Integrand size = 34, antiderivative size = 331 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A c \left (c d^2+a e^2\right )+a \left (a C e^2+c d (C d+2 B e)\right )+c \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right ) x^2\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {C e^2 x \sqrt {a-c x^4}}{3 c^2}-\frac {\left (B c d^2+2 A c d e+6 a C d e+3 a B e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (3 A \left (c d^2-a e^2\right )+\frac {3 \sqrt {a} \left (B c d^2+2 A c d e+6 a C d e+3 a B e^2\right )}{\sqrt {c}}-\frac {a \left (5 a C e^2+3 c d (C d+2 B e)\right )}{c}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{6 a^{3/4} c^{5/4} \sqrt {a-c x^4}} \] Output:
1/2*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)^(1/2)+1/3*C*e^2*x*(-c*x^4+a)^(1/2)/c^ 2-1/2*(2*A*c*d*e+3*B*a*e^2+B*c*d^2+6*C*a*d*e)*(1-c*x^4/a)^(1/2)*EllipticE( c^(1/4)*x/a^(1/4),I)/a^(1/4)/c^(7/4)/(-c*x^4+a)^(1/2)+1/6*(3*A*(-a*e^2+c*d ^2)+3*a^(1/2)*(2*A*c*d*e+3*B*a*e^2+B*c*d^2+6*C*a*d*e)/c^(1/2)-a*(5*C*a*e^2 +3*c*d*(2*B*e+C*d))/c)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^ (3/4)/c^(5/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.35 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 A c \left (c d^2+a e^2\right ) x+a x \left (5 a C e^2+6 B c e \left (d-e x^2\right )+c C \left (3 d^2-12 d e x^2-2 e^2 x^4\right )\right )-\left (3 A c \left (-c d^2+a e^2\right )+a \left (5 a C e^2+3 c d (C d+2 B e)\right )\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 c \left (B c d^2+2 A c d e+6 a C d e+3 a B e^2\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{6 a c^2 \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x]
Output:
(3*A*c*(c*d^2 + a*e^2)*x + a*x*(5*a*C*e^2 + 6*B*c*e*(d - e*x^2) + c*C*(3*d ^2 - 12*d*e*x^2 - 2*e^2*x^4)) - (3*A*c*(-(c*d^2) + a*e^2) + a*(5*a*C*e^2 + 3*c*d*(C*d + 2*B*e)))*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5 /4, (c*x^4)/a] + 2*c*(B*c*d^2 + 2*A*c*d*e + 6*a*C*d*e + 3*a*B*e^2)*x^3*Sqr t[1 - (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (c*x^4)/a])/(6*a*c^2*Sqr t[a - c*x^4])
Time = 0.92 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.66, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (-\frac {a C e^2+c e (A e+2 B d)+c C d^2}{c^2 \sqrt {a-c x^4}}+\frac {c x^2 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )}{c^2 \left (a-c x^4\right )^{3/2}}-\frac {e x^2 (B e+2 C d)}{c \sqrt {a-c x^4}}-\frac {C e^2 x^4}{c \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (\sqrt {a} B \sqrt {c}+a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{9/4} \sqrt {a-c x^4}}+\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} (B e+2 C d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{7/4} \sqrt {a-c x^4}}-\frac {a^{3/4} e \sqrt {1-\frac {c x^4}{a}} (B e+2 C d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} \sqrt {a-c x^4}}-\frac {a^{5/4} C e^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{9/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (a C e^2+c e (A e+2 B d)+c C d^2\right )}{c^{9/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {x \left (c x^2 \left (2 d e (a C+A c)+B \left (a e^2+c d^2\right )\right )+A c \left (a e^2+c d^2\right )+a \left (a C e^2+c d (2 B e+C d)\right )\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {C e^2 x \sqrt {a-c x^4}}{3 c^2}\) |
Input:
Int[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),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))/(2*a*c^2*Sqrt[a - c*x^4]) + (C*e^2*x*Sqr t[a - c*x^4])/(3*c^2) - (a^(3/4)*e*(2*C*d + B*e)*Sqrt[1 - (c*x^4)/a]*Ellip ticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(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])/(2*a^(1/4)*c^(7/4)*Sqrt[a - c*x^4]) - (a^(5/4)*C*e ^2*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(9 /4)*Sqrt[a - c*x^4]) + ((Sqrt[a]*B*Sqrt[c] + A*c + a*C)*(Sqrt[c]*d + Sqrt[ a]*e)^2*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]) + (a^(3/4)*e*(2*C*d + B*e)*Sqrt[1 - (c*x ^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(7/4)*Sqrt[a - c*x^4 ]) - (a^(1/4)*(c*C*d^2 + a*C*e^2 + c*e*(2*B*d + A*e))*Sqrt[1 - (c*x^4)/a]* EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(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 = 5.73 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.24
| method | result | size |
| elliptic | \(\frac {2 c \left (\frac {\left (2 A c d e +B a \,e^{2}+B c \,d^{2}+2 C a d e \right ) x^{3}}{4 c^{2} a}+\frac {\left (A a c \,e^{2}+A \,c^{2} d^{2}+2 B a c d e +a^{2} C \,e^{2}+C a c \,d^{2}\right ) x}{4 a \,c^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {C \,e^{2} x \sqrt {-c \,x^{4}+a}}{3 c^{2}}+\frac {\left (-\frac {A c \,e^{2}+2 B c d e +C a \,e^{2}+C c \,d^{2}}{c^{2}}+\frac {A a c \,e^{2}+A \,c^{2} d^{2}+2 B a c d e +a^{2} C \,e^{2}+C a c \,d^{2}}{2 c^{2} a}-\frac {C \,e^{2} a}{3 c^{2}}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (-\frac {e \left (B e +2 C d \right )}{c}-\frac {2 A c d e +B a \,e^{2}+B c \,d^{2}+2 C a d e}{2 c a}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(409\) |
| default | \(A \,d^{2} \left (\frac {x}{2 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 )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+d \left (2 A e +B d \right ) \left (\frac {x^{3}}{2 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 )}{2 \sqrt {a}\, \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}}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {3 \sqrt {a}\, \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 )}{2 c^{\frac {3}{2}} \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}{2 c \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 )}{2 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C \,e^{2} \left (\frac {a x}{2 c^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {x \sqrt {-c \,x^{4}+a}}{3 c^{2}}-\frac {5 a \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 )}{6 c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(556\) |
| risch | \(\frac {C \,e^{2} x \sqrt {-c \,x^{4}+a}}{3 c^{2}}-\frac {\frac {6 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}}{4 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}{4 a c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {3 \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 ) \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 )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, \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 )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 A c \,e^{2} \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 {4 C a \,e^{2} \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 {3 C c \,d^{2} \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 {3 B \sqrt {c}\, e^{2} \sqrt {a}\, \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 )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {6 B c d e \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 {6 C \sqrt {c}\, d e \sqrt {a}\, \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 )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c^{2}}\) | \(800\) |
Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4/c^2*(2*A*c*d*e+B*a*e^2+B*c*d^2+2*C*a*d*e)/a*x^3+1/4/a/c^3*(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)/(-(x^4-a/c)*c)^(1/2)+1/3 *C*e^2*x*(-c*x^4+a)^(1/2)/c^2+(-(A*c*e^2+2*B*c*d*e+C*a*e^2+C*c*d^2)/c^2+1/ 2/c^2/a*(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)-1/3*C/c^2*e^ 2*a)/(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 /c*e*(B*e+2*C*d)-1/2/c*(2*A*c*d*e+B*a*e^2+B*c*d^2+2*C*a*d*e)/a)*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)/c^(1/2)*(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-E llipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.10 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.32 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 \, {\left ({\left (B a c^{2} d^{2} + 3 \, B a^{2} c e^{2} + 2 \, {\left (3 \, C a^{2} c + A a c^{2}\right )} d e\right )} x^{5} - {\left (B a^{2} c d^{2} + 3 \, B a^{3} e^{2} + 2 \, {\left (3 \, C a^{3} + A a^{2} c\right )} d e\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (3 \, {\left ({\left (B + C\right )} a c^{2} - A c^{3}\right )} d^{2} + 6 \, {\left (3 \, C a^{2} c + {\left (A + B\right )} a c^{2}\right )} d e + {\left ({\left (9 \, B + 5 \, C\right )} a^{2} c + 3 \, A a c^{2}\right )} e^{2}\right )} x^{5} - {\left (3 \, {\left ({\left (B + C\right )} a^{2} c - A a c^{2}\right )} d^{2} + 6 \, {\left (3 \, C a^{3} + {\left (A + B\right )} a^{2} c\right )} d e + {\left ({\left (9 \, B + 5 \, C\right )} a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, C a^{2} c e^{2} x^{6} - 3 \, B a^{2} c d^{2} - 9 \, B a^{3} e^{2} + 6 \, {\left (2 \, C a^{2} c d e + B a^{2} c e^{2}\right )} x^{4} - 6 \, {\left (3 \, C a^{3} + A a^{2} c\right )} d e - {\left (6 \, B a^{2} c d e + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} + {\left (5 \, C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{6 \, {\left (a^{2} c^{3} x^{5} - a^{3} c^{2} x\right )}} \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="frica s")
Output:
1/6*(3*((B*a*c^2*d^2 + 3*B*a^2*c*e^2 + 2*(3*C*a^2*c + A*a*c^2)*d*e)*x^5 - (B*a^2*c*d^2 + 3*B*a^3*e^2 + 2*(3*C*a^3 + A*a^2*c)*d*e)*x)*sqrt(-c)*(a/c)^ (3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - ((3*((B + C)*a*c^2 - A*c^3)* d^2 + 6*(3*C*a^2*c + (A + B)*a*c^2)*d*e + ((9*B + 5*C)*a^2*c + 3*A*a*c^2)* e^2)*x^5 - (3*((B + C)*a^2*c - A*a*c^2)*d^2 + 6*(3*C*a^3 + (A + B)*a^2*c)* d*e + ((9*B + 5*C)*a^3 + 3*A*a^2*c)*e^2)*x)*sqrt(-c)*(a/c)^(3/4)*elliptic_ f(arcsin((a/c)^(1/4)/x), -1) + (2*C*a^2*c*e^2*x^6 - 3*B*a^2*c*d^2 - 9*B*a^ 3*e^2 + 6*(2*C*a^2*c*d*e + B*a^2*c*e^2)*x^4 - 6*(3*C*a^3 + A*a^2*c)*d*e - (6*B*a^2*c*d*e + 3*(C*a^2*c + A*a*c^2)*d^2 + (5*C*a^3 + 3*A*a^2*c)*e^2)*x^ 2)*sqrt(-c*x^4 + a))/(a^2*c^3*x^5 - a^3*c^2*x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{2} \left (A + B x^{2} + C x^{4}\right )}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**2*(C*x**4+B*x**2+A)/(-c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x**2)**2*(A + B*x**2 + C*x**4)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^2/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^2/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x)
Output:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {8 \sqrt {-c \,x^{4}+a}\, a \,e^{2} x +6 \sqrt {-c \,x^{4}+a}\, b d e x -3 \sqrt {-c \,x^{4}+a}\, b \,e^{2} x^{3}+3 \sqrt {-c \,x^{4}+a}\, c \,d^{2} x -6 \sqrt {-c \,x^{4}+a}\, c d e \,x^{3}-\sqrt {-c \,x^{4}+a}\, c \,e^{2} x^{5}-8 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{3} e^{2}-6 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b d e +8 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c \,e^{2} x^{4}+6 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b c d e \,x^{4}+9 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b \,e^{2}+24 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c d e +3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b c \,d^{2}-9 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b c \,e^{2} x^{4}-24 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} d e \,x^{4}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) b \,c^{2} d^{2} x^{4}}{3 c \left (-c \,x^{4}+a \right )} \] Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x)
Output:
(8*sqrt(a - c*x**4)*a*e**2*x + 6*sqrt(a - c*x**4)*b*d*e*x - 3*sqrt(a - c*x **4)*b*e**2*x**3 + 3*sqrt(a - c*x**4)*c*d**2*x - 6*sqrt(a - c*x**4)*c*d*e* x**3 - sqrt(a - c*x**4)*c*e**2*x**5 - 8*int(sqrt(a - c*x**4)/(a**2 - 2*a*c *x**4 + c**2*x**8),x)*a**3*e**2 - 6*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x** 4 + c**2*x**8),x)*a**2*b*d*e + 8*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c*e**2*x**4 + 6*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x** 4 + c**2*x**8),x)*a*b*c*d*e*x**4 + 9*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2 *a*c*x**4 + c**2*x**8),x)*a**2*b*e**2 + 24*int((sqrt(a - c*x**4)*x**2)/(a* *2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c*d*e + 3*int((sqrt(a - c*x**4)*x**2) /(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c*d**2 - 9*int((sqrt(a - c*x**4)*x **2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c*e**2*x**4 - 24*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c**2*d*e*x**4 - 3*int( (sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*b*c**2*d**2*x** 4)/(3*c*(a - c*x**4))