Integrand size = 34, antiderivative size = 512 \[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {2 a x \left (65 \left (3 A c \left (11 c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right )+77 c \left (13 B c d^2+26 A c d e+6 a C d e+3 a B e^2\right ) x^2\right ) \sqrt {a-c x^4}}{15015 c^2}+\frac {x \left (39 \left (3 A c \left (11 c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right )+77 c \left (13 B c d^2+26 A c d e+6 a C d e+3 a B e^2\right ) x^2\right ) \left (a-c x^4\right )^{3/2}}{9009 c^2}-\frac {\left (a C e^2+3 c \left (C d^2+e (2 B d+A e)\right )\right ) x \left (a-c x^4\right )^{5/2}}{33 c^2}-\frac {e (2 C d+B e) x^3 \left (a-c x^4\right )^{5/2}}{13 c}-\frac {C e^2 x^5 \left (a-c x^4\right )^{5/2}}{15 c}+\frac {4 a^{11/4} \left (13 B c d^2+26 A c d e+6 a C d e+3 a B e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{195 c^{7/4} \sqrt {a-c x^4}}-\frac {4 a^{9/4} \left (77 \sqrt {a} \sqrt {c} \left (13 B c d^2+26 A c d e+6 a C d e+3 a B e^2\right )-65 \left (3 A c \left (11 c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15015 c^{9/4} \sqrt {a-c x^4}} \] Output:
2/15015*a*x*(195*A*c*(a*e^2+11*c*d^2)+65*a*(C*a*e^2+3*c*d*(2*B*e+C*d))+77* c*(26*A*c*d*e+3*B*a*e^2+13*B*c*d^2+6*C*a*d*e)*x^2)*(-c*x^4+a)^(1/2)/c^2+1/ 9009*x*(117*A*c*(a*e^2+11*c*d^2)+39*a*(C*a*e^2+3*c*d*(2*B*e+C*d))+77*c*(26 *A*c*d*e+3*B*a*e^2+13*B*c*d^2+6*C*a*d*e)*x^2)*(-c*x^4+a)^(3/2)/c^2-1/33*(C *a*e^2+3*c*(C*d^2+e*(A*e+2*B*d)))*x*(-c*x^4+a)^(5/2)/c^2-1/13*e*(B*e+2*C*d )*x^3*(-c*x^4+a)^(5/2)/c-1/15*C*e^2*x^5*(-c*x^4+a)^(5/2)/c+4/195*a^(11/4)* (26*A*c*d*e+3*B*a*e^2+13*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)/c^(7/4)/(-c*x^4+a)^(1/2)-4/15015*a^(9/4)*(77*a^(1/2)*c^ (1/2)*(26*A*c*d*e+3*B*a*e^2+13*B*c*d^2+6*C*a*d*e)-195*A*c*(a*e^2+11*c*d^2) -65*a*(C*a*e^2+3*c*d*(2*B*e+C*d)))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a ^(1/4),I)/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 = 10.41 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.69 \[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-c x^4} \left (-195 c \left (C d^2+e (2 B d+A e)\right ) \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-165 c e (2 C d+B e) x^2 \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-143 c C e^2 x^4 \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}+2145 a A c^2 d^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+195 a^2 c \left (C d^2+e (2 B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )-65 a C e^2 \left (\left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )\right )+715 a c^2 d (B d+2 A e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )+165 a^2 c e (2 C d+B e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{2145 c^2 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)^2*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - c*x^4]*(-195*c*(C*d^2 + e*(2*B*d + A*e))*(a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] - 165*c*e*(2*C*d + B*e)*x^2*(a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] - 143*c*C*e^2*x^4*(a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] + 2145*a*A*c^2*d^2*Hy pergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/a] + 195*a^2*c*(C*d^2 + e*(2*B*d + A*e))*Hypergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/a] - 65*a*C*e^2*((a - c*x ^4)^2*Sqrt[1 - (c*x^4)/a] - a^2*Hypergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/ a]) + 715*a*c^2*d*(B*d + 2*A*e)*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, (c*x ^4)/a] + 165*a^2*c*e*(2*C*d + B*e)*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, ( c*x^4)/a]))/(2145*c^2*Sqrt[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(1607\) vs. \(2(512)=1024\).
Time = 2.02 (sec) , antiderivative size = 1607, normalized size of antiderivative = 3.14, 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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {a^2 d x^2 (2 A e+B d)}{\sqrt {a-c x^4}}+\frac {a^2 A d^2}{\sqrt {a-c x^4}}+\frac {a x^4 \left (a d (2 B e+C d)-A \left (2 c d^2-a e^2\right )\right )}{\sqrt {a-c x^4}}+\frac {c x^{12} \left (-2 a C e^2+c e (A e+2 B d)+c C d^2\right )}{\sqrt {a-c x^4}}+\frac {c x^{10} \left (-2 a B e^2-4 a C d e+2 A c d e+B c d^2\right )}{\sqrt {a-c x^4}}+\frac {x^8 \left (A c \left (c d^2-2 a e^2\right )+a \left (a C e^2-2 c d (2 B e+C d)\right )\right )}{\sqrt {a-c x^4}}+\frac {a x^6 \left (a B e^2+2 a C d e-4 A c d e-2 B c d^2\right )}{\sqrt {a-c x^4}}+\frac {c^2 e x^{14} (B e+2 C d)}{\sqrt {a-c x^4}}+\frac {c^2 C e^2 x^{16}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{15} c C e^2 \sqrt {a-c x^4} x^{13}-\frac {1}{13} c e (2 C d+B e) \sqrt {a-c x^4} x^{11}-\frac {13}{165} a C e^2 \sqrt {a-c x^4} x^9-\frac {1}{11} \left (c C d^2-2 a C e^2+c e (2 B d+A e)\right ) \sqrt {a-c x^4} x^9-\frac {11}{117} a e (2 C d+B e) \sqrt {a-c x^4} x^7-\frac {1}{9} \left (B c d^2+2 A c e d-4 a C e d-2 a B e^2\right ) \sqrt {a-c x^4} x^7-\frac {39 a^2 C e^2 \sqrt {a-c x^4} x^5}{385 c}-\frac {9 a \left (c C d^2-2 a C e^2+c e (2 B d+A e)\right ) \sqrt {a-c x^4} x^5}{77 c}-\frac {\left (A c \left (c d^2-2 a e^2\right )+a \left (a C e^2-2 c d (C d+2 B e)\right )\right ) \sqrt {a-c x^4} x^5}{7 c}-\frac {77 a^2 e (2 C d+B e) \sqrt {a-c x^4} x^3}{585 c}-\frac {7 a \left (B c d^2+2 A c e d-4 a C e d-2 a B e^2\right ) \sqrt {a-c x^4} x^3}{45 c}+\frac {a \left (2 B c d^2+4 A c e d-2 a C e d-a B e^2\right ) \sqrt {a-c x^4} x^3}{5 c}-\frac {13 a^3 C e^2 \sqrt {a-c x^4} x}{77 c^2}-\frac {15 a^2 \left (c C d^2-2 a C e^2+c e (2 B d+A e)\right ) \sqrt {a-c x^4} x}{77 c^2}-\frac {a \left (a d (C d+2 B e)-A \left (2 c d^2-a e^2\right )\right ) \sqrt {a-c x^4} x}{3 c}-\frac {5 a \left (A c \left (c d^2-2 a e^2\right )+a \left (a C e^2-2 c d (C d+2 B e)\right )\right ) \sqrt {a-c x^4} x}{21 c^2}+\frac {a^{11/4} d (B d+2 A 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 )}{c^{3/4} \sqrt {a-c x^4}}+\frac {77 a^{15/4} e (2 C d+B 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 )}{195 c^{7/4} \sqrt {a-c x^4}}+\frac {7 a^{11/4} \left (B c d^2+2 A c e d-4 a C e d-2 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 )}{15 c^{7/4} \sqrt {a-c x^4}}-\frac {3 a^{11/4} \left (2 B c d^2+4 A c e d-2 a C e d-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 )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {a^{9/4} A d^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}+\frac {13 a^{17/4} 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 )}{77 c^{9/4} \sqrt {a-c x^4}}-\frac {a^{11/4} d (B d+2 A e) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}-\frac {77 a^{15/4} e (2 C d+B e) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{195 c^{7/4} \sqrt {a-c x^4}}-\frac {7 a^{11/4} \left (B c d^2+2 A c e d-4 a C e d-2 a B 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 )}{15 c^{7/4} \sqrt {a-c x^4}}+\frac {3 a^{11/4} \left (2 B c d^2+4 A c e d-2 a C e d-a B 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 )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {15 a^{13/4} \left (c C d^2-2 a C e^2+c e (2 B d+A e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{77 c^{9/4} \sqrt {a-c x^4}}+\frac {a^{9/4} \left (a d (C d+2 B e)-A \left (2 c d^2-a 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 )}{3 c^{5/4} \sqrt {a-c x^4}}+\frac {5 a^{9/4} \left (A c \left (c d^2-2 a e^2\right )+a \left (a C e^2-2 c d (C d+2 B 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 )}{21 c^{9/4} \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)^2*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
(-13*a^3*C*e^2*x*Sqrt[a - c*x^4])/(77*c^2) - (15*a^2*(c*C*d^2 - 2*a*C*e^2 + c*e*(2*B*d + A*e))*x*Sqrt[a - c*x^4])/(77*c^2) - (a*(a*d*(C*d + 2*B*e) - A*(2*c*d^2 - a*e^2))*x*Sqrt[a - c*x^4])/(3*c) - (5*a*(A*c*(c*d^2 - 2*a*e^ 2) + a*(a*C*e^2 - 2*c*d*(C*d + 2*B*e)))*x*Sqrt[a - c*x^4])/(21*c^2) - (77* a^2*e*(2*C*d + B*e)*x^3*Sqrt[a - c*x^4])/(585*c) - (7*a*(B*c*d^2 + 2*A*c*d *e - 4*a*C*d*e - 2*a*B*e^2)*x^3*Sqrt[a - c*x^4])/(45*c) + (a*(2*B*c*d^2 + 4*A*c*d*e - 2*a*C*d*e - a*B*e^2)*x^3*Sqrt[a - c*x^4])/(5*c) - (39*a^2*C*e^ 2*x^5*Sqrt[a - c*x^4])/(385*c) - (9*a*(c*C*d^2 - 2*a*C*e^2 + c*e*(2*B*d + A*e))*x^5*Sqrt[a - c*x^4])/(77*c) - ((A*c*(c*d^2 - 2*a*e^2) + a*(a*C*e^2 - 2*c*d*(C*d + 2*B*e)))*x^5*Sqrt[a - c*x^4])/(7*c) - (11*a*e*(2*C*d + B*e)* x^7*Sqrt[a - c*x^4])/117 - ((B*c*d^2 + 2*A*c*d*e - 4*a*C*d*e - 2*a*B*e^2)* x^7*Sqrt[a - c*x^4])/9 - (13*a*C*e^2*x^9*Sqrt[a - c*x^4])/165 - ((c*C*d^2 - 2*a*C*e^2 + c*e*(2*B*d + A*e))*x^9*Sqrt[a - c*x^4])/11 - (c*e*(2*C*d + B *e)*x^11*Sqrt[a - c*x^4])/13 - (c*C*e^2*x^13*Sqrt[a - c*x^4])/15 + (a^(11/ 4)*d*(B*d + 2*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]) + (77*a^(15/4)*e*(2*C*d + B*e)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(195*c^(7/4)*Sqrt[ a - c*x^4]) + (7*a^(11/4)*(B*c*d^2 + 2*A*c*d*e - 4*a*C*d*e - 2*a*B*e^2)*Sq rt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(7/4)* Sqrt[a - c*x^4]) - (3*a^(11/4)*(2*B*c*d^2 + 4*A*c*d*e - 2*a*C*d*e - a*B...
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 = 3.16 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(A \,d^{2} \left (-\frac {c \,x^{5} \sqrt {-c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {-c \,x^{4}+a}}{7}+\frac {4 a^{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 )}{7 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+d \left (2 A e +B d \right ) \left (-\frac {c \,x^{7} \sqrt {-c \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{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 )}{15 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+e \left (B e +2 C d \right ) \left (-\frac {c \,x^{11} \sqrt {-c \,x^{4}+a}}{13}+\frac {5 a \,x^{7} \sqrt {-c \,x^{4}+a}}{39}-\frac {4 a^{2} x^{3} \sqrt {-c \,x^{4}+a}}{195 c}-\frac {4 a^{\frac {7}{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 )}{65 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 {c \,x^{9} \sqrt {-c \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {-c \,x^{4}+a}}{77}-\frac {4 a^{2} x \sqrt {-c \,x^{4}+a}}{77 c}+\frac {4 a^{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 )}{77 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C \,e^{2} \left (-\frac {c \,x^{13} \sqrt {-c \,x^{4}+a}}{15}+\frac {17 a \,x^{9} \sqrt {-c \,x^{4}+a}}{165}-\frac {4 a^{2} x^{5} \sqrt {-c \,x^{4}+a}}{385 c}-\frac {4 a^{3} x \sqrt {-c \,x^{4}+a}}{231 c^{2}}+\frac {4 a^{4} \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 )}{231 c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(673\) |
| risch | \(-\frac {x \left (3003 C \,e^{2} x^{12} c^{3}+3465 B \,c^{3} e^{2} x^{10}+6930 C \,c^{3} d e \,x^{10}+4095 A \,c^{3} e^{2} x^{8}+8190 B \,c^{3} d e \,x^{8}-4641 C a \,c^{2} e^{2} x^{8}+4095 C \,c^{3} d^{2} x^{8}+10010 A \,c^{3} d e \,x^{6}-5775 B a \,c^{2} e^{2} x^{6}+5005 B \,c^{3} d^{2} x^{6}-11550 C a \,c^{2} d e \,x^{6}-7605 A a \,c^{2} e^{2} x^{4}+6435 A \,c^{3} d^{2} x^{4}-15210 B a \,c^{2} d e \,x^{4}+468 C \,a^{2} c \,e^{2} x^{4}-7605 C a \,c^{2} d^{2} x^{4}-22022 A a \,c^{2} d e \,x^{2}+924 B \,a^{2} e^{2} x^{2} c -11011 B a \,c^{2} d^{2} x^{2}+1848 C \,a^{2} d e \,x^{2} c +2340 A \,a^{2} c \,e^{2}-19305 A a \,c^{2} d^{2}+4680 B \,a^{2} c d e +780 a^{3} C \,e^{2}+2340 C \,a^{2} c \,d^{2}\right ) \sqrt {-c \,x^{4}+a}}{45045 c^{2}}+\frac {4 a^{2} \left (-\frac {77 \sqrt {c}\, \left (26 A c d e +3 B a \,e^{2}+13 B c \,d^{2}+6 C a d e \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}}+\frac {2145 A \,c^{2} 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 {65 a^{2} 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 {195 A 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 {195 C a 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 {390 B a 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}}\right )}{15015 c^{2}}\) | \(785\) |
| elliptic | \(-\frac {C c \,e^{2} x^{13} \sqrt {-c \,x^{4}+a}}{15}-\frac {\left (B \,c^{2} e^{2}+2 c^{2} d e C \right ) x^{11} \sqrt {-c \,x^{4}+a}}{13 c}-\frac {\left (A \,c^{2} e^{2}+2 B \,c^{2} d e -\frac {17}{15} C a c \,e^{2}+C \,c^{2} d^{2}\right ) x^{9} \sqrt {-c \,x^{4}+a}}{11 c}-\frac {\left (2 A \,c^{2} d e -2 B a c \,e^{2}+B \,c^{2} d^{2}-4 C a d e c +\frac {11 a \left (B \,c^{2} e^{2}+2 c^{2} d e C \right )}{13 c}\right ) x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {\left (-2 A a c \,e^{2}+A \,c^{2} d^{2}-4 B a c d e +a^{2} C \,e^{2}-2 C a c \,d^{2}+\frac {9 a \left (A \,c^{2} e^{2}+2 B \,c^{2} d e -\frac {17}{15} C a c \,e^{2}+C \,c^{2} d^{2}\right )}{11 c}\right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (-4 A a c d e +B \,a^{2} e^{2}-2 B a c \,d^{2}+2 C \,a^{2} d e +\frac {7 a \left (2 A \,c^{2} d e -2 B a c \,e^{2}+B \,c^{2} d^{2}-4 C a d e c +\frac {11 a \left (B \,c^{2} e^{2}+2 c^{2} d e C \right )}{13 c}\right )}{9 c}\right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (A \,a^{2} e^{2}-2 A \,d^{2} a c +2 B \,a^{2} d e +C \,a^{2} d^{2}+\frac {5 a \left (-2 A a c \,e^{2}+A \,c^{2} d^{2}-4 B a c d e +a^{2} C \,e^{2}-2 C a c \,d^{2}+\frac {9 a \left (A \,c^{2} e^{2}+2 B \,c^{2} d e -\frac {17}{15} C a c \,e^{2}+C \,c^{2} d^{2}\right )}{11 c}\right )}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A \,a^{2} d^{2}+\frac {a \left (A \,a^{2} e^{2}-2 A \,d^{2} a c +2 B \,a^{2} d e +C \,a^{2} d^{2}+\frac {5 a \left (-2 A a c \,e^{2}+A \,c^{2} d^{2}-4 B a c d e +a^{2} C \,e^{2}-2 C a c \,d^{2}+\frac {9 a \left (A \,c^{2} e^{2}+2 B \,c^{2} d e -\frac {17}{15} C a c \,e^{2}+C \,c^{2} d^{2}\right )}{11 c}\right )}{7 c}\right )}{3 c}\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 \,a^{2} d e +B \,a^{2} d^{2}+\frac {3 a \left (-4 A a c d e +B \,a^{2} e^{2}-2 B a c \,d^{2}+2 C \,a^{2} d e +\frac {7 a \left (2 A \,c^{2} d e -2 B a c \,e^{2}+B \,c^{2} d^{2}-4 C a d e c +\frac {11 a \left (B \,c^{2} e^{2}+2 c^{2} d e C \right )}{13 c}\right )}{9 c}\right )}{5 c}\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}}\) | \(922\) |
Input:
int((e*x^2+d)^2*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
A*d^2*(-1/7*c*x^5*(-c*x^4+a)^(1/2)+3/7*a*x*(-c*x^4+a)^(1/2)+4/7*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))+d*(2*A*e+B*d) *(-1/9*c*x^7*(-c*x^4+a)^(1/2)+11/45*a*x^3*(-c*x^4+a)^(1/2)-4/15*a^(5/2)/(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)))+e*(B*e+2*C*d)*(-1/13*c*x^11*(-c*x^ 4+a)^(1/2)+5/39*a*x^7*(-c*x^4+a)^(1/2)-4/195*a^2/c*x^3*(-c*x^4+a)^(1/2)-4/ 65*a^(7/2)/c^(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)))+(A*e^2+2*B*d*e+C*d^2) *(-1/11*c*x^9*(-c*x^4+a)^(1/2)+13/77*a*x^5*(-c*x^4+a)^(1/2)-4/77*a^2/c*x*( -c*x^4+a)^(1/2)+4/77*a^3/c/(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))+C*e^2*(-1/15*c*x^13*(-c*x^4+a)^(1/2)+17/165*a*x^9*(-c* x^4+a)^(1/2)-4/385*a^2/c*x^5*(-c*x^4+a)^(1/2)-4/231*a^3/c^2*x*(-c*x^4+a)^( 1/2)+4/231*a^4/c^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))
Time = 0.08 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.97 \[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=-\frac {924 \, {\left (13 \, B a^{2} c d^{2} + 3 \, B a^{3} e^{2} + 2 \, {\left (3 \, C a^{3} + 13 \, A a^{2} c\right )} d e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 12 \, {\left (13 \, {\left ({\left (77 \, B + 15 \, C\right )} a^{2} c + 165 \, A a c^{2}\right )} d^{2} + 2 \, {\left (231 \, C a^{3} + 13 \, {\left (77 \, A + 15 \, B\right )} a^{2} c\right )} d e + {\left ({\left (231 \, B + 65 \, C\right )} a^{3} + 195 \, A a^{2} c\right )} e^{2}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3003 \, C c^{3} e^{2} x^{14} + 3465 \, {\left (2 \, C c^{3} d e + B c^{3} e^{2}\right )} x^{12} + 273 \, {\left (15 \, C c^{3} d^{2} + 30 \, B c^{3} d e - {\left (17 \, C a c^{2} - 15 \, A c^{3}\right )} e^{2}\right )} x^{10} + 385 \, {\left (13 \, B c^{3} d^{2} - 15 \, B a c^{2} e^{2} - 2 \, {\left (15 \, C a c^{2} - 13 \, A c^{3}\right )} d e\right )} x^{8} - 117 \, {\left (130 \, B a c^{2} d e + 5 \, {\left (13 \, C a c^{2} - 11 \, A c^{3}\right )} d^{2} - {\left (4 \, C a^{2} c - 65 \, A a c^{2}\right )} e^{2}\right )} x^{6} + 12012 \, B a^{2} c d^{2} + 2772 \, B a^{3} e^{2} - 77 \, {\left (143 \, B a c^{2} d^{2} - 12 \, B a^{2} c e^{2} - 2 \, {\left (12 \, C a^{2} c - 143 \, A a c^{2}\right )} d e\right )} x^{4} + 1848 \, {\left (3 \, C a^{3} + 13 \, A a^{2} c\right )} d e + 195 \, {\left (24 \, B a^{2} c d e + 3 \, {\left (4 \, C a^{2} c - 33 \, A a c^{2}\right )} d^{2} + 4 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{45045 \, c^{2} x} \] Input:
integrate((e*x^2+d)^2*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="frica s")
Output:
-1/45045*(924*(13*B*a^2*c*d^2 + 3*B*a^3*e^2 + 2*(3*C*a^3 + 13*A*a^2*c)*d*e )*sqrt(-c)*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - 12*(13*(( 77*B + 15*C)*a^2*c + 165*A*a*c^2)*d^2 + 2*(231*C*a^3 + 13*(77*A + 15*B)*a^ 2*c)*d*e + ((231*B + 65*C)*a^3 + 195*A*a^2*c)*e^2)*sqrt(-c)*x*(a/c)^(3/4)* elliptic_f(arcsin((a/c)^(1/4)/x), -1) + (3003*C*c^3*e^2*x^14 + 3465*(2*C*c ^3*d*e + B*c^3*e^2)*x^12 + 273*(15*C*c^3*d^2 + 30*B*c^3*d*e - (17*C*a*c^2 - 15*A*c^3)*e^2)*x^10 + 385*(13*B*c^3*d^2 - 15*B*a*c^2*e^2 - 2*(15*C*a*c^2 - 13*A*c^3)*d*e)*x^8 - 117*(130*B*a*c^2*d*e + 5*(13*C*a*c^2 - 11*A*c^3)*d ^2 - (4*C*a^2*c - 65*A*a*c^2)*e^2)*x^6 + 12012*B*a^2*c*d^2 + 2772*B*a^3*e^ 2 - 77*(143*B*a*c^2*d^2 - 12*B*a^2*c*e^2 - 2*(12*C*a^2*c - 143*A*a*c^2)*d* e)*x^4 + 1848*(3*C*a^3 + 13*A*a^2*c)*d*e + 195*(24*B*a^2*c*d*e + 3*(4*C*a^ 2*c - 33*A*a*c^2)*d^2 + 4*(C*a^3 + 3*A*a^2*c)*e^2)*x^2)*sqrt(-c*x^4 + a))/ (c^2*x)
Time = 7.89 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.70 \[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x**2+d)**2*(-c*x**4+a)**(3/2)*(C*x**4+B*x**2+A),x)
Output:
A*a**(3/2)*d**2*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(2 *I*pi)/a)/(4*gamma(5/4)) + A*a**(3/2)*d*e*x**3*gamma(3/4)*hyper((-1/2, 3/4 ), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*gamma(7/4)) + A*a**(3/2)*e**2*x* *5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*ga mma(9/4)) - A*sqrt(a)*c*d**2*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c* x**4*exp_polar(2*I*pi)/a)/(4*gamma(9/4)) - A*sqrt(a)*c*d*e*x**7*gamma(7/4) *hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*gamma(11/4)) - A*sqrt(a)*c*e**2*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**4*exp_p olar(2*I*pi)/a)/(4*gamma(13/4)) + B*a**(3/2)*d**2*x**3*gamma(3/4)*hyper((- 1/2, 3/4), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) + B*a**(3/2) *d*e*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a )/(2*gamma(9/4)) + B*a**(3/2)*e**2*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/ 4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(11/4)) - B*sqrt(a)*c*d**2*x**7*g amma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma (11/4)) - B*sqrt(a)*c*d*e*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x* *4*exp_polar(2*I*pi)/a)/(2*gamma(13/4)) - B*sqrt(a)*c*e**2*x**11*gamma(11/ 4)*hyper((-1/2, 11/4), (15/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(15/4) ) + C*a**(3/2)*d**2*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_ polar(2*I*pi)/a)/(4*gamma(9/4)) + C*a**(3/2)*d*e*x**7*gamma(7/4)*hyper((-1 /2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*gamma(11/4)) + C*a**(...
\[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*(-c*x^4 + a)^(3/2)*(e*x^2 + d)^2, x)
\[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*(-c*x^4 + a)^(3/2)*(e*x^2 + d)^2, x)
Timed out. \[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int {\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(3/2)*(d + e*x^2)^2*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(3/2)*(d + e*x^2)^2*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right )^2 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {-3120 \sqrt {-c \,x^{4}+a}\, a^{3} e^{2} x -4680 \sqrt {-c \,x^{4}+a}\, a^{2} b d e x -924 \sqrt {-c \,x^{4}+a}\, a^{2} b \,e^{2} x^{3}+16965 \sqrt {-c \,x^{4}+a}\, a^{2} c \,d^{2} x +20174 \sqrt {-c \,x^{4}+a}\, a^{2} c d e \,x^{3}+7137 \sqrt {-c \,x^{4}+a}\, a^{2} c \,e^{2} x^{5}+11011 \sqrt {-c \,x^{4}+a}\, a b c \,d^{2} x^{3}+15210 \sqrt {-c \,x^{4}+a}\, a b c d e \,x^{5}+5775 \sqrt {-c \,x^{4}+a}\, a b c \,e^{2} x^{7}+1170 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d^{2} x^{5}+1540 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d e \,x^{7}+546 \sqrt {-c \,x^{4}+a}\, a \,c^{2} e^{2} x^{9}-5005 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d^{2} x^{7}-8190 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d e \,x^{9}-3465 \sqrt {-c \,x^{4}+a}\, b \,c^{2} e^{2} x^{11}-4095 \sqrt {-c \,x^{4}+a}\, c^{3} d^{2} x^{9}-6930 \sqrt {-c \,x^{4}+a}\, c^{3} d e \,x^{11}-3003 \sqrt {-c \,x^{4}+a}\, c^{3} e^{2} x^{13}+3120 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{4} e^{2}+4680 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} b d e +28080 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} c \,d^{2}+2772 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{3} b \,e^{2}+29568 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{3} c d e +12012 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} b c \,d^{2}}{45045 c} \] Input:
int((e*x^2+d)^2*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x)
Output:
( - 3120*sqrt(a - c*x**4)*a**3*e**2*x - 4680*sqrt(a - c*x**4)*a**2*b*d*e*x - 924*sqrt(a - c*x**4)*a**2*b*e**2*x**3 + 16965*sqrt(a - c*x**4)*a**2*c*d **2*x + 20174*sqrt(a - c*x**4)*a**2*c*d*e*x**3 + 7137*sqrt(a - c*x**4)*a** 2*c*e**2*x**5 + 11011*sqrt(a - c*x**4)*a*b*c*d**2*x**3 + 15210*sqrt(a - c* x**4)*a*b*c*d*e*x**5 + 5775*sqrt(a - c*x**4)*a*b*c*e**2*x**7 + 1170*sqrt(a - c*x**4)*a*c**2*d**2*x**5 + 1540*sqrt(a - c*x**4)*a*c**2*d*e*x**7 + 546* sqrt(a - c*x**4)*a*c**2*e**2*x**9 - 5005*sqrt(a - c*x**4)*b*c**2*d**2*x**7 - 8190*sqrt(a - c*x**4)*b*c**2*d*e*x**9 - 3465*sqrt(a - c*x**4)*b*c**2*e* *2*x**11 - 4095*sqrt(a - c*x**4)*c**3*d**2*x**9 - 6930*sqrt(a - c*x**4)*c* *3*d*e*x**11 - 3003*sqrt(a - c*x**4)*c**3*e**2*x**13 + 3120*int(sqrt(a - c *x**4)/(a - c*x**4),x)*a**4*e**2 + 4680*int(sqrt(a - c*x**4)/(a - c*x**4), x)*a**3*b*d*e + 28080*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*c*d**2 + 2 772*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**3*b*e**2 + 29568*int((s qrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**3*c*d*e + 12012*int((sqrt(a - c*x **4)*x**2)/(a - c*x**4),x)*a**2*b*c*d**2)/(45045*c)