Integrand size = 34, antiderivative size = 577 \[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (65 \left (11 A c d \left (7 c d^2+3 a e^2\right )+a \left (5 a e^2 (3 C d+B e)+11 c d^2 (C d+3 B e)\right )\right )+77 \left (39 B c d \left (c d^2+a e^2\right )+e \left (13 A c \left (9 c d^2+a e^2\right )+a C \left (39 c d^2+7 a e^2\right )\right )\right ) x^2\right ) \sqrt {a-c x^4}}{15015 c^2}-\frac {\left (5 a e^2 (3 C d+B e)+11 c d \left (C d^2+3 e (B d+A e)\right )\right ) x \left (a-c x^4\right )^{3/2}}{77 c^2}-\frac {e \left (7 a C e^2+13 c \left (3 C d^2+e (3 B d+A e)\right )\right ) x^3 \left (a-c x^4\right )^{3/2}}{117 c^2}-\frac {e^2 (3 C d+B e) x^5 \left (a-c x^4\right )^{3/2}}{11 c}-\frac {C e^3 x^7 \left (a-c x^4\right )^{3/2}}{13 c}+\frac {2 a^{7/4} \left (39 B c d \left (c d^2+a e^2\right )+e \left (13 A c \left (9 c d^2+a e^2\right )+a C \left (39 c d^2+7 a e^2\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{195 c^{11/4} \sqrt {a-c x^4}}+\frac {2 a^{5/4} \left (65 \sqrt {c} \left (11 A c d \left (7 c d^2+3 a e^2\right )+a \left (5 a e^2 (3 C d+B e)+11 c d^2 (C d+3 B e)\right )\right )-77 \sqrt {a} \left (39 B c d \left (c d^2+a e^2\right )+e \left (13 A c \left (9 c d^2+a e^2\right )+a C \left (39 c d^2+7 a e^2\right )\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15015 c^{11/4} \sqrt {a-c x^4}} \] Output:
1/15015*x*(715*A*c*d*(3*a*e^2+7*c*d^2)+65*a*(5*a*e^2*(B*e+3*C*d)+11*c*d^2* (3*B*e+C*d))+77*(39*B*c*d*(a*e^2+c*d^2)+e*(13*A*c*(a*e^2+9*c*d^2)+a*C*(7*a *e^2+39*c*d^2)))*x^2)*(-c*x^4+a)^(1/2)/c^2-1/77*(5*a*e^2*(B*e+3*C*d)+11*c* d*(C*d^2+3*e*(A*e+B*d)))*x*(-c*x^4+a)^(3/2)/c^2-1/117*e*(7*C*a*e^2+13*c*(3 *C*d^2+e*(A*e+3*B*d)))*x^3*(-c*x^4+a)^(3/2)/c^2-1/11*e^2*(B*e+3*C*d)*x^5*( -c*x^4+a)^(3/2)/c-1/13*C*e^3*x^7*(-c*x^4+a)^(3/2)/c+2/195*a^(7/4)*(39*B*c* d*(a*e^2+c*d^2)+e*(13*A*c*(a*e^2+9*c*d^2)+a*C*(7*a*e^2+39*c*d^2)))*(1-c*x^ 4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(11/4)/(-c*x^4+a)^(1/2)+2/1501 5*a^(5/4)*(65*c^(1/2)*(11*A*c*d*(3*a*e^2+7*c*d^2)+a*(5*a*e^2*(B*e+3*C*d)+1 1*c*d^2*(3*B*e+C*d)))-77*a^(1/2)*(39*B*c*d*(a*e^2+c*d^2)+e*(13*A*c*(a*e^2+ 9*c*d^2)+a*C*(7*a*e^2+39*c*d^2))))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a ^(1/4),I)/c^(11/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.56 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.80 \[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-c x^4} \left (-1287 c d \left (C d^2+3 e (B d+A e)\right ) \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}-1001 c e \left (3 C d^2+e (3 B d+A e)\right ) x^2 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}-819 c e^2 (3 C d+B e) x^4 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}-693 c C e^3 x^6 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}+9009 A c^2 d^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+1287 a c d \left (C d^2+3 e (B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )-585 a e^2 (3 C d+B e) \left (\left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )\right )+3003 c^2 d^2 (B d+3 A e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )+1001 a c e \left (3 C d^2+e (3 B d+A e)\right ) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )-539 a C e^3 x^2 \left (\left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}}-a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )\right )}{9009 c^2 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)^3*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - c*x^4]*(-1287*c*d*(C*d^2 + 3*e*(B*d + A*e))*(a - c*x^4)*Sqrt[1 - (c*x^4)/a] - 1001*c*e*(3*C*d^2 + e*(3*B*d + A*e))*x^2*(a - c*x^4)*Sqrt[ 1 - (c*x^4)/a] - 819*c*e^2*(3*C*d + B*e)*x^4*(a - c*x^4)*Sqrt[1 - (c*x^4)/ a] - 693*c*C*e^3*x^6*(a - c*x^4)*Sqrt[1 - (c*x^4)/a] + 9009*A*c^2*d^3*Hype rgeometric2F1[-1/2, 1/4, 5/4, (c*x^4)/a] + 1287*a*c*d*(C*d^2 + 3*e*(B*d + A*e))*Hypergeometric2F1[-1/2, 1/4, 5/4, (c*x^4)/a] - 585*a*e^2*(3*C*d + B* e)*((a - c*x^4)*Sqrt[1 - (c*x^4)/a] - a*Hypergeometric2F1[-1/2, 1/4, 5/4, (c*x^4)/a]) + 3003*c^2*d^2*(B*d + 3*A*e)*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, (c*x^4)/a] + 1001*a*c*e*(3*C*d^2 + e*(3*B*d + A*e))*x^2*Hypergeometri c2F1[-1/2, 3/4, 7/4, (c*x^4)/a] - 539*a*C*e^3*x^2*((a - c*x^4)*Sqrt[1 - (c *x^4)/a] - a*Hypergeometric2F1[-1/2, 3/4, 7/4, (c*x^4)/a])))/(9009*c^2*Sqr t[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(1401\) vs. \(2(577)=1154\).
Time = 1.84 (sec) , antiderivative size = 1401, normalized size of antiderivative = 2.43, 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.
| \(\displaystyle \int \sqrt {a-c x^4} \left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^8 \left (a e^2 (B e+3 C d)-c \left (3 d e (A e+B d)+C d^3\right )\right )}{\sqrt {a-c x^4}}+\frac {d x^4 \left (a d (3 B e+C d)-A \left (c d^2-3 a e^2\right )\right )}{\sqrt {a-c x^4}}+\frac {e x^{10} \left (a C e^2-c e (A e+3 B d)-3 c C d^2\right )}{\sqrt {a-c x^4}}-\frac {x^6 \left (-a A e^3-3 a B d e^2-3 a C d^2 e+3 A c d^2 e+B c d^3\right )}{\sqrt {a-c x^4}}+\frac {a d^2 x^2 (3 A e+B d)}{\sqrt {a-c x^4}}+\frac {a A d^3}{\sqrt {a-c x^4}}-\frac {c e^2 x^{12} (B e+3 C d)}{\sqrt {a-c x^4}}-\frac {c C e^3 x^{14}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{13} C e^3 \sqrt {a-c x^4} x^{11}+\frac {1}{11} e^2 (3 C d+B e) \sqrt {a-c x^4} x^9+\frac {11 a C e^3 \sqrt {a-c x^4} x^7}{117 c}-\frac {e \left (a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) \sqrt {a-c x^4} x^7}{9 c}+\frac {9 a e^2 (3 C d+B e) \sqrt {a-c x^4} x^5}{77 c}-\frac {\left (a e^2 (3 C d+B e)-c \left (C d^3+3 e (B d+A e) d\right )\right ) \sqrt {a-c x^4} x^5}{7 c}+\frac {77 a^2 C e^3 \sqrt {a-c x^4} x^3}{585 c^2}+\frac {\left (B c d^3+3 A c e d^2-3 a C e d^2-3 a B e^2 d-a A e^3\right ) \sqrt {a-c x^4} x^3}{5 c}-\frac {7 a e \left (a C e^2-c \left (3 C d^2+e (3 B d+A e)\right )\right ) \sqrt {a-c x^4} x^3}{45 c^2}+\frac {15 a^2 e^2 (3 C d+B e) \sqrt {a-c x^4} x}{77 c^2}-\frac {d \left (a d (C d+3 B e)-A \left (c d^2-3 a e^2\right )\right ) \sqrt {a-c x^4} x}{3 c}-\frac {5 a \left (a e^2 (3 C d+B e)-c \left (C d^3+3 e (B d+A e) d\right )\right ) \sqrt {a-c x^4} x}{21 c^2}-\frac {77 a^{15/4} 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 )}{195 c^{11/4} \sqrt {a-c x^4}}+\frac {a^{7/4} d^2 (B d+3 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 {3 a^{7/4} \left (B c d^3+3 A c e d^2-3 a C e d^2-3 a B e^2 d-a A e^3\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 {7 a^{11/4} e \left (a C e^2-c \left (3 C d^2+e (3 B d+A e)\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 )}{15 c^{11/4} \sqrt {a-c x^4}}+\frac {a^{5/4} A d^3 \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 {77 a^{15/4} 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 )}{195 c^{11/4} \sqrt {a-c x^4}}-\frac {a^{7/4} d^2 (B d+3 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 {15 a^{13/4} e^2 (3 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 )}{77 c^{9/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} \left (B c d^3+3 A c e d^2-3 a C e d^2-3 a B e^2 d-a A e^3\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 {a^{5/4} d \left (a d (C d+3 B e)-A \left (c d^2-3 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 e^2 (3 C d+B e)-c \left (C d^3+3 e (B d+A e) d\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}}-\frac {7 a^{11/4} e \left (a C e^2-c \left (3 C d^2+e (3 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 )}{15 c^{11/4} \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)^3*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(15*a^2*e^2*(3*C*d + B*e)*x*Sqrt[a - c*x^4])/(77*c^2) - (d*(a*d*(C*d + 3*B *e) - A*(c*d^2 - 3*a*e^2))*x*Sqrt[a - c*x^4])/(3*c) - (5*a*(a*e^2*(3*C*d + B*e) - c*(C*d^3 + 3*d*e*(B*d + A*e)))*x*Sqrt[a - c*x^4])/(21*c^2) + (77*a ^2*C*e^3*x^3*Sqrt[a - c*x^4])/(585*c^2) + ((B*c*d^3 + 3*A*c*d^2*e - 3*a*C* d^2*e - 3*a*B*d*e^2 - a*A*e^3)*x^3*Sqrt[a - c*x^4])/(5*c) - (7*a*e*(a*C*e^ 2 - c*(3*C*d^2 + e*(3*B*d + A*e)))*x^3*Sqrt[a - c*x^4])/(45*c^2) + (9*a*e^ 2*(3*C*d + B*e)*x^5*Sqrt[a - c*x^4])/(77*c) - ((a*e^2*(3*C*d + B*e) - c*(C *d^3 + 3*d*e*(B*d + A*e)))*x^5*Sqrt[a - c*x^4])/(7*c) + (11*a*C*e^3*x^7*Sq rt[a - c*x^4])/(117*c) - (e*(a*C*e^2 - c*(3*C*d^2 + e*(3*B*d + A*e)))*x^7* Sqrt[a - c*x^4])/(9*c) + (e^2*(3*C*d + B*e)*x^9*Sqrt[a - c*x^4])/11 + (C*e ^3*x^11*Sqrt[a - c*x^4])/13 - (77*a^(15/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*Ellip ticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(195*c^(11/4)*Sqrt[a - c*x^4]) + (a ^(7/4)*d^2*(B*d + 3*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]) - (3*a^(7/4)*(B*c*d^3 + 3*A*c*d^2 *e - 3*a*C*d^2*e - 3*a*B*d*e^2 - a*A*e^3)*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar cSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(7/4)*Sqrt[a - c*x^4]) + (7*a^(11/4)* e*(a*C*e^2 - c*(3*C*d^2 + e*(3*B*d + A*e)))*Sqrt[1 - (c*x^4)/a]*EllipticE[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(11/4)*Sqrt[a - c*x^4]) + (a^(5/4) *A*d^3*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^ (1/4)*Sqrt[a - c*x^4]) + (77*a^(15/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*Ellipti...
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 = 4.92 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.29
| method | result | size |
| elliptic | \(\frac {C \,e^{3} x^{11} \sqrt {-c \,x^{4}+a}}{13}-\frac {\left (-B \,e^{3} c -3 d \,e^{2} c C \right ) x^{9} \sqrt {-c \,x^{4}+a}}{11 c}-\frac {\left (-A c \,e^{3}-3 B c d \,e^{2}+\frac {2}{13} C \,e^{3} a -3 e C c \,d^{2}\right ) x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {\left (-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}+\frac {9 \left (-B \,e^{3} c -3 d \,e^{2} c C \right ) a}{11 c}\right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}+3 C a \,d^{2} e +\frac {7 \left (-A c \,e^{3}-3 B c d \,e^{2}+\frac {2}{13} C \,e^{3} a -3 e C c \,d^{2}\right ) a}{9 c}\right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (3 a d A \,e^{2}-A \,d^{3} c +3 B a \,d^{2} e +C a \,d^{3}+\frac {5 \left (-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}+\frac {9 \left (-B \,e^{3} c -3 d \,e^{2} c C \right ) a}{11 c}\right ) a}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A a \,d^{3}+\frac {\left (3 a d A \,e^{2}-A \,d^{3} c +3 B a \,d^{2} e +C a \,d^{3}+\frac {5 \left (-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}+\frac {9 \left (-B \,e^{3} c -3 d \,e^{2} c C \right ) a}{11 c}\right ) a}{7 c}\right ) a}{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 (3 A a \,d^{2} e +B a \,d^{3}+\frac {3 \left (A a \,e^{3}-3 A c \,d^{2} e +3 B a d \,e^{2}-B c \,d^{3}+3 C a \,d^{2} e +\frac {7 \left (-A c \,e^{3}-3 B c d \,e^{2}+\frac {2}{13} C \,e^{3} a -3 e C c \,d^{2}\right ) a}{9 c}\right ) a}{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}}\) | \(742\) |
| default | \(A \,d^{3} \left (\frac {x \sqrt {-c \,x^{4}+a}}{3}+\frac {2 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 )}{3 \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} \sqrt {-c \,x^{4}+a}}{5}-\frac {2 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 )}{5 \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 {x^{9} \sqrt {-c \,x^{4}+a}}{11}-\frac {2 a \,x^{5} \sqrt {-c \,x^{4}+a}}{77 c}-\frac {10 a^{2} x \sqrt {-c \,x^{4}+a}}{231 c^{2}}+\frac {10 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 )}{231 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^{7} \sqrt {-c \,x^{4}+a}}{9}-\frac {2 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45 c}-\frac {2 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 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^{5} \sqrt {-c \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-c \,x^{4}+a}}{21 c}+\frac {2 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 )}{21 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C \,e^{3} \left (\frac {x^{11} \sqrt {-c \,x^{4}+a}}{13}-\frac {2 a \,x^{7} \sqrt {-c \,x^{4}+a}}{117 c}-\frac {14 a^{2} x^{3} \sqrt {-c \,x^{4}+a}}{585 c^{2}}-\frac {14 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 )}{195 c^{\frac {5}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(748\) |
| risch | \(-\frac {x \left (-3465 C \,e^{3} x^{10} c^{2}-4095 B \,c^{2} e^{3} x^{8}-12285 C \,c^{2} d \,e^{2} x^{8}-5005 A \,c^{2} e^{3} x^{6}-15015 B \,c^{2} d \,e^{2} x^{6}+770 C \,e^{3} a \,x^{6} c -15015 C \,c^{2} d^{2} e \,x^{6}-19305 A \,c^{2} d \,e^{2} x^{4}+1170 B a c \,e^{3} x^{4}-19305 B \,c^{2} d^{2} e \,x^{4}+3510 C a c d \,e^{2} x^{4}-6435 C \,c^{2} d^{3} x^{4}+2002 A a c \,e^{3} x^{2}-27027 A \,c^{2} d^{2} e \,x^{2}+6006 B a c d \,e^{2} x^{2}-9009 B \,c^{2} d^{3} x^{2}+1078 C \,a^{2} e^{3} x^{2}+6006 C a c \,d^{2} e \,x^{2}+12870 A a c d \,e^{2}-15015 A \,c^{2} d^{3}+1950 a^{2} B \,e^{3}+12870 B a c \,d^{2} e +5850 C \,a^{2} d \,e^{2}+4290 C a c \,d^{3}\right ) \sqrt {-c \,x^{4}+a}}{45045 c^{2}}+\frac {2 a \left (-\frac {\left (1001 A a c \,e^{3}+9009 A \,d^{2} e \,c^{2}+3003 B a c d \,e^{2}+3003 B \,d^{3} c^{2}+539 C \,a^{2} e^{3}+3003 C a c \,d^{2} 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}\, \sqrt {c}}+\frac {5005 A \,c^{2} 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 {325 a^{2} B \,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 {715 C a 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 {975 C \,a^{2} 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 {2145 A 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 {2145 B a 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}}\right )}{15015 c^{2}}\) | \(867\) |
Input:
int((e*x^2+d)^3*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/13*C*e^3*x^11*(-c*x^4+a)^(1/2)-1/11*(-B*c*e^3-3*C*c*d*e^2)/c*x^9*(-c*x^4 +a)^(1/2)-1/9*(-A*c*e^3-3*B*c*d*e^2+2/13*C*e^3*a-3*e*C*c*d^2)/c*x^7*(-c*x^ 4+a)^(1/2)-1/7*(-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+9/11* (-B*c*e^3-3*C*c*d*e^2)/c*a)/c*x^5*(-c*x^4+a)^(1/2)-1/5*(A*a*e^3-3*A*c*d^2* e+3*B*a*d*e^2-B*c*d^3+3*C*a*d^2*e+7/9*(-A*c*e^3-3*B*c*d*e^2+2/13*C*e^3*a-3 *e*C*c*d^2)/c*a)/c*x^3*(-c*x^4+a)^(1/2)-1/3*(3*a*d*A*e^2-A*d^3*c+3*B*a*d^2 *e+C*a*d^3+5/7*(-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+9/11* (-B*c*e^3-3*C*c*d*e^2)/c*a)/c*a)/c*x*(-c*x^4+a)^(1/2)+(A*a*d^3+1/3*(3*a*d* A*e^2-A*d^3*c+3*B*a*d^2*e+C*a*d^3+5/7*(-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+9/11*(-B*c*e^3-3*C*c*d*e^2)/c*a)/c*a)/c*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)-(3*A*a*d^2*e+B*a*d^3+ 3/5*(A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2-B*c*d^3+3*C*a*d^2*e+7/9*(-A*c*e^3-3*B *c*d*e^2+2/13*C*e^3*a-3*e*C*c*d^2)/c*a)/c*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 = 0.09 (sec) , antiderivative size = 570, normalized size of antiderivative = 0.99 \[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=-\frac {462 \, {\left (39 \, B a c^{2} d^{3} + 39 \, B a^{2} c d e^{2} + 39 \, {\left (C a^{2} c + 3 \, A a c^{2}\right )} d^{2} e + {\left (7 \, C a^{3} + 13 \, A a^{2} c\right )} e^{3}\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) - 6 \, {\left (143 \, {\left ({\left (21 \, B + 5 \, C\right )} a c^{2} + 35 \, A c^{3}\right )} d^{3} + 429 \, {\left (7 \, C a^{2} c + {\left (21 \, A + 5 \, B\right )} a c^{2}\right )} d^{2} e + 39 \, {\left ({\left (77 \, B + 25 \, C\right )} a^{2} c + 55 \, A a c^{2}\right )} d e^{2} + {\left (539 \, C a^{3} + 13 \, {\left (77 \, A + 25 \, B\right )} a^{2} c\right )} e^{3}\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 (3465 \, C c^{3} e^{3} x^{12} + 4095 \, {\left (3 \, C c^{3} d e^{2} + B c^{3} e^{3}\right )} x^{10} + 385 \, {\left (39 \, C c^{3} d^{2} e + 39 \, B c^{3} d e^{2} - {\left (2 \, C a c^{2} - 13 \, A c^{3}\right )} e^{3}\right )} x^{8} - 18018 \, B a c^{2} d^{3} - 18018 \, B a^{2} c d e^{2} + 585 \, {\left (11 \, C c^{3} d^{3} + 33 \, B c^{3} d^{2} e - 2 \, B a c^{2} e^{3} - 3 \, {\left (2 \, C a c^{2} - 11 \, A c^{3}\right )} d e^{2}\right )} x^{6} + 77 \, {\left (117 \, B c^{3} d^{3} - 78 \, B a c^{2} d e^{2} - 39 \, {\left (2 \, C a c^{2} - 9 \, A c^{3}\right )} d^{2} e - 2 \, {\left (7 \, C a^{2} c + 13 \, A a c^{2}\right )} e^{3}\right )} x^{4} - 18018 \, {\left (C a^{2} c + 3 \, A a c^{2}\right )} d^{2} e - 462 \, {\left (7 \, C a^{3} + 13 \, A a^{2} c\right )} e^{3} - 195 \, {\left (66 \, B a c^{2} d^{2} e + 10 \, B a^{2} c e^{3} + 11 \, {\left (2 \, C a c^{2} - 7 \, A c^{3}\right )} d^{3} + 6 \, {\left (5 \, C a^{2} c + 11 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{45045 \, c^{3} x} \] Input:
integrate((e*x^2+d)^3*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="frica s")
Output:
-1/45045*(462*(39*B*a*c^2*d^3 + 39*B*a^2*c*d*e^2 + 39*(C*a^2*c + 3*A*a*c^2 )*d^2*e + (7*C*a^3 + 13*A*a^2*c)*e^3)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_e(ar csin((a/c)^(1/4)/x), -1) - 6*(143*((21*B + 5*C)*a*c^2 + 35*A*c^3)*d^3 + 42 9*(7*C*a^2*c + (21*A + 5*B)*a*c^2)*d^2*e + 39*((77*B + 25*C)*a^2*c + 55*A* a*c^2)*d*e^2 + (539*C*a^3 + 13*(77*A + 25*B)*a^2*c)*e^3)*sqrt(-c)*x*(a/c)^ (3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) - (3465*C*c^3*e^3*x^12 + 4095* (3*C*c^3*d*e^2 + B*c^3*e^3)*x^10 + 385*(39*C*c^3*d^2*e + 39*B*c^3*d*e^2 - (2*C*a*c^2 - 13*A*c^3)*e^3)*x^8 - 18018*B*a*c^2*d^3 - 18018*B*a^2*c*d*e^2 + 585*(11*C*c^3*d^3 + 33*B*c^3*d^2*e - 2*B*a*c^2*e^3 - 3*(2*C*a*c^2 - 11*A *c^3)*d*e^2)*x^6 + 77*(117*B*c^3*d^3 - 78*B*a*c^2*d*e^2 - 39*(2*C*a*c^2 - 9*A*c^3)*d^2*e - 2*(7*C*a^2*c + 13*A*a*c^2)*e^3)*x^4 - 18018*(C*a^2*c + 3* A*a*c^2)*d^2*e - 462*(7*C*a^3 + 13*A*a^2*c)*e^3 - 195*(66*B*a*c^2*d^2*e + 10*B*a^2*c*e^3 + 11*(2*C*a*c^2 - 7*A*c^3)*d^3 + 6*(5*C*a^2*c + 11*A*a*c^2) *d*e^2)*x^2)*sqrt(-c*x^4 + a))/(c^3*x)
Time = 5.03 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)**3*(-c*x**4+a)**(1/2)*(C*x**4+B*x**2+A),x)
Output:
A*sqrt(a)*d**3*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), c*x**4*exp_polar(2* I*pi)/a)/(4*gamma(5/4)) + 3*A*sqrt(a)*d**2*e*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)) + 3*A*sqrt(a)*d*e **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)*e**3*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)*d**3*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) ) + 3*B*sqrt(a)*d**2*e*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*e xp_polar(2*I*pi)/a)/(4*gamma(9/4)) + 3*B*sqrt(a)*d*e**2*x**7*gamma(7/4)*hy per((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(11/4)) + B* sqrt(a)*e**3*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**4*exp_polar( 2*I*pi)/a)/(4*gamma(13/4)) + C*sqrt(a)*d**3*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)) + 3*C*sqrt(a)*d**2 *e*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)) + 3*C*sqrt(a)*d*e**2*x**9*gamma(9/4)*hyper((-1/2, 9/4), ( 13/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(13/4)) + C*sqrt(a)*e**3*x**11 *gamma(11/4)*hyper((-1/2, 11/4), (15/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*g amma(15/4))
\[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*(e*x^2 + d)^3, x)
\[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(-c*x^4 + a)*(e*x^2 + d)^3, x)
Timed out. \[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int \sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^3\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(1/2)*(d + e*x^2)^3*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(1/2)*(d + e*x^2)^3*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right )^3 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {1950 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} b \,e^{3}+34320 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} c^{2} d^{3}+9240 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{3} c \,e^{3}-18720 \sqrt {-c \,x^{4}+a}\, a^{2} c d \,e^{2} x -1170 \sqrt {-c \,x^{4}+a}\, a b c \,e^{3} x^{5}+21021 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d^{2} e \,x^{3}+6435 \sqrt {-c \,x^{4}+a}\, c^{3} d^{3} x^{5}+3465 \sqrt {-c \,x^{4}+a}\, c^{3} e^{3} x^{11}+15015 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d \,e^{2} x^{7}+18720 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} c d \,e^{2}+72072 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} c^{2} d^{2} e +18018 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a b \,c^{2} d^{3}-1950 \sqrt {-c \,x^{4}+a}\, a^{2} b \,e^{3} x -3080 \sqrt {-c \,x^{4}+a}\, a^{2} c \,e^{3} x^{3}+10725 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d^{3} x +4235 \sqrt {-c \,x^{4}+a}\, a \,c^{2} e^{3} x^{7}+9009 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d^{3} x^{3}+4095 \sqrt {-c \,x^{4}+a}\, b \,c^{2} e^{3} x^{9}+15015 \sqrt {-c \,x^{4}+a}\, c^{3} d^{2} e \,x^{7}+12285 \sqrt {-c \,x^{4}+a}\, c^{3} d \,e^{2} x^{9}+18018 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} b c d \,e^{2}+12870 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} b c \,d^{2} e +15795 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d \,e^{2} x^{5}+19305 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d^{2} e \,x^{5}-12870 \sqrt {-c \,x^{4}+a}\, a b c \,d^{2} e x -6006 \sqrt {-c \,x^{4}+a}\, a b c d \,e^{2} x^{3}}{45045 c^{2}} \] Input:
int((e*x^2+d)^3*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x)
Output:
( - 1950*sqrt(a - c*x**4)*a**2*b*e**3*x - 18720*sqrt(a - c*x**4)*a**2*c*d* e**2*x - 3080*sqrt(a - c*x**4)*a**2*c*e**3*x**3 - 12870*sqrt(a - c*x**4)*a *b*c*d**2*e*x - 6006*sqrt(a - c*x**4)*a*b*c*d*e**2*x**3 - 1170*sqrt(a - c* x**4)*a*b*c*e**3*x**5 + 10725*sqrt(a - c*x**4)*a*c**2*d**3*x + 21021*sqrt( a - c*x**4)*a*c**2*d**2*e*x**3 + 15795*sqrt(a - c*x**4)*a*c**2*d*e**2*x**5 + 4235*sqrt(a - c*x**4)*a*c**2*e**3*x**7 + 9009*sqrt(a - c*x**4)*b*c**2*d **3*x**3 + 19305*sqrt(a - c*x**4)*b*c**2*d**2*e*x**5 + 15015*sqrt(a - c*x* *4)*b*c**2*d*e**2*x**7 + 4095*sqrt(a - c*x**4)*b*c**2*e**3*x**9 + 6435*sqr t(a - c*x**4)*c**3*d**3*x**5 + 15015*sqrt(a - c*x**4)*c**3*d**2*e*x**7 + 1 2285*sqrt(a - c*x**4)*c**3*d*e**2*x**9 + 3465*sqrt(a - c*x**4)*c**3*e**3*x **11 + 1950*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*b*e**3 + 18720*int(s qrt(a - c*x**4)/(a - c*x**4),x)*a**3*c*d*e**2 + 12870*int(sqrt(a - c*x**4) /(a - c*x**4),x)*a**2*b*c*d**2*e + 34320*int(sqrt(a - c*x**4)/(a - c*x**4) ,x)*a**2*c**2*d**3 + 9240*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**3 *c*e**3 + 18018*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*b*c*d*e** 2 + 72072*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*c**2*d**2*e + 1 8018*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a*b*c**2*d**3)/(45045*c** 2)