*((B*c^4*d^2 - B*a*c^3*e^2 - 2*(C*a*c^3 - A*c^4)*d*e)*x^8 + B*a^2* c^2*d^2 - B*a^3*c*e^2 - 2*(B*a*c^3*d^2 - B*a^2*c^2*e^2 - 2*(C*a^2*c^2 - A* a*c^3)*d*e)*x^4 - 2*(C*a^3*c - A*a^2*c^2)*d*e)*sqrt(a)*(c/a)^(3/4)*ellipti c_e(arcsin(x*(c/a)^(1/4)), -1) + (((C*a*c^3 - (5*A + 3*B)*c^4)*d^2 + 2*((B + 3*C)*a*c^3 - 3*A*c^4)*d*e - (5*C*a^2*c^2 - (A + 3*B)*a*c^3)*e^2)*x^8 - 2*((C*a^2*c^2 - (5*A + 3*B)*a*c^3)*d^2 + 2*((B + 3*C)*a^2*c^2 - 3*A*a*c^3) *d*e - (5*C*a^3*c - (A + 3*B)*a^2*c^2)*e^2)*x^4 + (C*a^3*c - (5*A + 3*B)*a ^2*c^2)*d^2 + 2*((B + 3*C)*a^3*c - 3*A*a^2*c^2)*d*e - (5*C*a^4 - (A + 3*B) *a^3*c)*e^2)*sqrt(a)*(c/a)^(3/4)*elliptic_f(arcsin(x*(c/a)^(1/4)), -1) + ( 3*(B*c^4*d^2 - B*a*c^3*e^2 - 2*(C*a*c^3 - A*c^4)*d*e)*x^7 - (2*B*a*c^3*d*e + (C*a*c^3 - 5*A*c^4)*d^2 + (7*C*a^2*c^2 + A*a*c^3)*e^2)*x^5 - (5*B*a*c^3 *d^2 - B*a^2*c^2*e^2 - 2*(C*a^2*c^2 - 5*A*a*c^3)*d*e)*x^3 - (2*B*a^2*c^2*d *e + (C*a^2*c^2 + 7*A*a*c^3)*d^2 - (5*C*a^3*c - A*a^2*c^2)*e^2)*x)*sqrt(-c *x^4 + a))/(a^2*c^5*x^8 - 2*a^3*c^4*x^4 + a^4*c^3)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{2} \left (A + B x^{2} + C x^{4}\right )}{\left (a - c x^{4}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x**2+d)**2*(C*x**4+B*x**2+A)/(-c*x**4+a)**(5/2),x)
Output:
Integral((d + e*x**2)**2*(A + B*x**2 + C*x**4)/(a - c*x**4)**(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^2/(-c*x^4 + a)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^2/(-c*x^4 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{5/2}} \,d x \] Input:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x)
Output:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x)
Output:
( - 12*sqrt(a - c*x**4)*a*e**2*x + 6*sqrt(a - c*x**4)*b*d*e*x + 5*sqrt(a - c*x**4)*b*e**2*x**3 + 3*sqrt(a - c*x**4)*c*d**2*x + 10*sqrt(a - c*x**4)*c *d*e*x**3 + 15*sqrt(a - c*x**4)*c*e**2*x**5 + 12*int(sqrt(a - c*x**4)/(a** 3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**4*e**2 - 6*int(sqrt( a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**3*b* d*e + 12*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3 *x**12),x)*a**3*c*d**2 - 24*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3 *a*c**2*x**8 - c**3*x**12),x)*a**3*c*e**2*x**4 + 12*int(sqrt(a - c*x**4)/( a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2*b*c*d*e*x**4 - 24*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12 ),x)*a**2*c**2*d**2*x**4 + 12*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2*c**2*e**2*x**8 - 6*int(sqrt(a - c*x** 4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a*b*c**2*d*e*x** 8 + 12*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x **12),x)*a*c**3*d**2*x**8 - 15*int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2* c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**3*b*e**2 + 15*int((sqrt(a - c*x **4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2*b*c *d**2 + 30*int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x* *8 - c**3*x**12),x)*a**2*b*c*e**2*x**4 - 30*int((sqrt(a - c*x**4)*x**2)/(a **3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a*b*c**2*d**2*x**4...