Integrand size = 32, antiderivative size = 342 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {2 a x \left (195 (11 A c d+a C d+a B e)+77 (13 B c d+13 A c e+3 a C e) x^2\right ) \sqrt {a-c x^4}}{15015 c}+\frac {x \left (117 (11 A c d+a C d+a B e)+77 (13 B c d+13 A c e+3 a C e) x^2\right ) \left (a-c x^4\right )^{3/2}}{9009 c}-\frac {(C d+B e) x \left (a-c x^4\right )^{5/2}}{11 c}-\frac {C e x^3 \left (a-c x^4\right )^{5/2}}{13 c}+\frac {4 a^{11/4} (13 B c d+13 A c e+3 a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{195 c^{7/4} \sqrt {a-c x^4}}+\frac {4 a^{9/4} \left (195 \sqrt {c} (11 A c d+a C d+a B e)-77 \sqrt {a} (13 B c d+13 A c e+3 a C e)\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{15015 c^{7/4} \sqrt {a-c x^4}} \] Output:
2/15015*a*x*(2145*A*c*d+195*B*a*e+195*C*a*d+77*(13*A*c*e+13*B*c*d+3*C*a*e) *x^2)*(-c*x^4+a)^(1/2)/c+1/9009*x*(1287*A*c*d+117*B*a*e+117*C*a*d+77*(13*A *c*e+13*B*c*d+3*C*a*e)*x^2)*(-c*x^4+a)^(3/2)/c-1/11*(B*e+C*d)*x*(-c*x^4+a) ^(5/2)/c-1/13*C*e*x^3*(-c*x^4+a)^(5/2)/c+4/195*a^(11/4)*(13*A*c*e+13*B*c*d +3*C*a*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)*(195*c^(1/2)*(11*A*c*d+B*a*e+C*a*d)-77*a^(1/2)*( 13*A*c*e+13*B*c*d+3*C*a*e))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4), I)/c^(7/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.45 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=-\frac {x \sqrt {a-c x^4} \left (3 \left (13 C d+13 B e+11 C e x^2\right ) \left (a-c x^4\right )^2 \sqrt {1-\frac {c x^4}{a}}-39 a (11 A c d+a C d+a B e) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {c x^4}{a}\right )-11 a (13 B c d+13 A c e+3 a C e) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{429 c \sqrt {1-\frac {c x^4}{a}}} \] Input:
Integrate[(d + e*x^2)*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
-1/429*(x*Sqrt[a - c*x^4]*(3*(13*C*d + 13*B*e + 11*C*e*x^2)*(a - c*x^4)^2* Sqrt[1 - (c*x^4)/a] - 39*a*(11*A*c*d + a*C*d + a*B*e)*Hypergeometric2F1[-3 /2, 1/4, 5/4, (c*x^4)/a] - 11*a*(13*B*c*d + 13*A*c*e + 3*a*C*e)*x^2*Hyperg eometric2F1[-3/2, 3/4, 7/4, (c*x^4)/a]))/(c*Sqrt[1 - (c*x^4)/a])
Leaf count is larger than twice the leaf count of optimal. \(1154\) vs. \(2(342)=684\).
Time = 1.37 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, 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.
| \(\displaystyle \int \left (a-c x^4\right )^{3/2} \left (d+e x^2\right ) \left (A+B x^2+C x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {a^2 x^2 (A e+B d)}{\sqrt {a-c x^4}}+\frac {a^2 A d}{\sqrt {a-c x^4}}+\frac {a x^4 (a B e+a C d-2 A c d)}{\sqrt {a-c x^4}}+\frac {c x^{10} (-2 a C e+A c e+B c d)}{\sqrt {a-c x^4}}+\frac {c x^8 (A c d-2 a (B e+C d))}{\sqrt {a-c x^4}}+\frac {a x^6 (a C e-2 A c e-2 B c d)}{\sqrt {a-c x^4}}+\frac {c^2 x^{12} (B e+C d)}{\sqrt {a-c x^4}}+\frac {c^2 C e x^{14}}{\sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{13} c C e \sqrt {a-c x^4} x^{11}-\frac {1}{11} c (C d+B e) \sqrt {a-c x^4} x^9-\frac {11}{117} a C e \sqrt {a-c x^4} x^7-\frac {1}{9} (B c d+A c e-2 a C e) \sqrt {a-c x^4} x^7-\frac {9}{77} a (C d+B e) \sqrt {a-c x^4} x^5-\frac {1}{7} (A c d-2 a (C d+B e)) \sqrt {a-c x^4} x^5-\frac {77 a^2 C e \sqrt {a-c x^4} x^3}{585 c}-\frac {7 a (B c d+A c e-2 a C e) \sqrt {a-c x^4} x^3}{45 c}+\frac {a (2 B c d+2 A c e-a C e) \sqrt {a-c x^4} x^3}{5 c}-\frac {15 a^2 (C d+B e) \sqrt {a-c x^4} x}{77 c}+\frac {a (2 A c d-a C d-a B e) \sqrt {a-c x^4} x}{3 c}-\frac {5 a (A c d-2 a (C d+B e)) \sqrt {a-c x^4} x}{21 c}+\frac {77 a^{15/4} C 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 {a^{11/4} (B d+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} (B c d+A c e-2 a C 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^{11/4} (2 B c d+2 A c e-a C 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 )}{5 c^{7/4} \sqrt {a-c x^4}}+\frac {a^{9/4} A d \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 \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 {a^{11/4} (B d+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} (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^{5/4} \sqrt {a-c x^4}}-\frac {a^{9/4} (2 A c d-a C d-a 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 )}{3 c^{5/4} \sqrt {a-c x^4}}-\frac {7 a^{11/4} (B c d+A c e-2 a C 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^{11/4} (2 B c d+2 A c e-a C e) \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 {5 a^{9/4} (A c d-2 a (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 )}{21 c^{5/4} \sqrt {a-c x^4}}\) |
Input:
Int[(d + e*x^2)*(a - c*x^4)^(3/2)*(A + B*x^2 + C*x^4),x]
Output:
(-15*a^2*(C*d + B*e)*x*Sqrt[a - c*x^4])/(77*c) + (a*(2*A*c*d - a*C*d - a*B *e)*x*Sqrt[a - c*x^4])/(3*c) - (5*a*(A*c*d - 2*a*(C*d + B*e))*x*Sqrt[a - c *x^4])/(21*c) - (77*a^2*C*e*x^3*Sqrt[a - c*x^4])/(585*c) - (7*a*(B*c*d + A *c*e - 2*a*C*e)*x^3*Sqrt[a - c*x^4])/(45*c) + (a*(2*B*c*d + 2*A*c*e - a*C* e)*x^3*Sqrt[a - c*x^4])/(5*c) - (9*a*(C*d + B*e)*x^5*Sqrt[a - c*x^4])/77 - ((A*c*d - 2*a*(C*d + B*e))*x^5*Sqrt[a - c*x^4])/7 - (11*a*C*e*x^7*Sqrt[a - c*x^4])/117 - ((B*c*d + A*c*e - 2*a*C*e)*x^7*Sqrt[a - c*x^4])/9 - (c*(C* d + B*e)*x^9*Sqrt[a - c*x^4])/11 - (c*C*e*x^11*Sqrt[a - c*x^4])/13 + (77*a ^(15/4)*C*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]) + (a^(11/4)*(B*d + 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)*(B*c*d + A*c*e - 2*a*C*e)*Sqrt[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*A*c*e - a*C*e)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/ a^(1/4)], -1])/(5*c^(7/4)*Sqrt[a - c*x^4]) + (a^(9/4)*A*d*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*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4 )], -1])/(195*c^(7/4)*Sqrt[a - c*x^4]) - (a^(11/4)*(B*d + 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]) + (15*a^(13/4)*(C*d + B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(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 = 1.89 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.37
| method | result | size |
| risch | \(\frac {x \left (-3465 e C \,x^{10} c^{2}-4095 B \,c^{2} e \,x^{8}-4095 C \,c^{2} d \,x^{8}-5005 A \,c^{2} e \,x^{6}-5005 B \,c^{2} d \,x^{6}+5775 C a c e \,x^{6}-6435 A \,c^{2} d \,x^{4}+7605 B a c e \,x^{4}+7605 C a c d \,x^{4}+11011 A a c e \,x^{2}+11011 B a c d \,x^{2}-924 C \,a^{2} e \,x^{2}+19305 a A c d -2340 B \,a^{2} e -2340 a^{2} d C \right ) \sqrt {-c \,x^{4}+a}}{45045 c}+\frac {4 a^{2} \left (-\frac {\left (1001 A c e +1001 B c d +231 C a e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+\frac {2145 A c d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {195 B a e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {195 C a d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )}{15015 c}\) | \(467\) |
| default | \(A d \left (-\frac {c \,x^{5} \sqrt {-c \,x^{4}+a}}{7}+\frac {3 a x \sqrt {-c \,x^{4}+a}}{7}+\frac {4 a^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{7 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A e +B d \right ) \left (-\frac {c \,x^{7} \sqrt {-c \,x^{4}+a}}{9}+\frac {11 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45}-\frac {4 a^{\frac {5}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{15 \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+\left (B e +C d \right ) \left (-\frac {c \,x^{9} \sqrt {-c \,x^{4}+a}}{11}+\frac {13 a \,x^{5} \sqrt {-c \,x^{4}+a}}{77}-\frac {4 a^{2} x \sqrt {-c \,x^{4}+a}}{77 c}+\frac {4 a^{3} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{77 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e C \left (-\frac {c \,x^{11} \sqrt {-c \,x^{4}+a}}{13}+\frac {5 a \,x^{7} \sqrt {-c \,x^{4}+a}}{39}-\frac {4 a^{2} x^{3} \sqrt {-c \,x^{4}+a}}{195 c}-\frac {4 a^{\frac {7}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{65 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(505\) |
| elliptic | \(-\frac {C e c \,x^{11} \sqrt {-c \,x^{4}+a}}{13}-\frac {\left (B \,c^{2} e +d \,c^{2} C \right ) x^{9} \sqrt {-c \,x^{4}+a}}{11 c}-\frac {\left (A \,c^{2} e +d \,c^{2} B -\frac {15}{13} C a e c \right ) x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {\left (A \,c^{2} d -2 B a c e -2 a c C d +\frac {9 \left (B \,c^{2} e +d \,c^{2} C \right ) a}{11 c}\right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (-2 a A c e -2 a B c d +a^{2} C e +\frac {7 \left (A \,c^{2} e +d \,c^{2} B -\frac {15}{13} C a e c \right ) a}{9 c}\right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (-2 a A c d +B \,a^{2} e +a^{2} d C +\frac {5 \left (A \,c^{2} d -2 B a c e -2 a c C d +\frac {9 \left (B \,c^{2} e +d \,c^{2} C \right ) a}{11 c}\right ) a}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A \,a^{2} d +\frac {\left (-2 a A c d +B \,a^{2} e +a^{2} d C +\frac {5 \left (A \,c^{2} d -2 B a c e -2 a c C d +\frac {9 \left (B \,c^{2} e +d \,c^{2} C \right ) a}{11 c}\right ) a}{7 c}\right ) a}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (A \,a^{2} e +B \,a^{2} d +\frac {3 \left (-2 a A c e -2 a B c d +a^{2} C e +\frac {7 \left (A \,c^{2} e +d \,c^{2} B -\frac {15}{13} C a e c \right ) a}{9 c}\right ) a}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(574\) |
Input:
int((e*x^2+d)*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
Output:
1/45045/c*x*(-3465*C*c^2*e*x^10-4095*B*c^2*e*x^8-4095*C*c^2*d*x^8-5005*A*c ^2*e*x^6-5005*B*c^2*d*x^6+5775*C*a*c*e*x^6-6435*A*c^2*d*x^4+7605*B*a*c*e*x ^4+7605*C*a*c*d*x^4+11011*A*a*c*e*x^2+11011*B*a*c*d*x^2-924*C*a^2*e*x^2+19 305*A*a*c*d-2340*B*a^2*e-2340*C*a^2*d)*(-c*x^4+a)^(1/2)+4/15015*a^2/c*(-(1 001*A*c*e+1001*B*c*d+231*C*a*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) *(EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/ 2),I))+2145*A*c*d/(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)+195*B*a*e/(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)+195*C*a*d/(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 = 326, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=-\frac {924 \, {\left (13 \, B a^{2} c d + {\left (3 \, C a^{3} + 13 \, A a^{2} c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 12 \, {\left (13 \, {\left ({\left (77 \, B + 15 \, C\right )} a^{2} c + 165 \, A a c^{2}\right )} d + {\left (231 \, C a^{3} + 13 \, {\left (77 \, A + 15 \, B\right )} a^{2} c\right )} e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3465 \, C c^{3} e x^{12} + 4095 \, {\left (C c^{3} d + B c^{3} e\right )} x^{10} + 385 \, {\left (13 \, B c^{3} d - {\left (15 \, C a c^{2} - 13 \, A c^{3}\right )} e\right )} x^{8} - 585 \, {\left (13 \, B a c^{2} e + {\left (13 \, C a c^{2} - 11 \, A c^{3}\right )} d\right )} x^{6} + 12012 \, B a^{2} c d - 77 \, {\left (143 \, B a c^{2} d - {\left (12 \, C a^{2} c - 143 \, A a c^{2}\right )} e\right )} x^{4} + 585 \, {\left (4 \, B a^{2} c e + {\left (4 \, C a^{2} c - 33 \, A a c^{2}\right )} d\right )} x^{2} + 924 \, {\left (3 \, C a^{3} + 13 \, A a^{2} c\right )} e\right )} \sqrt {-c x^{4} + a}}{45045 \, c^{2} x} \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="fricas" )
Output:
-1/45045*(924*(13*B*a^2*c*d + (3*C*a^3 + 13*A*a^2*c)*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 + (231*C*a^3 + 13*(77*A + 15*B)*a^2*c)*e)*sqrt(-c)*x*(a/c) ^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + (3465*C*c^3*e*x^12 + 4095*( C*c^3*d + B*c^3*e)*x^10 + 385*(13*B*c^3*d - (15*C*a*c^2 - 13*A*c^3)*e)*x^8 - 585*(13*B*a*c^2*e + (13*C*a*c^2 - 11*A*c^3)*d)*x^6 + 12012*B*a^2*c*d - 77*(143*B*a*c^2*d - (12*C*a^2*c - 143*A*a*c^2)*e)*x^4 + 585*(4*B*a^2*c*e + (4*C*a^2*c - 33*A*a*c^2)*d)*x^2 + 924*(3*C*a^3 + 13*A*a^2*c)*e)*sqrt(-c*x ^4 + a))/(c^2*x)
Time = 5.17 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.63 \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)*(-c*x**4+a)**(3/2)*(C*x**4+B*x**2+A),x)
Output:
A*a**(3/2)*d*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)*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)) - A*sqrt(a)*c*d*x**5*gamm a(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*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)) + B*a**(3/2)*d*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)*e *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)*c*d*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*e*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*a**(3/2)*d*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)*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)) - C*sqrt(a)*c*d*x** 9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*ga mma(13/4)) - C*sqrt(a)*c*e*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))
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*(-c*x^4 + a)^(3/2)*(e*x^2 + d), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (-c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )} \,d x } \] Input:
integrate((e*x^2+d)*(-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), x)
Timed out. \[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\int {\left (a-c\,x^4\right )}^{3/2}\,\left (e\,x^2+d\right )\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:
int((a - c*x^4)^(3/2)*(d + e*x^2)*(A + B*x^2 + C*x^4),x)
Output:
int((a - c*x^4)^(3/2)*(d + e*x^2)*(A + B*x^2 + C*x^4), x)
\[ \int \left (d+e x^2\right ) \left (a-c x^4\right )^{3/2} \left (A+B x^2+C x^4\right ) \, dx=\frac {-2340 \sqrt {-c \,x^{4}+a}\, a^{2} b e x +16965 \sqrt {-c \,x^{4}+a}\, a^{2} c d x +10087 \sqrt {-c \,x^{4}+a}\, a^{2} c e \,x^{3}+11011 \sqrt {-c \,x^{4}+a}\, a b c d \,x^{3}+7605 \sqrt {-c \,x^{4}+a}\, a b c e \,x^{5}+1170 \sqrt {-c \,x^{4}+a}\, a \,c^{2} d \,x^{5}+770 \sqrt {-c \,x^{4}+a}\, a \,c^{2} e \,x^{7}-5005 \sqrt {-c \,x^{4}+a}\, b \,c^{2} d \,x^{7}-4095 \sqrt {-c \,x^{4}+a}\, b \,c^{2} e \,x^{9}-4095 \sqrt {-c \,x^{4}+a}\, c^{3} d \,x^{9}-3465 \sqrt {-c \,x^{4}+a}\, c^{3} e \,x^{11}+2340 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} b e +28080 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{3} c d +14784 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{3} c e +12012 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} b c d}{45045 c} \] Input:
int((e*x^2+d)*(-c*x^4+a)^(3/2)*(C*x^4+B*x^2+A),x)
Output:
( - 2340*sqrt(a - c*x**4)*a**2*b*e*x + 16965*sqrt(a - c*x**4)*a**2*c*d*x + 10087*sqrt(a - c*x**4)*a**2*c*e*x**3 + 11011*sqrt(a - c*x**4)*a*b*c*d*x** 3 + 7605*sqrt(a - c*x**4)*a*b*c*e*x**5 + 1170*sqrt(a - c*x**4)*a*c**2*d*x* *5 + 770*sqrt(a - c*x**4)*a*c**2*e*x**7 - 5005*sqrt(a - c*x**4)*b*c**2*d*x **7 - 4095*sqrt(a - c*x**4)*b*c**2*e*x**9 - 4095*sqrt(a - c*x**4)*c**3*d*x **9 - 3465*sqrt(a - c*x**4)*c**3*e*x**11 + 2340*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*b*e + 28080*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**3*c*d + 14784*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**3*c*e + 12012*int(( sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*b*c*d)/(45045*c)