Integrand size = 34, antiderivative size = 325 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {\left (5 a C e^2+7 c \left (C d^2+e (2 B d+A e)\right )\right ) x \sqrt {a-c x^4}}{21 c^2}-\frac {e (2 C d+B e) x^3 \sqrt {a-c x^4}}{5 c}-\frac {C e^2 x^5 \sqrt {a-c x^4}}{7 c}+\frac {a^{3/4} \left (5 B c d^2+10 A c d e+6 a C d e+3 a B e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \left (21 \sqrt {a} \sqrt {c} \left (5 B c d^2+10 A c d e+6 a C d e+3 a B e^2\right )-5 \left (7 A c \left (3 c d^2+a e^2\right )+a \left (5 a C e^2+7 c d (C d+2 B e)\right )\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{105 c^{9/4} \sqrt {a-c x^4}} \] Output:
-1/21*(5*C*a*e^2+7*c*(C*d^2+e*(A*e+2*B*d)))*x*(-c*x^4+a)^(1/2)/c^2-1/5*e*( B*e+2*C*d)*x^3*(-c*x^4+a)^(1/2)/c-1/7*C*e^2*x^5*(-c*x^4+a)^(1/2)/c+1/5*a^( 3/4)*(10*A*c*d*e+3*B*a*e^2+5*B*c*d^2+6*C*a*d*e)*(1-c*x^4/a)^(1/2)*Elliptic E(c^(1/4)*x/a^(1/4),I)/c^(7/4)/(-c*x^4+a)^(1/2)-1/105*a^(1/4)*(21*a^(1/2)* c^(1/2)*(10*A*c*d*e+3*B*a*e^2+5*B*c*d^2+6*C*a*d*e)-35*A*c*(a*e^2+3*c*d^2)- 5*a*(5*C*a*e^2+7*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 = 10.34 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {x \left (-a+c x^4\right ) \left (25 a C e^2+7 c e \left (10 B d+5 A e+3 B e x^2\right )+c C \left (35 d^2+42 d e x^2+15 e^2 x^4\right )\right )+5 \left (7 A c \left (3 c d^2+a e^2\right )+a \left (5 a C e^2+7 c d (C d+2 B e)\right )\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+7 c \left (5 B c d^2+10 A c d e+6 a C d e+3 a B e^2\right ) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )}{105 c^2 \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
(x*(-a + c*x^4)*(25*a*C*e^2 + 7*c*e*(10*B*d + 5*A*e + 3*B*e*x^2) + c*C*(35 *d^2 + 42*d*e*x^2 + 15*e^2*x^4)) + 5*(7*A*c*(3*c*d^2 + a*e^2) + a*(5*a*C*e ^2 + 7*c*d*(C*d + 2*B*e)))*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/ 2, 5/4, (c*x^4)/a] + 7*c*(5*B*c*d^2 + 10*A*c*d*e + 6*a*C*d*e + 3*a*B*e^2)* x^3*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, (c*x^4)/a])/(105* c^2*Sqrt[a - c*x^4])
Time = 0.94 (sec) , antiderivative size = 535, normalized size of antiderivative = 1.65, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx\) |
| \(\Big \downarrow \) 2209 |
| \(\displaystyle -\frac {\int -\frac {\left (e x^2+d\right ) \left (c (4 C d+7 B e) x^4+(7 B c d+7 A c e+5 a C e) x^2+(7 A c+a C) d\right )}{\sqrt {a-c x^4}}dx}{7 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^2}{7 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (e x^2+d\right ) \left (c (4 C d+7 B e) x^4+(7 B c d+7 A c e+5 a C e) x^2+(7 A c+a C) d\right )}{\sqrt {a-c x^4}}dx}{7 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^2}{7 c}\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \frac {\int \left (\frac {c e (4 C d+7 B e) x^6}{\sqrt {a-c x^4}}+\frac {\left (4 c C d^2+5 a C e^2+7 c e (2 B d+A e)\right ) x^4}{\sqrt {a-c x^4}}+\frac {d (7 B c d+14 A c e+6 a C e) x^2}{\sqrt {a-c x^4}}+\frac {(7 A c+a C) d^2}{\sqrt {a-c x^4}}\right )dx}{7 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^2}{7 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^{5/4} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (5 a C e^2+7 c e (A e+2 B d)+4 c C d^2\right )}{3 c^{5/4} \sqrt {a-c x^4}}-\frac {a^{3/4} d \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) (6 a C e+14 A c e+7 B c d)}{c^{3/4} \sqrt {a-c x^4}}+\frac {a^{3/4} d \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (6 a C e+14 A c e+7 B c d)}{c^{3/4} \sqrt {a-c x^4}}-\frac {3 a^{7/4} e \sqrt {1-\frac {c x^4}{a}} (7 B e+4 C d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{5 c^{3/4} \sqrt {a-c x^4}}+\frac {3 a^{7/4} e \sqrt {1-\frac {c x^4}{a}} (7 B e+4 C d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} d^2 \sqrt {1-\frac {c x^4}{a}} (a C+7 A c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}-\frac {x \sqrt {a-c x^4} \left (5 a C e^2+7 c e (A e+2 B d)+4 c C d^2\right )}{3 c}-\frac {1}{5} e x^3 \sqrt {a-c x^4} (7 B e+4 C d)}{7 c}-\frac {C x \sqrt {a-c x^4} \left (d+e x^2\right )^2}{7 c}\) |
Input:
Int[((d + e*x^2)^2*(A + B*x^2 + C*x^4))/Sqrt[a - c*x^4],x]
Output:
-1/7*(C*x*(d + e*x^2)^2*Sqrt[a - c*x^4])/c + (-1/3*((4*c*C*d^2 + 5*a*C*e^2 + 7*c*e*(2*B*d + A*e))*x*Sqrt[a - c*x^4])/c - (e*(4*C*d + 7*B*e)*x^3*Sqrt [a - c*x^4])/5 + (3*a^(7/4)*e*(4*C*d + 7*B*e)*Sqrt[1 - (c*x^4)/a]*Elliptic E[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*Sqrt[a - c*x^4]) + (a^(3/4) *d*(7*B*c*d + 14*A*c*e + 6*a*C*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]) + (a^(1/4)*(7*A*c + a*C) *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]) - (3*a^(7/4)*e*(4*C*d + 7*B*e)*Sqrt[1 - (c*x^4)/a]*El lipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(5*c^(3/4)*Sqrt[a - c*x^4]) - (a ^(3/4)*d*(7*B*c*d + 14*A*c*e + 6*a*C*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcS in[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(5/4)*(4*c*C* d^2 + 5*a*C*e^2 + 7*c*e*(2*B*d + A*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSi n[(c^(1/4)*x)/a^(1/4)], -1])/(3*c^(5/4)*Sqrt[a - c*x^4]))/(7*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 = 3.04 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.96
| method | result | size |
| elliptic | \(-\frac {C \,e^{2} x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {\left (B \,e^{2}+2 C d e \right ) x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {\left (A \,e^{2}+2 B d e +C \,d^{2}+\frac {5 a C \,e^{2}}{7 c}\right ) x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (A \,d^{2}+\frac {a \left (A \,e^{2}+2 B d e +C \,d^{2}+\frac {5 a C \,e^{2}}{7 c}\right )}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (2 d e A +B \,d^{2}+\frac {3 a \left (B \,e^{2}+2 C d e \right )}{5 c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(311\) |
| default | \(\frac {A \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {d \left (2 A e +B d \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}+e \left (B e +2 C d \right ) \left (-\frac {x^{3} \sqrt {-c \,x^{4}+a}}{5 c}-\frac {3 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A \,e^{2}+2 B d e +C \,d^{2}\right ) \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+C \,e^{2} \left (-\frac {x^{5} \sqrt {-c \,x^{4}+a}}{7 c}-\frac {5 a x \sqrt {-c \,x^{4}+a}}{21 c^{2}}+\frac {5 a^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{21 c^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(494\) |
| risch | \(-\frac {x \left (15 C \,x^{4} c \,e^{2}+21 B c \,e^{2} x^{2}+42 C c d e \,x^{2}+35 A c \,e^{2}+70 B c d e +25 C a \,e^{2}+35 C c \,d^{2}\right ) \sqrt {-c \,x^{4}+a}}{105 c^{2}}+\frac {-\frac {21 \sqrt {c}\, \left (10 A c d e +3 B a \,e^{2}+5 B c \,d^{2}+6 C a d e \right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {105 A \,c^{2} d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {25 a^{2} C \,e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {35 A a c \,e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {35 C a c \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {70 B a c d e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{105 c^{2}}\) | \(546\) |
Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/7*C*e^2*x^5*(-c*x^4+a)^(1/2)/c-1/5*(B*e^2+2*C*d*e)/c*x^3*(-c*x^4+a)^(1/ 2)-1/3*(A*e^2+2*B*d*e+C*d^2+5/7*a/c*C*e^2)/c*x*(-c*x^4+a)^(1/2)+(A*d^2+1/3 *a/c*(A*e^2+2*B*d*e+C*d^2+5/7*a/c*C*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)*Ellip ticF(x*(c^(1/2)/a^(1/2))^(1/2),I)-(2*d*e*A+B*d^2+3/5*a/c*(B*e^2+2*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)*(EllipticF(x*(c^(1/2)/a^(1/2))^ (1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=-\frac {21 \, {\left (5 \, B a c d^{2} + 3 \, B a^{2} e^{2} + 2 \, {\left (3 \, C a^{2} + 5 \, A a c\right )} d e\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (35 \, {\left ({\left (3 \, B + C\right )} a c + 3 \, A c^{2}\right )} d^{2} + 14 \, {\left (9 \, C a^{2} + 5 \, {\left (3 \, A + B\right )} a c\right )} d e + {\left ({\left (63 \, B + 25 \, C\right )} a^{2} + 35 \, A a c\right )} e^{2}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (15 \, C a c e^{2} x^{6} + 105 \, B a c d^{2} + 63 \, B a^{2} e^{2} + 21 \, {\left (2 \, C a c d e + B a c e^{2}\right )} x^{4} + 42 \, {\left (3 \, C a^{2} + 5 \, A a c\right )} d e + 5 \, {\left (7 \, C a c d^{2} + 14 \, B a c d e + {\left (5 \, C a^{2} + 7 \, A a c\right )} e^{2}\right )} x^{2}\right )} \sqrt {-c x^{4} + a}}{105 \, a c^{2} x} \] Input:
integrate((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x, algorithm="frica s")
Output:
-1/105*(21*(5*B*a*c*d^2 + 3*B*a^2*e^2 + 2*(3*C*a^2 + 5*A*a*c)*d*e)*sqrt(-c )*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - (35*((3*B + C)*a*c + 3*A*c^2)*d^2 + 14*(9*C*a^2 + 5*(3*A + B)*a*c)*d*e + ((63*B + 25*C)*a^2 + 35*A*a*c)*e^2)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + (15*C*a*c*e^2*x^6 + 105*B*a*c*d^2 + 63*B*a^2*e^2 + 21*(2*C*a*c*d*e + B*a*c*e^2)*x^4 + 42*(3*C*a^2 + 5*A*a*c)*d*e + 5*(7*C*a*c*d^2 + 14*B*a*c*d *e + (5*C*a^2 + 7*A*a*c)*e^2)*x^2)*sqrt(-c*x^4 + a))/(a*c^2*x)
Time = 4.34 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx =\text {Too large to display} \] Input:
integrate((e*x**2+d)**2*(C*x**4+B*x**2+A)/(-c*x**4+a)**(1/2),x)
Output:
A*d**2*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)) + 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*sqrt(a)*gamma(7/4)) + 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*sqrt(a)*gamma(9/ 4)) + B*d**2*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)) + B*d*e*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*sqrt(a)*gamma(9/4)) + B*e**2*x**7*g amma(7/4)*hyper((1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a )*gamma(11/4)) + C*d**2*x**5*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**4*e xp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(9/4)) + C*d*e*x**7*gamma(7/4)*hyper(( 1/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(2*sqrt(a)*gamma(11/4)) + C*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))
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^2*(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)^2/sqrt(-c*x^4 + a), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{\sqrt {-c x^{4} + a}} \,d x } \] Input:
integrate((e*x^2+d)^2*(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)^2/sqrt(-c*x^4 + a), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {a-c\,x^4}} \,d x \] Input:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2),x)
Output:
int(((d + e*x^2)^2*(A + B*x^2 + C*x^4))/(a - c*x^4)^(1/2), x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (A+B x^2+C x^4\right )}{\sqrt {a-c x^4}} \, dx=\frac {-60 \sqrt {-c \,x^{4}+a}\, a \,e^{2} x -70 \sqrt {-c \,x^{4}+a}\, b d e x -21 \sqrt {-c \,x^{4}+a}\, b \,e^{2} x^{3}-35 \sqrt {-c \,x^{4}+a}\, c \,d^{2} x -42 \sqrt {-c \,x^{4}+a}\, c d e \,x^{3}-15 \sqrt {-c \,x^{4}+a}\, c \,e^{2} x^{5}+60 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a^{2} e^{2}+70 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a b d e +140 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,x^{4}+a}d x \right ) a c \,d^{2}+63 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a b \,e^{2}+336 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) a c d e +105 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c \,x^{4}+a}d x \right ) b c \,d^{2}}{105 c} \] Input:
int((e*x^2+d)^2*(C*x^4+B*x^2+A)/(-c*x^4+a)^(1/2),x)
Output:
( - 60*sqrt(a - c*x**4)*a*e**2*x - 70*sqrt(a - c*x**4)*b*d*e*x - 21*sqrt(a - c*x**4)*b*e**2*x**3 - 35*sqrt(a - c*x**4)*c*d**2*x - 42*sqrt(a - c*x**4 )*c*d*e*x**3 - 15*sqrt(a - c*x**4)*c*e**2*x**5 + 60*int(sqrt(a - c*x**4)/( a - c*x**4),x)*a**2*e**2 + 70*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*b*d*e + 140*int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*c*d**2 + 63*int((sqrt(a - c* x**4)*x**2)/(a - c*x**4),x)*a*b*e**2 + 336*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*a*c*d*e + 105*int((sqrt(a - c*x**4)*x**2)/(a - c*x**4),x)*b*c *d**2)/(105*c)