Integrand size = 34, antiderivative size = 449 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {\left (5 a e^2 (3 C d+B e)+7 c d \left (C d^2+3 e (B d+A e)\right )\right ) x \sqrt {a-c x^4}}{21 c^2}-\frac {e \left (7 a C e^2+9 c \left (3 C d^2+e (3 B d+A e)\right )\right ) x^3 \sqrt {a-c x^4}}{45 c^2}-\frac {e^2 (3 C d+B e) x^5 \sqrt {a-c x^4}}{7 c}-\frac {C e^3 x^7 \sqrt {a-c x^4}}{9 c}+\frac {a^{3/4} \left (3 B c d \left (5 c d^2+9 a e^2\right )+e \left (9 A c \left (5 c d^2+a e^2\right )+a C \left (27 c d^2+7 a e^2\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 c^{11/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (5 \sqrt {c} \left (21 A c d \left (c d^2+a e^2\right )+a \left (5 a e^2 (3 C d+B e)+7 c d^2 (C d+3 B e)\right )\right )-7 \sqrt {a} \left (3 B c d \left (5 c d^2+9 a e^2\right )+e \left (9 A c \left (5 c d^2+a e^2\right )+a C \left (27 c d^2+7 a e^2\right )\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 c^{11/4} \sqrt {a-c x^4}} \] Output:
-1/21*(5*a*e^2*(B*e+3*C*d)+7*c*d*(C*d^2+3*e*(A*e+B*d)))*x*(-c*x^4+a)^(1/2) /c^2-1/45*e*(7*C*a*e^2+9*c*(3*C*d^2+e*(A*e+3*B*d)))*x^3*(-c*x^4+a)^(1/2)/c ^2-1/7*e^2*(B*e+3*C*d)*x^5*(-c*x^4+a)^(1/2)/c-1/9*C*e^3*x^7*(-c*x^4+a)^(1/ 2)/c+1/15*a^(3/4)*(3*B*c*d*(9*a*e^2+5*c*d^2)+e*(9*A*c*(a*e^2+5*c*d^2)+a*C* (7*a*e^2+27*c*d^2)))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/c^(1 1/4)/(-c*x^4+a)^(1/2)+1/105*a^(1/4)*(5*c^(1/2)*(21*A*c*d*(a*e^2+c*d^2)+a*( 5*a*e^2*(B*e+3*C*d)+7*c*d^2*(3*B*e+C*d)))-7*a^(1/2)*(3*B*c*d*(9*a*e^2+5*c* d^2)+e*(9*A*c*(a*e^2+5*c*d^2)+a*C*(7*a*e^2+27*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 = 10.51 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {x \left (-a+c x^4\right ) \left (a e^2 \left (225 C d+75 B e+49 C e x^2\right )+c C \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )+9 c e \left (7 A e \left (5 d+e x^2\right )+B \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )\right )+15 \left (21 A c d \left (c d^2+a e^2\right )+a \left (5 a e^2 (3 C d+B e)+7 c d^2 (C d+3 B e)\right )\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+7 \left (9 A c e \left (5 c d^2+a e^2\right )+a C e \left (27 c d^2+7 a e^2\right )+3 B c d \left (5 c d^2+9 a e^2\right )\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )}{315 c^2 \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
(x*(-a + c*x^4)*(a*e^2*(225*C*d + 75*B*e + 49*C*e*x^2) + c*C*(105*d^3 + 18 9*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) + 9*c*e*(7*A*e*(5*d + e*x^2) + B *(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 15*(21*A*c*d*(c*d^2 + a*e^2) + a*(5 *a*e^2*(3*C*d + B*e) + 7*c*d^2*(C*d + 3*B*e)))*x*Sqrt[1 - (c*x^4)/a]*Hyper geometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 7*(9*A*c*e*(5*c*d^2 + a*e^2) + a* C*e*(27*c*d^2 + 7*a*e^2) + 3*B*c*d*(5*c*d^2 + 9*a*e^2))*x^3*Sqrt[1 - (c*x^ 4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a])/(315*c^2*Sqrt[a - c*x^4 ])
Time = 1.51 (sec) , antiderivative size = 676, normalized size of antiderivative = 1.51, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2209, 25, 2209, 25, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx\) |
| \(\Big \downarrow \) 2209 |
| \(\displaystyle -\frac {\int -\frac {\left (e x^2+d\right )^2 \left (3 c (2 C d+3 B e) x^4+(9 B c d+9 A c e+7 a C e) x^2+(9 A c+a C) d\right )}{\sqrt {a-c x^4}}dx}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (e x^2+d\right )^2 \left (3 c (2 C d+3 B e) x^4+(9 B c d+9 A c e+7 a C e) x^2+(9 A c+a C) d\right )}{\sqrt {a-c x^4}}dx}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
| \(\Big \downarrow \) 2209 |
| \(\displaystyle \frac {-\frac {\int -\frac {\left (e x^2+d\right ) \left (c \left (24 c C d^2+49 a C e^2+9 c e (11 B d+7 A e)\right ) x^4+c \left (63 B c d^2+126 A c e d+86 a C e d+45 a B e^2\right ) x^2+c d (63 A c d+13 a C d+9 a B e)\right )}{\sqrt {a-c x^4}}dx}{7 c}-\frac {3}{7} x \sqrt {a-c x^4} \left (d+e x^2\right )^2 (3 B e+2 C d)}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (e x^2+d\right ) \left (c \left (24 c C d^2+49 a C e^2+9 c e (11 B d+7 A e)\right ) x^4+c \left (63 B c d^2+126 A c e d+86 a C e d+45 a B e^2\right ) x^2+c d (63 A c d+13 a C d+9 a B e)\right )}{\sqrt {a-c x^4}}dx}{7 c}-\frac {3}{7} x \sqrt {a-c x^4} \left (d+e x^2\right )^2 (3 B e+2 C d)}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \frac {\frac {\int \left (\frac {c e \left (24 c C d^2+49 a C e^2+9 c e (11 B d+7 A e)\right ) x^6}{\sqrt {a-c x^4}}+\frac {3 c \left (8 c C d^3+9 c e (6 B d+7 A e) d+15 a e^2 (3 C d+B e)\right ) x^4}{\sqrt {a-c x^4}}+\frac {9 c d \left (7 B c d^2+21 A c e d+11 a C e d+6 a B e^2\right ) x^2}{\sqrt {a-c x^4}}+\frac {c d^2 (63 A c d+13 a C d+9 a B e)}{\sqrt {a-c x^4}}\right )dx}{7 c}-\frac {3}{7} x \sqrt {a-c x^4} \left (d+e x^2\right )^2 (3 B e+2 C d)}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {3 a^{7/4} e \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (49 a C e^2+9 c e (7 A e+11 B d)+24 c C d^2\right )}{5 c^{3/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (49 a C e^2+9 c e (7 A e+11 B d)+24 c C d^2\right )}{5 c^{3/4} \sqrt {a-c x^4}}+\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (15 a e^2 (B e+3 C d)+9 c d e (7 A e+6 B d)+8 c C d^3\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {9 a^{3/4} \sqrt [4]{c} d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (6 a B e^2+11 a C d e+21 A c d e+7 B c d^2\right )}{\sqrt {a-c x^4}}+\frac {9 a^{3/4} \sqrt [4]{c} d \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) \left (6 a B e^2+11 a C d e+21 A c d e+7 B c d^2\right )}{\sqrt {a-c x^4}}+\frac {\sqrt [4]{a} c^{3/4} d^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) (9 a B e+13 a C d+63 A c d)}{\sqrt {a-c x^4}}-x \sqrt {a-c x^4} \left (15 a e^2 (B e+3 C d)+9 c d e (7 A e+6 B d)+8 c C d^3\right )-\frac {1}{5} e x^3 \sqrt {a-c x^4} \left (49 a C e^2+9 c e (7 A e+11 B d)+24 c C d^2\right )}{7 c}-\frac {3}{7} x \sqrt {a-c x^4} \left (d+e x^2\right )^2 (3 B e+2 C d)}{9 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^3}{9 c}\) |
Input:
Int[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
-1/9*(C*x*(d + e*x^2)^3*Sqrt[a - c*x^4])/c + ((-3*(2*C*d + 3*B*e)*x*(d + e *x^2)^2*Sqrt[a - c*x^4])/7 + (-((8*c*C*d^3 + 9*c*d*e*(6*B*d + 7*A*e) + 15* a*e^2*(3*C*d + B*e))*x*Sqrt[a - c*x^4]) - (e*(24*c*C*d^2 + 49*a*C*e^2 + 9* c*e*(11*B*d + 7*A*e))*x^3*Sqrt[a - c*x^4])/5 + (9*a^(3/4)*c^(1/4)*d*(7*B*c *d^2 + 21*A*c*d*e + 11*a*C*d*e + 6*a*B*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] + (3*a^(7/4)*e*(24*c*C*d ^2 + 49*a*C*e^2 + 9*c*e*(11*B*d + 7*A*e))*Sqrt[1 - (c*x^4)/a]*EllipticE[Ar cSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*c^( 3/4)*d^2*(63*A*c*d + 13*a*C*d + 9*a*B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[Arc Sin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] - (9*a^(3/4)*c^(1/4)*d*(7*B *c*d^2 + 21*A*c*d*e + 11*a*C*d*e + 6*a*B*e^2)*Sqrt[1 - (c*x^4)/a]*Elliptic F[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] - (3*a^(7/4)*e*(24*c*C *d^2 + 49*a*C*e^2 + 9*c*e*(11*B*d + 7*A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*Sqrt[a - c*x^4]) + (a^(5/4)*( 8*c*C*d^3 + 9*c*d*e*(6*B*d + 7*A*e) + 15*a*e^2*(3*C*d + B*e))*Sqrt[1 - (c* x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^ 4]))/(7*c))/(9*c)
Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol ] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] }, Simp[C*x*(d + e*x^2)^q*(Sqrt[a + c*x^4]/(c*(2*q + 3))), x] + Simp[1/(c*( 2*q + 3)) Int[((d + e*x^2)^(q - 1)/Sqrt[a + c*x^4])*Simp[A*c*d*(2*q + 3) - a*C*d + (c*(B*d + A*e)*(2*q + 3) - a*C*e*(2*q + 1))*x^2 + (B*c*e*(2*q + 3 ) + 2*c*C*d*q)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2 ] && EqQ[Expon[P4x, x], 4] && IGtQ[q, 0]
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 = 4.85 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.91
| method | result | size |
| elliptic | \(-\frac {C \,e^{3} x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {\left (e^{3} B +3 d \,e^{2} C \right ) x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (A \,e^{3}+3 B d \,e^{2}+3 C \,d^{2} e +\frac {7 a C \,e^{3}}{9 c}\right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (3 A d \,e^{2}+3 B e \,d^{2}+C \,d^{3}+\frac {5 a \left (e^{3} B +3 d \,e^{2} C \right )}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A \,d^{3}+\frac {a \left (3 A d \,e^{2}+3 B e \,d^{2}+C \,d^{3}+\frac {5 a \left (e^{3} B +3 d \,e^{2} C \right )}{7 c}\right )}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (3 A \,d^{2} e +B \,d^{3}+\frac {3 a \left (A \,e^{3}+3 B d \,e^{2}+3 C \,d^{2} e +\frac {7 a C \,e^{3}}{9 c}\right )}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(408\) |
| default | \(\frac {A \,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 {d^{2} \left (3 A e +B d \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}}+e^{2} \left (B e +3 C d \right ) \left (-\frac {x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {5 a x \sqrt {-c \,x^{4}+a}}{21 c^{2}}+\frac {5 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^{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^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 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 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 \sqrt {-c \,x^{4}+a}}{3 c}+\frac {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 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e^{3} C \left (-\frac {x^{7} \sqrt {-c \,x^{4}+a}}{9 c}-\frac {7 a \,x^{3} \sqrt {-c \,x^{4}+a}}{45 c^{2}}-\frac {7 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 {5}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(644\) |
| risch | \(-\frac {x \left (35 C \,x^{6} c \,e^{3}+45 B c \,e^{3} x^{4}+135 C c d \,e^{2} x^{4}+63 A c \,e^{3} x^{2}+189 B c d \,e^{2} x^{2}+49 C a \,e^{3} x^{2}+189 C c \,d^{2} e \,x^{2}+315 A c d \,e^{2}+75 B a \,e^{3}+315 B c \,d^{2} e +225 a C d \,e^{2}+105 C c \,d^{3}\right ) \sqrt {-c \,x^{4}+a}}{315 c^{2}}+\frac {-\frac {\left (63 A a c \,e^{3}+315 A \,d^{2} e \,c^{2}+189 B a c d \,e^{2}+105 B \,d^{3} c^{2}+49 C \,a^{2} e^{3}+189 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 {105 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 {25 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 {35 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 {75 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 {105 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 {105 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}}}{105 c^{2}}\) | \(704\) |
Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/9*C*e^3*x^7*(-c*x^4+a)^(1/2)/c-1/7*(B*e^3+3*C*d*e^2)/c*x^5*(-c*x^4+a)^( 1/2)-1/5*(A*e^3+3*B*d*e^2+3*C*d^2*e+7/9*a/c*C*e^3)/c*x^3*(-c*x^4+a)^(1/2)- 1/3*(3*A*d*e^2+3*B*e*d^2+C*d^3+5/7*a/c*(B*e^3+3*C*d*e^2))/c*x*(-c*x^4+a)^( 1/2)+(A*d^3+1/3*a/c*(3*A*d*e^2+3*B*e*d^2+C*d^3+5/7*a/c*(B*e^3+3*C*d*e^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)-(3*A*d^ 2*e+B*d^3+3/5*a/c*(A*e^3+3*B*d*e^2+3*C*d^2*e+7/9*a/c*C*e^3))*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)-Elli pticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {21 \, {\left (15 \, B a c^{2} d^{3} + 27 \, B a^{2} c d e^{2} + 9 \, {\left (3 \, C a^{2} c + 5 \, A a c^{2}\right )} d^{2} e + {\left (7 \, C a^{3} + 9 \, A a^{2} c\right )} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 3 \, {\left (35 \, {\left ({\left (3 \, B + C\right )} a c^{2} + 3 \, A c^{3}\right )} d^{3} + 21 \, {\left (9 \, C a^{2} c + 5 \, {\left (3 \, A + B\right )} a c^{2}\right )} d^{2} e + 3 \, {\left ({\left (63 \, B + 25 \, C\right )} a^{2} c + 35 \, A a c^{2}\right )} d e^{2} + {\left (49 \, C a^{3} + {\left (63 \, A + 25 \, B\right )} a^{2} c\right )} e^{3}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (35 \, C a c^{2} e^{3} x^{8} + 315 \, B a c^{2} d^{3} + 567 \, B a^{2} c d e^{2} + 45 \, {\left (3 \, C a c^{2} d e^{2} + B a c^{2} e^{3}\right )} x^{6} + 7 \, {\left (27 \, C a c^{2} d^{2} e + 27 \, B a c^{2} d e^{2} + {\left (7 \, C a^{2} c + 9 \, A a c^{2}\right )} e^{3}\right )} x^{4} + 189 \, {\left (3 \, C a^{2} c + 5 \, A a c^{2}\right )} d^{2} e + 21 \, {\left (7 \, C a^{3} + 9 \, A a^{2} c\right )} e^{3} + 15 \, {\left (7 \, C a c^{2} d^{3} + 21 \, B a c^{2} d^{2} e + 5 \, B a^{2} c e^{3} + 3 \, {\left (5 \, C a^{2} c + 7 \, A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{315 \, a c^{3} x} \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="frica s")
Output:
-1/315*(21*(15*B*a*c^2*d^3 + 27*B*a^2*c*d*e^2 + 9*(3*C*a^2*c + 5*A*a*c^2)* d^2*e + (7*C*a^3 + 9*A*a^2*c)*e^3)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_e(arcsi n((a/c)^(1/4)/x), -1) - 3*(35*((3*B + C)*a*c^2 + 3*A*c^3)*d^3 + 21*(9*C*a^ 2*c + 5*(3*A + B)*a*c^2)*d^2*e + 3*((63*B + 25*C)*a^2*c + 35*A*a*c^2)*d*e^ 2 + (49*C*a^3 + (63*A + 25*B)*a^2*c)*e^3)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_ f(arcsin((a/c)^(1/4)/x), -1) + (35*C*a*c^2*e^3*x^8 + 315*B*a*c^2*d^3 + 567 *B*a^2*c*d*e^2 + 45*(3*C*a*c^2*d*e^2 + B*a*c^2*e^3)*x^6 + 7*(27*C*a*c^2*d^ 2*e + 27*B*a*c^2*d*e^2 + (7*C*a^2*c + 9*A*a*c^2)*e^3)*x^4 + 189*(3*C*a^2*c + 5*A*a*c^2)*d^2*e + 21*(7*C*a^3 + 9*A*a^2*c)*e^3 + 15*(7*C*a*c^2*d^3 + 2 1*B*a*c^2*d^2*e + 5*B*a^2*c*e^3 + 3*(5*C*a^2*c + 7*A*a*c^2)*d*e^2)*x^2)*sq rt(-c*x^4 + a))/(a*c^3*x)
Time = 6.06 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.27 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)**3*(C*x**4+B*x**2+A)/(-c*x**4+a)**(1/2),x)
Output:
A*d**3*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/ (4*sqrt(a)*gamma(5/4)) + 3*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*sqrt(a)*gamma(7/4)) + 3*A*d*e**2*x**5*g amma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a) *gamma(9/4)) + A*e**3*x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), c*x**4*ex p_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(11/4)) + B*d**3*x**3*gamma(3/4)*hyper( (1/2, 3/4), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + 3 *B*d**2*e*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*p i)/a)/(4*sqrt(a)*gamma(9/4)) + 3*B*d*e**2*x**7*gamma(7/4)*hyper((1/2, 7/4) , (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(11/4)) + B*e**3*x* *9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sq rt(a)*gamma(13/4)) + 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*sqrt(a)*gamma(9/4)) + 3*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*sqrt(a)*gamma( 11/4)) + 3*C*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*sqrt(a)*gamma(13/4)) + 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*sqrt(a)*gamma(15/4))
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^3/sqrt(-c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^3/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {a-c\,x^4}} \,d x \] Input:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2),x)
Output:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {-75 \sqrt {-c \,x^{4}+a}\, a b \,e^{3} x -540 \sqrt {-c \,x^{4}+a}\, a c d \,e^{2} x -112 \sqrt {-c \,x^{4}+a}\, a c \,e^{3} x^{3}-315 \sqrt {-c \,x^{4}+a}\, b c \,d^{2} e x -189 \sqrt {-c \,x^{4}+a}\, b c d \,e^{2} x^{3}-45 \sqrt {-c \,x^{4}+a}\, b c \,e^{3} x^{5}-105 \sqrt {-c \,x^{4}+a}\, c^{2} d^{3} x -189 \sqrt {-c \,x^{4}+a}\, c^{2} d^{2} e \,x^{3}-135 \sqrt {-c \,x^{4}+a}\, c^{2} d \,e^{2} x^{5}-35 \sqrt {-c \,x^{4}+a}\, c^{2} e^{3} x^{7}+75 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} b \,e^{3}+540 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} c d \,e^{2}+315 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a b c \,d^{2} e +420 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a \,c^{2} d^{3}+336 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a^{2} c \,e^{3}+567 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a b c d \,e^{2}+1512 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a \,c^{2} d^{2} e +315 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) b \,c^{2} d^{3}}{315 c^{2}} \] Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x)
Output:
( - 75*sqrt(a - c*x**4)*a*b*e**3*x - 540*sqrt(a - c*x**4)*a*c*d*e**2*x - 1 12*sqrt(a - c*x**4)*a*c*e**3*x**3 - 315*sqrt(a - c*x**4)*b*c*d**2*e*x - 18 9*sqrt(a - c*x**4)*b*c*d*e**2*x**3 - 45*sqrt(a - c*x**4)*b*c*e**3*x**5 - 1 05*sqrt(a - c*x**4)*c**2*d**3*x - 189*sqrt(a - c*x**4)*c**2*d**2*e*x**3 - 135*sqrt(a - c*x**4)*c**2*d*e**2*x**5 - 35*sqrt(a - c*x**4)*c**2*e**3*x**7 + 75*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**2*b*e**3 + 540*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a**2*c*d*e**2 + 315*int(sqrt(a - c*x**4)/(a - c*x* *4),x)*a*b*c*d**2*e + 420*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*c**2*d**3 + 336*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a**2*c*e**3 + 567*int(( sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a*b*c*d*e**2 + 1512*int((sqrt(a - c *x**4)*x**2)/(a - c*x**4),x)*a*c**2*d**2*e + 315*int((sqrt(a - c*x**4)*x** 2)/(a - c*x**4),x)*b*c**2*d**3)/(315*c**2)