Integrand size = 34, antiderivative size = 452 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d \left (c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )+\left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {e^2 (3 C d+B e) x \sqrt {a-c x^4}}{3 c^2}+\frac {C e^3 x^3 \sqrt {a-c x^4}}{5 c^2}-\frac {\left (5 B c d \left (c d^2+9 a e^2\right )+3 e \left (5 A c \left (c d^2+a e^2\right )+a C \left (15 c d^2+7 a e^2\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{10 \sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}+\frac {\left (5 \left (3 A c d \left (c d^2-3 a e^2\right )-a \left (5 a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right )+\frac {3 \sqrt {a} \left (5 B c d \left (c d^2+9 a e^2\right )+3 e \left (5 A c \left (c d^2+a e^2\right )+a C \left (15 c d^2+7 a e^2\right )\right )\right )}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{30 a^{3/4} c^{9/4} \sqrt {a-c x^4}} \] Output:
1/2*x*(A*c*d*(3*a*e^2+c*d^2)+a*(a*e^2*(B*e+3*C*d)+c*d^2*(3*B*e+C*d))+((A*c +C*a)*e*(a*e^2+3*c*d^2)+B*c*d*(3*a*e^2+c*d^2))*x^2)/a/c^2/(-c*x^4+a)^(1/2) +1/3*e^2*(B*e+3*C*d)*x*(-c*x^4+a)^(1/2)/c^2+1/5*C*e^3*x^3*(-c*x^4+a)^(1/2) /c^2-1/10*(5*B*c*d*(9*a*e^2+c*d^2)+3*e*(5*A*c*(a*e^2+c*d^2)+a*C*(7*a*e^2+1 5*c*d^2)))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/c^(11/ 4)/(-c*x^4+a)^(1/2)+1/30*(15*A*c*d*(-3*a*e^2+c*d^2)-5*a*(5*a*e^2*(B*e+3*C* d)+3*c*d^2*(3*B*e+C*d))+3*a^(1/2)*(5*B*c*d*(9*a*e^2+c*d^2)+3*e*(5*A*c*(a*e ^2+c*d^2)+a*C*(7*a*e^2+15*c*d^2)))/c^(1/2))*(1-c*x^4/a)^(1/2)*EllipticF(c^ (1/4)*x/a^(1/4),I)/a^(3/4)/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 = 10.52 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (15 A c \left (c d^3+a e^2 \left (3 d-2 e x^2\right )\right )+a \left (a e^2 \left (75 C d+25 B e-42 C e x^2\right )-5 B c e \left (-9 d^2+18 d e x^2+2 e^2 x^4\right )+3 c C \left (5 d^3-30 d^2 e x^2-10 d e^2 x^4-2 e^3 x^6\right )\right )\right )-5 \left (-3 A c d \left (c d^2-3 a e^2\right )+a \left (5 a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 \left (5 B c d \left (c d^2+9 a e^2\right )+3 e \left (5 A c \left (c d^2+a e^2\right )+a C \left (15 c d^2+7 a e^2\right )\right )\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{30 a c^2 \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x]
Output:
(x*(15*A*c*(c*d^3 + a*e^2*(3*d - 2*e*x^2)) + a*(a*e^2*(75*C*d + 25*B*e - 4 2*C*e*x^2) - 5*B*c*e*(-9*d^2 + 18*d*e*x^2 + 2*e^2*x^4) + 3*c*C*(5*d^3 - 30 *d^2*e*x^2 - 10*d*e^2*x^4 - 2*e^3*x^6))) - 5*(-3*A*c*d*(c*d^2 - 3*a*e^2) + a*(5*a*e^2*(3*C*d + B*e) + 3*c*d^2*(C*d + 3*B*e)))*x*Sqrt[1 - (c*x^4)/a]* Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 2*(5*B*c*d*(c*d^2 + 9*a*e^2) + 3*e*(5*A*c*(c*d^2 + a*e^2) + a*C*(15*c*d^2 + 7*a*e^2)))*x^3*Sqrt[1 - (c *x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (c*x^4)/a])/(30*a*c^2*Sqrt[a - c *x^4])
Time = 1.26 (sec) , antiderivative size = 787, normalized size of antiderivative = 1.74, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (-\frac {a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3}{c^2 \sqrt {a-c x^4}}-\frac {e x^2 \left (a C e^2+c e (A e+3 B d)+3 c C d^2\right )}{c^2 \sqrt {a-c x^4}}+\frac {x^2 \left (e (a C+A c) \left (a e^2+3 c d^2\right )+B c d \left (3 a e^2+c d^2\right )\right )+A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )}{c^2 \left (a-c x^4\right )^{3/2}}-\frac {e^2 x^4 (B e+3 C d)}{c \sqrt {a-c x^4}}-\frac {C e^3 x^6}{c \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} e \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+3 B d)+3 c C d^2\right )}{c^{11/4} \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 C e^2+c e (A e+3 B d)+3 c C d^2\right )}{c^{11/4} \sqrt {a-c x^4}}+\frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} e+\sqrt {c} d\right )^3 \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^{11/4} \sqrt {a-c x^4}}-\frac {a^{5/4} e^2 \sqrt {1-\frac {c x^4}{a}} (B e+3 C d) \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 {3 a^{7/4} C e^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{11/4} \sqrt {a-c x^4}}-\frac {3 a^{7/4} C e^3 \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{11/4} \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 e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\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 (e (a C+A c) \left (a e^2+3 c d^2\right )+B c d \left (3 a e^2+c d^2\right )\right )}{2 \sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}+\frac {x \left (x^2 \left (e (a C+A c) \left (a e^2+3 c d^2\right )+B c d \left (3 a e^2+c d^2\right )\right )+A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {e^2 x \sqrt {a-c x^4} (B e+3 C d)}{3 c^2}+\frac {C e^3 x^3 \sqrt {a-c x^4}}{5 c^2}\) |
Input:
Int[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x]
Output:
(x*(A*c*d*(c*d^2 + 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e) ) + ((A*c + a*C)*e*(3*c*d^2 + a*e^2) + B*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2*a *c^2*Sqrt[a - c*x^4]) + (e^2*(3*C*d + B*e)*x*Sqrt[a - c*x^4])/(3*c^2) + (C *e^3*x^3*Sqrt[a - c*x^4])/(5*c^2) - (3*a^(7/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*E llipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(11/4)*Sqrt[a - c*x^4]) - (a^(3/4)*e*(3*c*C*d^2 + a*C*e^2 + c*e*(3*B*d + A*e))*Sqrt[1 - (c*x^4)/a]*E llipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(11/4)*Sqrt[a - c*x^4]) - (( (A*c + a*C)*e*(3*c*d^2 + a*e^2) + B*c*d*(c*d^2 + 3*a*e^2))*Sqrt[1 - (c*x^4 )/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(1/4)*c^(11/4)*Sqrt[ a - c*x^4]) + (3*a^(7/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/ 4)*x)/a^(1/4)], -1])/(5*c^(11/4)*Sqrt[a - c*x^4]) + ((Sqrt[a]*B*Sqrt[c] + A*c + a*C)*(Sqrt[c]*d + Sqrt[a]*e)^3*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[ (c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*c^(11/4)*Sqrt[a - c*x^4]) - (a^(5/4) *e^2*(3*C*d + B*e)*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]) + (a^(3/4)*e*(3*c*C*d^2 + a*C*e^2 + c *e*(3*B*d + A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4) ], -1])/(c^(11/4)*Sqrt[a - c*x^4]) - (a^(1/4)*(c*C*d^3 + 3*c*d*e*(B*d + A* e) + a*e^2*(3*C*d + B*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 = 8.01 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(\frac {2 c \left (\frac {\left (A a c \,e^{3}+3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}+B \,d^{3} c^{2}+C \,a^{2} e^{3}+3 C a c \,d^{2} e \right ) x^{3}}{4 a \,c^{3}}+\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}+a^{2} B \,e^{3}+3 B a c \,d^{2} e +3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{4 a \,c^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {C \,e^{3} x^{3} \sqrt {-c \,x^{4}+a}}{5 c^{2}}+\frac {e^{2} \left (B e +3 C d \right ) x \sqrt {-c \,x^{4}+a}}{3 c^{2}}+\frac {\left (-\frac {3 A c d \,e^{2}+B a \,e^{3}+3 B c \,d^{2} e +3 a C d \,e^{2}+C c \,d^{3}}{c^{2}}+\frac {3 A a c d \,e^{2}+A \,c^{2} d^{3}+a^{2} B \,e^{3}+3 B a c \,d^{2} e +3 C \,a^{2} d \,e^{2}+C a c \,d^{3}}{2 c^{2} a}-\frac {e^{2} \left (B e +3 C d \right ) a}{3 c^{2}}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (-\frac {e \left (A c \,e^{2}+3 B c d e +C a \,e^{2}+3 C c \,d^{2}\right )}{c^{2}}-\frac {A a c \,e^{3}+3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}+B \,d^{3} c^{2}+C \,a^{2} e^{3}+3 C a c \,d^{2} e}{2 c^{2} a}-\frac {3 C \,e^{3} a}{5 c^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(567\) |
| default | \(A \,d^{3} \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^{2} \left (3 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^{2} \left (B e +3 C d \right ) \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 )+e \left (A \,e^{2}+3 B d e +3 C \,d^{2}\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 )+d \left (3 A \,e^{2}+3 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 )+e^{3} C \left (\frac {a \,x^{3}}{2 c^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c^{2}}+\frac {21 a^{\frac {3}{2}} \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 )}{10 c^{\frac {5}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(711\) |
| risch | \(\frac {e^{2} x \left (3 C \,x^{2} e +5 B e +15 C d \right ) \sqrt {-c \,x^{4}+a}}{15 c^{2}}-\frac {-\frac {3 e \left (5 A c \,e^{2}+15 B c d e +8 C a \,e^{2}+15 C c \,d^{2}\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}}+\frac {30 c \left (-\frac {\left (A a c \,e^{3}+3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}+B \,d^{3} c^{2}+C \,a^{2} e^{3}+3 C a c \,d^{2} e \right ) x^{3}}{4 a c}-\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}+a^{2} B \,e^{3}+3 B a c \,d^{2} e +3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{4 a c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {15 \left (3 A a c d \,e^{2}+A \,c^{2} d^{3}+a^{2} B \,e^{3}+3 B a c \,d^{2} e +3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\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 {15 \left (A a c \,e^{3}+3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}+B \,d^{3} c^{2}+C \,a^{2} e^{3}+3 C a c \,d^{2} 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}\, \sqrt {c}}+\frac {20 B a \,e^{3} \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 {15 C c \,d^{3} \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 {45 A c d \,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 {45 B c \,d^{2} 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 {60 a C d \,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}}}{15 c^{2}}\) | \(908\) |
Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4/a/c^3*(A*a*c*e^3+3*A*c^2*d^2*e+3*B*a*c*d*e^2+B*c^2*d^3+C*a^2*e^3+ 3*C*a*c*d^2*e)*x^3+1/4/a/c^3*(3*A*a*c*d*e^2+A*c^2*d^3+B*a^2*e^3+3*B*a*c*d^ 2*e+3*C*a^2*d*e^2+C*a*c*d^3)*x)/(-(x^4-a/c)*c)^(1/2)+1/5*C*e^3*x^3*(-c*x^4 +a)^(1/2)/c^2+1/3*e^2*(B*e+3*C*d)*x*(-c*x^4+a)^(1/2)/c^2+(-(3*A*c*d*e^2+B* a*e^3+3*B*c*d^2*e+3*C*a*d*e^2+C*c*d^3)/c^2+1/2/c^2/a*(3*A*a*c*d*e^2+A*c^2* d^3+B*a^2*e^3+3*B*a*c*d^2*e+3*C*a^2*d*e^2+C*a*c*d^3)-1/3/c^2*e^2*(B*e+3*C* d)*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)-(- e*(A*c*e^2+3*B*c*d*e+C*a*e^2+3*C*c*d^2)/c^2-1/2/c^2/a*(A*a*c*e^3+3*A*c^2*d ^2*e+3*B*a*c*d*e^2+B*c^2*d^3+C*a^2*e^3+3*C*a*c*d^2*e)-3/5*C/c^2*e^3*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)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.10 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.50 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="frica s")
Output:
1/30*(3*((5*B*a*c^3*d^3 + 45*B*a^2*c^2*d*e^2 + 15*(3*C*a^2*c^2 + A*a*c^3)* d^2*e + 3*(7*C*a^3*c + 5*A*a^2*c^2)*e^3)*x^5 - (5*B*a^2*c^2*d^3 + 45*B*a^3 *c*d*e^2 + 15*(3*C*a^3*c + A*a^2*c^2)*d^2*e + 3*(7*C*a^4 + 5*A*a^3*c)*e^3) *x)*sqrt(-c)*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - ((15*((B + C)*a*c^3 - A*c^4)*d^3 + 45*(3*C*a^2*c^2 + (A + B)*a*c^3)*d^2*e + 15*((9* B + 5*C)*a^2*c^2 + 3*A*a*c^3)*d*e^2 + (63*C*a^3*c + 5*(9*A + 5*B)*a^2*c^2) *e^3)*x^5 - (15*((B + C)*a^2*c^2 - A*a*c^3)*d^3 + 45*(3*C*a^3*c + (A + B)* a^2*c^2)*d^2*e + 15*((9*B + 5*C)*a^3*c + 3*A*a^2*c^2)*d*e^2 + (63*C*a^4 + 5*(9*A + 5*B)*a^3*c)*e^3)*x)*sqrt(-c)*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^ (1/4)/x), -1) + (6*C*a^2*c^2*e^3*x^8 - 15*B*a^2*c^2*d^3 - 135*B*a^3*c*d*e^ 2 + 10*(3*C*a^2*c^2*d*e^2 + B*a^2*c^2*e^3)*x^6 + 6*(15*C*a^2*c^2*d^2*e + 1 5*B*a^2*c^2*d*e^2 + (7*C*a^3*c + 5*A*a^2*c^2)*e^3)*x^4 - 45*(3*C*a^3*c + A *a^2*c^2)*d^2*e - 9*(7*C*a^4 + 5*A*a^3*c)*e^3 - 5*(9*B*a^2*c^2*d^2*e + 5*B *a^3*c*e^3 + 3*(C*a^2*c^2 + A*a*c^3)*d^3 + 3*(5*C*a^3*c + 3*A*a^2*c^2)*d*e ^2)*x^2)*sqrt(-c*x^4 + a))/(a^2*c^4*x^5 - a^3*c^3*x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{3} \left (A + B x^{2} + C x^{4}\right )}{\left (a - c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((e*x**2+d)**3*(C*x**4+B*x**2+A)/(-c*x**4+a)**(3/2),x)
Output:
Integral((d + e*x**2)**3*(A + B*x**2 + C*x**4)/(a - c*x**4)**(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3*(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)^3/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3*(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)^3/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x)
Output:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x)
Output:
(25*sqrt(a - c*x**4)*a*b*e**3*x + 120*sqrt(a - c*x**4)*a*c*d*e**2*x - 36*s qrt(a - c*x**4)*a*c*e**3*x**3 + 45*sqrt(a - c*x**4)*b*c*d**2*e*x - 45*sqrt (a - c*x**4)*b*c*d*e**2*x**3 - 5*sqrt(a - c*x**4)*b*c*e**3*x**5 + 15*sqrt( a - c*x**4)*c**2*d**3*x - 45*sqrt(a - c*x**4)*c**2*d**2*e*x**3 - 15*sqrt(a - c*x**4)*c**2*d*e**2*x**5 - 3*sqrt(a - c*x**4)*c**2*e**3*x**7 - 25*int(s qrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**3*b*e**3 - 120*int(s qrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**3*c*d*e**2 - 45*int( sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*b*c*d**2*e + 25*i nt(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*b*c*e**3*x**4 + 120*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d* e**2*x**4 + 45*int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b *c**2*d**2*e*x**4 + 108*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c **2*x**8),x)*a**3*c*e**3 + 135*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x **4 + c**2*x**8),x)*a**2*b*c*d*e**2 + 180*int((sqrt(a - c*x**4)*x**2)/(a** 2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*d**2*e - 108*int((sqrt(a - c*x**4 )*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c**2*e**3*x**4 + 15*int((s qrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c**2*d**3 - 1 35*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c**2 *d*e**2*x**4 - 180*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x **8),x)*a*c**3*d**2*e*x**4 - 15*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a...