Integrand size = 34, antiderivative size = 716 \[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {2 a x \left (1105 \left (3 A c d \left (11 c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right )+77 \left (17 B c d \left (13 c d^2+9 a e^2\right )+3 e \left (17 A c \left (13 c d^2+a e^2\right )+a C \left (51 c d^2+7 a e^2\right )\right )\right ) x^2\right ) \sqrt {a-c x^4}}{255255 c^2}+\frac {x \left (663 \left (3 A c d \left (11 c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right )+77 \left (17 B c d \left (13 c d^2+9 a e^2\right )+3 e \left (17 A c \left (13 c d^2+a e^2\right )+a C \left (51 c d^2+7 a e^2\right )\right )\right ) x^2\right ) \left (a-c x^4\right )^{3/2}}{153153 c^2}-\frac {\left (a e^2 (3 C d+B e)+3 c d \left (C d^2+3 e (B d+A e)\right )\right ) x \left (a-c x^4\right )^{5/2}}{33 c^2}-\frac {e \left (7 a C e^2+17 c \left (3 C d^2+e (3 B d+A e)\right )\right ) x^3 \left (a-c x^4\right )^{5/2}}{221 c^2}-\frac {e^2 (3 C d+B e) x^5 \left (a-c x^4\right )^{5/2}}{15 c}-\frac {C e^3 x^7 \left (a-c x^4\right )^{5/2}}{17 c}+\frac {4 a^{11/4} \left (17 B c d \left (13 c d^2+9 a e^2\right )+3 e \left (17 A c \left (13 c d^2+a e^2\right )+a C \left (51 c d^2+7 a e^2\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3315 c^{11/4} \sqrt {a-c x^4}}+\frac {4 a^{9/4} \left (1105 \sqrt {c} \left (3 A c d \left (11 c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+3 c d^2 (C d+3 B e)\right )\right )-77 \sqrt {a} \left (17 B c d \left (13 c d^2+9 a e^2\right )+3 e \left (17 A c \left (13 c d^2+a e^2\right )+a C \left (51 c d^2+7 a e^2\right )\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{255255 c^{11/4} \sqrt {a-c x^4}} \] Output:
2/255255*a*x*(3315*A*c*d*(3*a*e^2+11*c*d^2)+1105*a*(a*e^2*(B*e+3*C*d)+3*c* d^2*(3*B*e+C*d))+77*(17*B*c*d*(9*a*e^2+13*c*d^2)+3*e*(17*A*c*(a*e^2+13*c*d ^2)+a*C*(7*a*e^2+51*c*d^2)))*x^2)*(-c*x^4+a)^(1/2)/c^2+1/153153*x*(1989*A* c*d*(3*a*e^2+11*c*d^2)+663*a*(a*e^2*(B*e+3*C*d)+3*c*d^2*(3*B*e+C*d))+77*(1 7*B*c*d*(9*a*e^2+13*c*d^2)+3*e*(17*A*c*(a*e^2+13*c*d^2)+a*C*(7*a*e^2+51*c* d^2)))*x^2)*(-c*x^4+a)^(3/2)/c^2-1/33*(a*e^2*(B*e+3*C*d)+3*c*d*(C*d^2+3*e* (A*e+B*d)))*x*(-c*x^4+a)^(5/2)/c^2-1/221*e*(7*C*a*e^2+17*c*(3*C*d^2+e*(A*e +3*B*d)))*x^3*(-c*x^4+a)^(5/2)/c^2-1/15*e^2*(B*e+3*C*d)*x^5*(-c*x^4+a)^(5/ 2)/c-1/17*C*e^3*x^7*(-c*x^4+a)^(5/2)/c+4/3315*a^(11/4)*(17*B*c*d*(9*a*e^2+ 13*c*d^2)+3*e*(17*A*c*(a*e^2+13*c*d^2)+a*C*(7*a*e^2+51*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)+4/255255* a^(9/4)*(1105*c^(1/2)*(3*A*c*d*(3*a*e^2+11*c*d^2)+a*(a*e^2*(B*e+3*C*d)+3*c *d^2*(3*B*e+C*d)))-77*a^(1/2)*(17*B*c*d*(9*a*e^2+13*c*d^2)+3*e*(17*A*c*(a* e^2+13*c*d^2)+a*C*(7*a*e^2+51*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 = 10.65 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.67 \[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-c x^4} \left (-3315 c d \left (C d^2+3 e (B d+A e)\right ) \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-2805 c e \left (3 C d^2+e (3 B d+A e)\right ) x^2 \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-2431 c e^2 (3 C d+B e) x^4 \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-2145 c C e^3 x^6 \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}+36465 a A c^2 d^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )+3315 a^2 c d \left (C d^2+3 e (B d+A e)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )-1105 a e^2 (3 C d+B e) \left (\left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )\right )+12155 a c^2 d^2 (B d+3 A e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )+2805 a^2 c e \left (3 C d^2+e (3 B d+A e)\right ) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )-1155 a C e^3 x^2 \left (\left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-a^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )\right )}{36465 c^2 \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)^3*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
(x*Sqrt[a - c*x^4]*(-3315*c*d*(C*d^2 + 3*e*(B*d + A*e))*(a - c*x^4)^2*Sqrt [1 - (c*x^4)/a] - 2805*c*e*(3*C*d^2 + e*(3*B*d + A*e))*x^2*(a - c*x^4)^2*S qrt[1 - (c*x^4)/a] - 2431*c*e^2*(3*C*d + B*e)*x^4*(a - c*x^4)^2*Sqrt[1 - ( c*x^4)/a] - 2145*c*C*e^3*x^6*(a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] + 36465*a*A *c^2*d^3*Hypergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/a] + 3315*a^2*c*d*(C*d^ 2 + 3*e*(B*d + A*e))*Hypergeometric2F1[-3/2, 1/4, 5/4, (c*x^4)/a] - 1105*a *e^2*(3*C*d + B*e)*((a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] - a^2*Hypergeometric 2F1[-3/2, 1/4, 5/4, (c*x^4)/a]) + 12155*a*c^2*d^2*(B*d + 3*A*e)*x^2*Hyperg eometric2F1[-3/2, 3/4, 7/4, (c*x^4)/a] + 2805*a^2*c*e*(3*C*d^2 + e*(3*B*d + A*e))*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, (c*x^4)/a] - 1155*a*C*e^3*x^ 2*((a - c*x^4)^2*Sqrt[1 - (c*x^4)/a] - a^2*Hypergeometric2F1[-3/2, 3/4, 7/ 4, (c*x^4)/a])))/(36465*c^2*Sqrt[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(2173\) vs. \(2(716)=1432\).
Time = 2.61 (sec) , antiderivative size = 2173, normalized size of antiderivative = 3.03, 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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {a^2 d^2 x^2 (3 A e+B d)}{\sqrt {a-c x^4}}+\frac {a^2 A d^3}{\sqrt {a-c x^4}}+\frac {c x^{12} \left (-2 a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )}{\sqrt {a-c x^4}}+\frac {a d x^4 \left (a d (3 B e+C d)-A \left (2 c d^2-3 a e^2\right )\right )}{\sqrt {a-c x^4}}+\frac {c e x^{14} \left (-2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )}{\sqrt {a-c x^4}}+\frac {x^{10} \left (A c e \left (3 c d^2-2 a e^2\right )+B c d \left (c d^2-6 a e^2\right )-a C e \left (6 c d^2-a e^2\right )\right )}{\sqrt {a-c x^4}}+\frac {x^8 \left (A c d \left (c d^2-6 a e^2\right )+a \left (a e^2 (B e+3 C d)-2 c d^2 (3 B e+C d)\right )\right )}{\sqrt {a-c x^4}}+\frac {a x^6 \left (a A e^3+3 a B d e^2+3 a C d^2 e-6 A c d^2 e-2 B c d^3\right )}{\sqrt {a-c x^4}}+\frac {c^2 e^2 x^{16} (B e+3 C d)}{\sqrt {a-c x^4}}+\frac {c^2 C e^3 x^{18}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{17} c C e^3 \sqrt {a-c x^4} x^{15}-\frac {1}{15} c e^2 (3 C d+B e) \sqrt {a-c x^4} x^{13}-\frac {15}{221} a C e^3 \sqrt {a-c x^4} x^{11}-\frac {1}{13} e \left (3 c C d^2-2 a C e^2+c e (3 B d+A e)\right ) \sqrt {a-c x^4} x^{11}-\frac {13}{165} a e^2 (3 C d+B e) \sqrt {a-c x^4} x^9+\frac {1}{11} \left (2 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^9-\frac {55 a^2 C e^3 \sqrt {a-c x^4} x^7}{663 c}-\frac {11 a e \left (3 c C d^2-2 a C e^2+c e (3 B d+A e)\right ) \sqrt {a-c x^4} x^7}{117 c}-\frac {\left (B c d \left (c d^2-6 a e^2\right )+A c e \left (3 c d^2-2 a e^2\right )-a C e \left (6 c d^2-a e^2\right )\right ) \sqrt {a-c x^4} x^7}{9 c}-\frac {39 a^2 e^2 (3 C d+B e) \sqrt {a-c x^4} x^5}{385 c}-\frac {9 a \left (c C d^3+3 c e (B d+A e) d-2 a e^2 (3 C d+B e)\right ) \sqrt {a-c x^4} x^5}{77 c}-\frac {\left (A c d \left (c d^2-6 a e^2\right )+a \left (a e^2 (3 C d+B e)-2 c d^2 (C d+3 B e)\right )\right ) \sqrt {a-c x^4} x^5}{7 c}-\frac {77 a^3 C e^3 \sqrt {a-c x^4} x^3}{663 c^2}+\frac {a \left (2 B c d^3+6 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 {77 a^2 e \left (3 c C d^2-2 a C e^2+c e (3 B d+A e)\right ) \sqrt {a-c x^4} x^3}{585 c^2}-\frac {7 a \left (B c d \left (c d^2-6 a e^2\right )+A c e \left (3 c d^2-2 a e^2\right )-a C e \left (6 c d^2-a e^2\right )\right ) \sqrt {a-c x^4} x^3}{45 c^2}-\frac {13 a^3 e^2 (3 C d+B e) \sqrt {a-c x^4} x}{77 c^2}-\frac {15 a^2 \left (c C d^3+3 c e (B d+A e) d-2 a e^2 (3 C d+B e)\right ) \sqrt {a-c x^4} x}{77 c^2}-\frac {a d \left (a d (C d+3 B e)-A \left (2 c d^2-3 a e^2\right )\right ) \sqrt {a-c x^4} x}{3 c}-\frac {5 a \left (A c d \left (c d^2-6 a e^2\right )+a \left (a e^2 (3 C d+B e)-2 c d^2 (C d+3 B e)\right )\right ) \sqrt {a-c x^4} x}{21 c^2}+\frac {77 a^{19/4} 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 )}{221 c^{11/4} \sqrt {a-c x^4}}+\frac {a^{11/4} 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^{11/4} \left (2 B c d^3+6 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 {77 a^{15/4} e \left (3 c C d^2-2 a C e^2+c e (3 B d+A e)\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 {7 a^{11/4} \left (B c d \left (c d^2-6 a e^2\right )+A c e \left (3 c d^2-2 a e^2\right )-a C e \left (6 c d^2-a e^2\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^{9/4} 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^{19/4} 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 )}{221 c^{11/4} \sqrt {a-c x^4}}-\frac {a^{11/4} 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 {13 a^{17/4} 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^{11/4} \left (2 B c d^3+6 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 {77 a^{15/4} e \left (3 c C d^2-2 a C e^2+c e (3 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 )}{195 c^{11/4} \sqrt {a-c x^4}}+\frac {15 a^{13/4} \left (c C d^3+3 c e (B d+A e) d-2 a e^2 (3 C d+B 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} d \left (a d (C d+3 B e)-A \left (2 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 {7 a^{11/4} \left (B c d \left (c d^2-6 a e^2\right )+A c e \left (3 c d^2-2 a e^2\right )-a C e \left (6 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 )}{15 c^{11/4} \sqrt {a-c x^4}}+\frac {5 a^{9/4} \left (A c d \left (c d^2-6 a e^2\right )+a \left (a e^2 (3 C d+B e)-2 c d^2 (C d+3 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)^3*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
(-13*a^3*e^2*(3*C*d + B*e)*x*Sqrt[a - c*x^4])/(77*c^2) - (15*a^2*(c*C*d^3 + 3*c*d*e*(B*d + A*e) - 2*a*e^2*(3*C*d + B*e))*x*Sqrt[a - c*x^4])/(77*c^2) - (a*d*(a*d*(C*d + 3*B*e) - A*(2*c*d^2 - 3*a*e^2))*x*Sqrt[a - c*x^4])/(3* c) - (5*a*(A*c*d*(c*d^2 - 6*a*e^2) + a*(a*e^2*(3*C*d + B*e) - 2*c*d^2*(C*d + 3*B*e)))*x*Sqrt[a - c*x^4])/(21*c^2) - (77*a^3*C*e^3*x^3*Sqrt[a - c*x^4 ])/(663*c^2) + (a*(2*B*c*d^3 + 6*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) - (77*a^2*e*(3*c*C*d^2 - 2*a*C*e^2 + c* e*(3*B*d + A*e))*x^3*Sqrt[a - c*x^4])/(585*c^2) - (7*a*(B*c*d*(c*d^2 - 6*a *e^2) + A*c*e*(3*c*d^2 - 2*a*e^2) - a*C*e*(6*c*d^2 - a*e^2))*x^3*Sqrt[a - c*x^4])/(45*c^2) - (39*a^2*e^2*(3*C*d + B*e)*x^5*Sqrt[a - c*x^4])/(385*c) - (9*a*(c*C*d^3 + 3*c*d*e*(B*d + A*e) - 2*a*e^2*(3*C*d + B*e))*x^5*Sqrt[a - c*x^4])/(77*c) - ((A*c*d*(c*d^2 - 6*a*e^2) + a*(a*e^2*(3*C*d + B*e) - 2* c*d^2*(C*d + 3*B*e)))*x^5*Sqrt[a - c*x^4])/(7*c) - (55*a^2*C*e^3*x^7*Sqrt[ a - c*x^4])/(663*c) - (11*a*e*(3*c*C*d^2 - 2*a*C*e^2 + c*e*(3*B*d + A*e))* x^7*Sqrt[a - c*x^4])/(117*c) - ((B*c*d*(c*d^2 - 6*a*e^2) + A*c*e*(3*c*d^2 - 2*a*e^2) - a*C*e*(6*c*d^2 - a*e^2))*x^7*Sqrt[a - c*x^4])/(9*c) - (13*a*e ^2*(3*C*d + B*e)*x^9*Sqrt[a - c*x^4])/165 + ((2*a*e^2*(3*C*d + B*e) - c*(C *d^3 + 3*d*e*(B*d + A*e)))*x^9*Sqrt[a - c*x^4])/11 - (15*a*C*e^3*x^11*Sqrt [a - c*x^4])/221 - (e*(3*c*C*d^2 - 2*a*C*e^2 + c*e*(3*B*d + A*e))*x^11*Sqr t[a - c*x^4])/13 - (c*e^2*(3*C*d + B*e)*x^13*Sqrt[a - c*x^4])/15 - (c*C...
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 = 4.94 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(A \,d^{3} \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^{2} \left (3 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^{2} \left (B e +3 C d \right ) \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 )+e \left (A \,e^{2}+3 B d e +3 C \,d^{2}\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 )+d \left (3 A \,e^{2}+3 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 )+e^{3} C \left (-\frac {c \,x^{15} \sqrt {-c \,x^{4}+a}}{17}+\frac {19 a \,x^{11} \sqrt {-c \,x^{4}+a}}{221}-\frac {4 a^{2} x^{7} \sqrt {-c \,x^{4}+a}}{663 c}-\frac {28 a^{3} x^{3} \sqrt {-c \,x^{4}+a}}{3315 c^{2}}-\frac {28 a^{\frac {9}{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 )}{1105 c^{\frac {5}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(860\) |
| risch | \(\text {Expression too large to display}\) | \(1049\) |
| elliptic | \(\text {Expression too large to display}\) | \(1331\) |
Input:
int((e*x^2+d)^3*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
A*d^3*(-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*(3*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) -EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I)))+e^2*(B*e+3*C*d)*(-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))+e*(A*e^2+3*B*d*e+3*C*d^2)*(-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)))+d*(3* A*e^2+3*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))+e^3*C*(-1/17*c*x^15*(-c*x^4+a)^(1/ 2)+19/221*a*x^11*(-c*x^4+a)^(1/2)-4/663*a^2/c*x^7*(-c*x^4+a)^(1/2)-28/3...
Time = 0.09 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.03 \[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^3*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="frica s")
Output:
-1/765765*(924*(221*B*a^2*c^2*d^3 + 153*B*a^3*c*d*e^2 + 51*(3*C*a^3*c + 13 *A*a^2*c^2)*d^2*e + 3*(7*C*a^4 + 17*A*a^3*c)*e^3)*sqrt(-c)*x*(a/c)^(3/4)*e lliptic_e(arcsin((a/c)^(1/4)/x), -1) - 12*(221*((77*B + 15*C)*a^2*c^2 + 16 5*A*a*c^3)*d^3 + 51*(231*C*a^3*c + 13*(77*A + 15*B)*a^2*c^2)*d^2*e + 51*(( 231*B + 65*C)*a^3*c + 195*A*a^2*c^2)*d*e^2 + (1617*C*a^4 + 17*(231*A + 65* B)*a^3*c)*e^3)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1 ) + (45045*C*c^4*e^3*x^16 + 51051*(3*C*c^4*d*e^2 + B*c^4*e^3)*x^14 + 3465* (51*C*c^4*d^2*e + 51*B*c^4*d*e^2 - (19*C*a*c^3 - 17*A*c^4)*e^3)*x^12 + 464 1*(15*C*c^4*d^3 + 45*B*c^4*d^2*e - 17*B*a*c^3*e^3 - 3*(17*C*a*c^3 - 15*A*c ^4)*d*e^2)*x^10 + 385*(221*B*c^4*d^3 - 765*B*a*c^3*d*e^2 - 51*(15*C*a*c^3 - 13*A*c^4)*d^2*e + 3*(4*C*a^2*c^2 - 85*A*a*c^3)*e^3)*x^8 + 204204*B*a^2*c ^2*d^3 + 141372*B*a^3*c*d*e^2 - 1989*(195*B*a*c^3*d^2*e - 4*B*a^2*c^2*e^3 + 5*(13*C*a*c^3 - 11*A*c^4)*d^3 - 3*(4*C*a^2*c^2 - 65*A*a*c^3)*d*e^2)*x^6 - 77*(2431*B*a*c^3*d^3 - 612*B*a^2*c^2*d*e^2 - 51*(12*C*a^2*c^2 - 143*A*a* c^3)*d^2*e - 12*(7*C*a^3*c + 17*A*a^2*c^2)*e^3)*x^4 + 47124*(3*C*a^3*c + 1 3*A*a^2*c^2)*d^2*e + 2772*(7*C*a^4 + 17*A*a^3*c)*e^3 + 3315*(36*B*a^2*c^2* d^2*e + 4*B*a^3*c*e^3 + 3*(4*C*a^2*c^2 - 33*A*a*c^3)*d^3 + 12*(C*a^3*c + 3 *A*a^2*c^2)*d*e^2)*x^2)*sqrt(-c*x^4 + a))/(c^3*x)
Time = 10.75 (sec) , antiderivative size = 1200, normalized size of antiderivative = 1.68 \[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate((e*x**2+d)**3*(-c*x**4+a)**(3/2)*(C*x**4+B*x**2+A),x)
Output:
A*a**(3/2)*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*a**(3/2)*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*a**(3/2)* 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*a**(3/2)*e**3*x**7*gamma(7/4)*hyper((-1/2, 7/4), (1 1/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma(11/4)) - A*sqrt(a)*c*d**3*x**5 *gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamm a(9/4)) - 3*A*sqrt(a)*c*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*A*sqrt(a)*c*d*e**2*x**9*g amma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*gamma (13/4)) - A*sqrt(a)*c*e**3*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)) + B*a**(3/2)*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*a**(3/2)*d**2*e*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*ex p_polar(2*I*pi)/a)/(4*gamma(9/4)) + 3*B*a**(3/2)*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* a**(3/2)*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)) - B*sqrt(a)*c*d**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)) - 3*B*sqrt(a) *c*d**2*e*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**4*exp_polar(...
\[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-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)^3, x)
\[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-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)^3, x)
Timed out. \[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int {\left (a-c\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^3\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(3/2)*(d + e*x^2)^3*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(3/2)*(d + e*x^2)^3*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right )^3 \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^3*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x)
Output:
( - 13260*sqrt(a - c*x**4)*a**3*b*e**3*x - 159120*sqrt(a - c*x**4)*a**3*c* d*e**2*x - 22176*sqrt(a - c*x**4)*a**3*c*e**3*x**3 - 119340*sqrt(a - c*x** 4)*a**2*b*c*d**2*e*x - 47124*sqrt(a - c*x**4)*a**2*b*c*d*e**2*x**3 - 7956* sqrt(a - c*x**4)*a**2*b*c*e**3*x**5 + 288405*sqrt(a - c*x**4)*a**2*c**2*d* *3*x + 514437*sqrt(a - c*x**4)*a**2*c**2*d**2*e*x**3 + 363987*sqrt(a - c*x **4)*a**2*c**2*d*e**2*x**5 + 93555*sqrt(a - c*x**4)*a**2*c**2*e**3*x**7 + 187187*sqrt(a - c*x**4)*a*b*c**2*d**3*x**3 + 387855*sqrt(a - c*x**4)*a*b*c **2*d**2*e*x**5 + 294525*sqrt(a - c*x**4)*a*b*c**2*d*e**2*x**7 + 78897*sqr t(a - c*x**4)*a*b*c**2*e**3*x**9 + 19890*sqrt(a - c*x**4)*a*c**3*d**3*x**5 + 39270*sqrt(a - c*x**4)*a*c**3*d**2*e*x**7 + 27846*sqrt(a - c*x**4)*a*c* *3*d*e**2*x**9 + 6930*sqrt(a - c*x**4)*a*c**3*e**3*x**11 - 85085*sqrt(a - c*x**4)*b*c**3*d**3*x**7 - 208845*sqrt(a - c*x**4)*b*c**3*d**2*e*x**9 - 17 6715*sqrt(a - c*x**4)*b*c**3*d*e**2*x**11 - 51051*sqrt(a - c*x**4)*b*c**3* e**3*x**13 - 69615*sqrt(a - c*x**4)*c**4*d**3*x**9 - 176715*sqrt(a - c*x** 4)*c**4*d**2*e*x**11 - 153153*sqrt(a - c*x**4)*c**4*d*e**2*x**13 - 45045*s qrt(a - c*x**4)*c**4*e**3*x**15 + 13260*int(sqrt(a - c*x**4)/(a - c*x**4), x)*a**4*b*e**3 + 159120*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**4*c*d*e**2 + 119340*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*b*c*d**2*e + 477360*in t(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*c**2*d**3 + 66528*int((sqrt(a - c* x**4)*x**2)/(a - c*x**4),x)*a**4*c*e**3 + 141372*int((sqrt(a - c*x**4)*...