Integrand size = 34, antiderivative size = 417 \[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (5 \left (11 A c \left (7 c d^2+a e^2\right )+a \left (5 a C e^2+11 c d (C d+2 B e)\right )\right )+77 c \left (3 B c d^2+6 A c d e+2 a C d e+a B e^2\right ) x^2\right ) \sqrt {a-c x^4}}{1155 c^2}-\frac {\left (5 a C e^2+11 c \left (C d^2+e (2 B d+A e)\right )\right ) x \left (a-c x^4\right )^{3/2}}{77 c^2}-\frac {e (2 C d+B e) x^3 \left (a-c x^4\right )^{3/2}}{9 c}-\frac {C e^2 x^5 \left (a-c x^4\right )^{3/2}}{11 c}+\frac {2 a^{7/4} \left (3 B c d^2+6 A c d e+2 a C d e+a B e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{7/4} \sqrt {a-c x^4}}-\frac {2 a^{5/4} \left (77 \sqrt {a} \sqrt {c} \left (3 B c d^2+6 A c d e+2 a C d e+a B e^2\right )-5 \left (11 A c \left (7 c d^2+a e^2\right )+a \left (5 a C e^2+11 c d (C d+2 B e)\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{1155 c^{9/4} \sqrt {a-c x^4}} \] Output:
1/1155*x*(55*A*c*(a*e^2+7*c*d^2)+5*a*(5*C*a*e^2+11*c*d*(2*B*e+C*d))+77*c*( 6*A*c*d*e+B*a*e^2+3*B*c*d^2+2*C*a*d*e)*x^2)*(-c*x^4+a)^(1/2)/c^2-1/77*(5*C *a*e^2+11*c*(C*d^2+e*(A*e+2*B*d)))*x*(-c*x^4+a)^(3/2)/c^2-1/9*e*(B*e+2*C*d )*x^3*(-c*x^4+a)^(3/2)/c-1/11*C*e^2*x^5*(-c*x^4+a)^(3/2)/c+2/15*a^(7/4)*(6 *A*c*d*e+B*a*e^2+3*B*c*d^2+2*C*a*d*e)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)* x/a^(1/4),I)/c^(7/4)/(-c*x^4+a)^(1/2)-2/1155*a^(5/4)*(77*a^(1/2)*c^(1/2)*( 6*A*c*d*e+B*a*e^2+3*B*c*d^2+2*C*a*d*e)-55*A*c*(a*e^2+7*c*d^2)-5*a*(5*C*a*e ^2+11*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 = 10.47 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.53 \[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-c x^4} \left (-\left (\left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \left (45 a C e^2+11 c e \left (18 B d+9 A e+7 B e x^2\right )+c C \left (99 d^2+154 d e x^2+63 e^2 x^4\right )\right )\right )+9 \left (11 A c \left (7 c d^2+a e^2\right )+a \left (5 a C e^2+11 c d (C d+2 B e)\right )\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+77 c \left (3 B c d^2+6 A c d e+2 a C d e+a B e^2\right ) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{693 c^2 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)^2*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - c*x^4]*(-((a - c*x^4)*Sqrt[1 - (c*x^4)/a]*(45*a*C*e^2 + 11*c*e *(18*B*d + 9*A*e + 7*B*e*x^2) + c*C*(99*d^2 + 154*d*e*x^2 + 63*e^2*x^4))) + 9*(11*A*c*(7*c*d^2 + a*e^2) + a*(5*a*C*e^2 + 11*c*d*(C*d + 2*B*e)))*Hype rgeometric2F1[-1/2, 1/4, 5/4, (c*x^4)/a] + 77*c*(3*B*c*d^2 + 6*A*c*d*e + 2 *a*C*d*e + a*B*e^2)*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, (c*x^4)/a]))/(69 3*c^2*Sqrt[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(1029\) vs. \(2(417)=834\).
Time = 1.28 (sec) , antiderivative size = 1029, normalized size of antiderivative = 2.47, 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.
| \(\displaystyle \int \sqrt {a-c x^4} \left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^4 \left (a d (2 B e+C d)-A \left (c d^2-a e^2\right )\right )}{\sqrt {a-c x^4}}+\frac {x^8 \left (a C e^2-c \left (e (A e+2 B d)+C d^2\right )\right )}{\sqrt {a-c x^4}}-\frac {x^6 \left (-a B e^2-2 a C d e+2 A c d e+B c d^2\right )}{\sqrt {a-c x^4}}+\frac {a d x^2 (2 A e+B d)}{\sqrt {a-c x^4}}+\frac {a A d^2}{\sqrt {a-c x^4}}-\frac {c e x^{10} (B e+2 C d)}{\sqrt {a-c x^4}}-\frac {c C e^2 x^{12}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{11} C e^2 \sqrt {a-c x^4} x^9+\frac {1}{9} e (2 C d+B e) \sqrt {a-c x^4} x^7+\frac {9 a C e^2 \sqrt {a-c x^4} x^5}{77 c}-\frac {\left (a C e^2-c \left (C d^2+e (2 B d+A e)\right )\right ) \sqrt {a-c x^4} x^5}{7 c}+\frac {7 a e (2 C d+B e) \sqrt {a-c x^4} x^3}{45 c}+\frac {\left (B c d^2+2 A c e d-2 a C e d-a B e^2\right ) \sqrt {a-c x^4} x^3}{5 c}+\frac {15 a^2 C e^2 \sqrt {a-c x^4} x}{77 c^2}-\frac {\left (a d (C d+2 B e)-A \left (c d^2-a e^2\right )\right ) \sqrt {a-c x^4} x}{3 c}-\frac {5 a \left (a C e^2-c \left (C d^2+e (2 B d+A e)\right )\right ) \sqrt {a-c x^4} x}{21 c^2}+\frac {a^{7/4} 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 {7 a^{11/4} 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 )}{15 c^{7/4} \sqrt {a-c x^4}}-\frac {3 a^{7/4} \left (B c d^2+2 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^{5/4} 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 {15 a^{13/4} 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^{7/4} 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 {7 a^{11/4} 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 )}{15 c^{7/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} \left (B c d^2+2 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 {a^{5/4} \left (a d (C d+2 B e)-A \left (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 e^2-c \left (C d^2+e (2 B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{21 c^{9/4} \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)^2*Sqrt[a - c*x^4]*(A + B*x^2 + C*x^4),x]
Output:
(15*a^2*C*e^2*x*Sqrt[a - c*x^4])/(77*c^2) - ((a*d*(C*d + 2*B*e) - A*(c*d^2 - a*e^2))*x*Sqrt[a - c*x^4])/(3*c) - (5*a*(a*C*e^2 - c*(C*d^2 + e*(2*B*d + A*e)))*x*Sqrt[a - c*x^4])/(21*c^2) + (7*a*e*(2*C*d + B*e)*x^3*Sqrt[a - c *x^4])/(45*c) + ((B*c*d^2 + 2*A*c*d*e - 2*a*C*d*e - a*B*e^2)*x^3*Sqrt[a - c*x^4])/(5*c) + (9*a*C*e^2*x^5*Sqrt[a - c*x^4])/(77*c) - ((a*C*e^2 - c*(C* d^2 + e*(2*B*d + A*e)))*x^5*Sqrt[a - c*x^4])/(7*c) + (e*(2*C*d + B*e)*x^7* Sqrt[a - c*x^4])/9 + (C*e^2*x^9*Sqrt[a - c*x^4])/11 + (a^(7/4)*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]) - (7*a^(11/4)*e*(2*C*d + B*e)*Sqrt[1 - (c*x^4)/a]*Ell ipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(7/4)*Sqrt[a - c*x^4]) - (3 *a^(7/4)*(B*c*d^2 + 2*A*c*d*e - 2*a*C*d*e - a*B*e^2)*Sqrt[1 - (c*x^4)/a]*E llipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(7/4)*Sqrt[a - c*x^4]) + ( a^(5/4)*A*d^2*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(c^(1/4)*Sqrt[a - c*x^4]) - (15*a^(13/4)*C*e^2*Sqrt[1 - (c*x^4)/a]*Ell ipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(77*c^(9/4)*Sqrt[a - c*x^4]) - (a ^(7/4)*d*(B*d + 2*A*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^ (1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (7*a^(11/4)*e*(2*C*d + B*e)*Sqrt[ 1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(15*c^(7/4)*Sqr t[a - c*x^4]) + (3*a^(7/4)*(B*c*d^2 + 2*A*c*d*e - 2*a*C*d*e - a*B*e^2)*Sqr t[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(7/4)...
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 = 3.10 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {C \,e^{2} x^{9} \sqrt {-c \,x^{4}+a}}{11}-\frac {\left (-B c \,e^{2}-2 C c d e \right ) x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {\left (-A c \,e^{2}-2 B c d e +\frac {2}{11} C a \,e^{2}-C c \,d^{2}\right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (-2 A c d e +B a \,e^{2}-B c \,d^{2}+2 C a d e +\frac {7 a \left (-B c \,e^{2}-2 C c d e \right )}{9 c}\right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (A a \,e^{2}-A c \,d^{2}+2 a B d e +C a \,d^{2}+\frac {5 a \left (-A c \,e^{2}-2 B c d e +\frac {2}{11} C a \,e^{2}-C c \,d^{2}\right )}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A a \,d^{2}+\frac {a \left (A a \,e^{2}-A c \,d^{2}+2 a B d e +C a \,d^{2}+\frac {5 a \left (-A c \,e^{2}-2 B c d e +\frac {2}{11} C a \,e^{2}-C c \,d^{2}\right )}{7 c}\right )}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (2 A a d e +B a \,d^{2}+\frac {3 a \left (-2 A c d e +B a \,e^{2}-B c \,d^{2}+2 C a d e +\frac {7 a \left (-B c \,e^{2}-2 C c d e \right )}{9 c}\right )}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(527\) |
| default | \(A \,d^{2} \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 \left (2 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 \left (B e +2 C d \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 )+\left (A \,e^{2}+2 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^{2} \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 )\) | \(580\) |
| risch | \(-\frac {x \left (-315 C \,c^{2} e^{2} x^{8}-385 B \,c^{2} e^{2} x^{6}-770 C \,c^{2} d e \,x^{6}-495 A \,c^{2} e^{2} x^{4}-990 B \,c^{2} d e \,x^{4}+90 C a c \,e^{2} x^{4}-495 C \,c^{2} d^{2} x^{4}-1386 A \,c^{2} d e \,x^{2}+154 B a c \,e^{2} x^{2}-693 B \,c^{2} d^{2} x^{2}+308 C a d e \,x^{2} c +330 A a c \,e^{2}-1155 A \,c^{2} d^{2}+660 B a c d e +150 a^{2} C \,e^{2}+330 C a c \,d^{2}\right ) \sqrt {-c \,x^{4}+a}}{3465 c^{2}}+\frac {2 a \left (-\frac {77 \sqrt {c}\, \left (6 A c d e +B a \,e^{2}+3 B c \,d^{2}+2 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 {385 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 {25 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 {55 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 {55 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 {110 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 )}{1155 c^{2}}\) | \(656\) |
Input:
int((e*x^2+d)^2*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/11*C*e^2*x^9*(-c*x^4+a)^(1/2)-1/9*(-B*c*e^2-2*C*c*d*e)/c*x^7*(-c*x^4+a)^ (1/2)-1/7*(-A*c*e^2-2*B*c*d*e+2/11*C*a*e^2-C*c*d^2)/c*x^5*(-c*x^4+a)^(1/2) -1/5*(-2*A*c*d*e+B*a*e^2-B*c*d^2+2*C*a*d*e+7/9*a/c*(-B*c*e^2-2*C*c*d*e))/c *x^3*(-c*x^4+a)^(1/2)-1/3*(A*a*e^2-A*c*d^2+2*a*B*d*e+C*a*d^2+5/7*a/c*(-A*c *e^2-2*B*c*d*e+2/11*C*a*e^2-C*c*d^2))/c*x*(-c*x^4+a)^(1/2)+(A*a*d^2+1/3*a/ c*(A*a*e^2-A*c*d^2+2*a*B*d*e+C*a*d^2+5/7*a/c*(-A*c*e^2-2*B*c*d*e+2/11*C*a* e^2-C*c*d^2)))/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^ (1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1 /2),I)-(2*A*a*d*e+B*a*d^2+3/5*a/c*(-2*A*c*d*e+B*a*e^2-B*c*d^2+2*C*a*d*e+7/ 9*a/c*(-B*c*e^2-2*C*c*d*e)))*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)*(E llipticF(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 = 369, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=-\frac {462 \, {\left (3 \, B a c d^{2} + B a^{2} e^{2} + 2 \, {\left (C a^{2} + 3 \, A a c\right )} d e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 6 \, {\left (11 \, {\left ({\left (21 \, B + 5 \, C\right )} a c + 35 \, A c^{2}\right )} d^{2} + 22 \, {\left (7 \, C a^{2} + {\left (21 \, A + 5 \, B\right )} a c\right )} d e + {\left ({\left (77 \, B + 25 \, C\right )} a^{2} + 55 \, A a c\right )} e^{2}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (315 \, C c^{2} e^{2} x^{10} + 385 \, {\left (2 \, C c^{2} d e + B c^{2} e^{2}\right )} x^{8} + 45 \, {\left (11 \, C c^{2} d^{2} + 22 \, B c^{2} d e - {\left (2 \, C a c - 11 \, A c^{2}\right )} e^{2}\right )} x^{6} - 1386 \, B a c d^{2} - 462 \, B a^{2} e^{2} + 77 \, {\left (9 \, B c^{2} d^{2} - 2 \, B a c e^{2} - 2 \, {\left (2 \, C a c - 9 \, A c^{2}\right )} d e\right )} x^{4} - 924 \, {\left (C a^{2} + 3 \, A a c\right )} d e - 15 \, {\left (44 \, B a c d e + 11 \, {\left (2 \, C a c - 7 \, A c^{2}\right )} d^{2} + 2 \, {\left (5 \, C a^{2} + 11 \, A a c\right )} e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{3465 \, c^{2} x} \] Input:
integrate((e*x^2+d)^2*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="frica s")
Output:
-1/3465*(462*(3*B*a*c*d^2 + B*a^2*e^2 + 2*(C*a^2 + 3*A*a*c)*d*e)*sqrt(-c)* x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - 6*(11*((21*B + 5*C)* a*c + 35*A*c^2)*d^2 + 22*(7*C*a^2 + (21*A + 5*B)*a*c)*d*e + ((77*B + 25*C) *a^2 + 55*A*a*c)*e^2)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4) /x), -1) - (315*C*c^2*e^2*x^10 + 385*(2*C*c^2*d*e + B*c^2*e^2)*x^8 + 45*(1 1*C*c^2*d^2 + 22*B*c^2*d*e - (2*C*a*c - 11*A*c^2)*e^2)*x^6 - 1386*B*a*c*d^ 2 - 462*B*a^2*e^2 + 77*(9*B*c^2*d^2 - 2*B*a*c*e^2 - 2*(2*C*a*c - 9*A*c^2)* d*e)*x^4 - 924*(C*a^2 + 3*A*a*c)*d*e - 15*(44*B*a*c*d*e + 11*(2*C*a*c - 7* A*c^2)*d^2 + 2*(5*C*a^2 + 11*A*a*c)*e^2)*x^2)*sqrt(-c*x^4 + a))/(c^2*x)
Time = 3.82 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.02 \[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)**2*(-c*x**4+a)**(1/2)*(C*x**4+B*x**2+A),x)
Output:
A*sqrt(a)*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*sqrt(a)*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*sqrt(a)*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)) + B*sqrt(a)*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*sqrt(a)*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*sqrt(a )*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)) + C*sqrt(a)*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*sqrt(a)*d*e*x**7*gam ma(7/4)*hyper((-1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*gamma(1 1/4)) + C*sqrt(a)*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))
\[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-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)^2, x)
\[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-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)^2, x)
Timed out. \[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\int \sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^2\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(1/2)*(d + e*x^2)^2*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(1/2)*(d + e*x^2)^2*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right )^2 \sqrt {a-c x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {-480 \sqrt {-c \,x^{4}+a}\, a^{2} e^{2} x -660 \sqrt {-c \,x^{4}+a}\, a b d e x -154 \sqrt {-c \,x^{4}+a}\, a b \,e^{2} x^{3}+825 \sqrt {-c \,x^{4}+a}\, a c \,d^{2} x +1078 \sqrt {-c \,x^{4}+a}\, a c d e \,x^{3}+405 \sqrt {-c \,x^{4}+a}\, a c \,e^{2} x^{5}+693 \sqrt {-c \,x^{4}+a}\, b c \,d^{2} x^{3}+990 \sqrt {-c \,x^{4}+a}\, b c d e \,x^{5}+385 \sqrt {-c \,x^{4}+a}\, b c \,e^{2} x^{7}+495 \sqrt {-c \,x^{4}+a}\, c^{2} d^{2} x^{5}+770 \sqrt {-c \,x^{4}+a}\, c^{2} d e \,x^{7}+315 \sqrt {-c \,x^{4}+a}\, c^{2} e^{2} x^{9}+480 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} e^{2}+660 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} b d e +2640 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} c \,d^{2}+462 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} b \,e^{2}+3696 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} c d e +1386 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a b c \,d^{2}}{3465 c} \] Input:
int((e*x^2+d)^2*(-c*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x)
Output:
( - 480*sqrt(a - c*x**4)*a**2*e**2*x - 660*sqrt(a - c*x**4)*a*b*d*e*x - 15 4*sqrt(a - c*x**4)*a*b*e**2*x**3 + 825*sqrt(a - c*x**4)*a*c*d**2*x + 1078* sqrt(a - c*x**4)*a*c*d*e*x**3 + 405*sqrt(a - c*x**4)*a*c*e**2*x**5 + 693*s qrt(a - c*x**4)*b*c*d**2*x**3 + 990*sqrt(a - c*x**4)*b*c*d*e*x**5 + 385*sq rt(a - c*x**4)*b*c*e**2*x**7 + 495*sqrt(a - c*x**4)*c**2*d**2*x**5 + 770*s qrt(a - c*x**4)*c**2*d*e*x**7 + 315*sqrt(a - c*x**4)*c**2*e**2*x**9 + 480* int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*e**2 + 660*int(sqrt(a - c*x**4)/ (a - c*x**4),x)*a**2*b*d*e + 2640*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a** 2*c*d**2 + 462*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*b*e**2 + 3 696*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*c*d*e + 1386*int((sqr t(a - c*x**4)*x**2)/(a - c*x**4),x)*a*b*c*d**2)/(3465*c)