Integrand size = 34, antiderivative size = 519 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (A c d \left (c d^2+3 a e^2\right )+a \left (a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )+\left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{6 a c^2 \left (a-c x^4\right )^{3/2}}+\frac {x \left (A c d \left (5 c d^2-3 a e^2\right )-a \left (7 a e^2 (3 C d+B e)+c d^2 (C d+3 B e)\right )+3 \left (B c d \left (c d^2-3 a e^2\right )+A c e \left (3 c d^2-a e^2\right )-3 a C e \left (c d^2+a e^2\right )\right ) x^2\right )}{12 a^2 c^2 \sqrt {a-c x^4}}-\frac {\left (B c d \left (c d^2-3 a e^2\right )+A c e \left (3 c d^2-a e^2\right )-a C e \left (3 c d^2+7 a e^2\right )\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{5/4} c^{11/4} \sqrt {a-c x^4}}+\frac {\left (5 A c^{5/2} d^3-21 a^{5/2} C e^3+3 \sqrt {a} c^2 d^2 (B d+3 A e)+5 a^2 \sqrt {c} e^2 (3 C d+B e)-a c^{3/2} d \left (C d^2+3 e (B d+A e)\right )-3 a^{3/2} c e \left (3 C d^2+e (3 B d+A e)\right )\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c^{11/4} \sqrt {a-c x^4}} \] Output:
1/6*x*(A*c*d*(3*a*e^2+c*d^2)+a*(a*e^2*(B*e+3*C*d)+c*d^2*(3*B*e+C*d))+((A*c +C*a)*e*(a*e^2+3*c*d^2)+B*c*d*(3*a*e^2+c*d^2))*x^2)/a/c^2/(-c*x^4+a)^(3/2) +1/12*x*(A*c*d*(-3*a*e^2+5*c*d^2)-a*(7*a*e^2*(B*e+3*C*d)+c*d^2*(3*B*e+C*d) )+3*(B*c*d*(-3*a*e^2+c*d^2)+A*c*e*(-a*e^2+3*c*d^2)-3*a*C*e*(a*e^2+c*d^2))* x^2)/a^2/c^2/(-c*x^4+a)^(1/2)-1/4*(B*c*d*(-3*a*e^2+c*d^2)+A*c*e*(-a*e^2+3* c*d^2)-a*C*e*(7*a*e^2+3*c*d^2))*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1 /4),I)/a^(5/4)/c^(11/4)/(-c*x^4+a)^(1/2)+1/12*(5*A*c^(5/2)*d^3-21*a^(5/2)* C*e^3+3*a^(1/2)*c^2*d^2*(3*A*e+B*d)+5*a^2*c^(1/2)*e^2*(B*e+3*C*d)-a*c^(3/2 )*d*(C*d^2+3*e*(A*e+B*d))-3*a^(3/2)*c*e*(3*C*d^2+e*(A*e+3*B*d)))*(1-c*x^4/ a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(7/4)/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 = 10.70 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (-5 A c^3 d^3 x^4+a^3 e^2 \left (-15 C d-5 B e+28 C e x^2\right )+a c^2 d \left (d (C d+3 B e) x^4+A \left (7 d^2+3 e^2 x^4\right )\right )+a^2 c \left (C \left (d^3+12 d^2 e x^2+21 d e^2 x^4-12 e^3 x^6\right )+e \left (A e \left (3 d+4 e x^2\right )+B \left (3 d^2+12 d e x^2+7 e^2 x^4\right )\right )\right )\right )+\left (A c d \left (5 c d^2-3 a e^2\right )+a \left (5 a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) x \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )-4 \left (A c e \left (-3 c d^2+a e^2\right )+B c d \left (-c d^2+3 a e^2\right )+a C e \left (3 c d^2+7 a e^2\right )\right ) x^3 \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{12 a^2 c^2 \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x]
Output:
(x*(-5*A*c^3*d^3*x^4 + a^3*e^2*(-15*C*d - 5*B*e + 28*C*e*x^2) + a*c^2*d*(d *(C*d + 3*B*e)*x^4 + A*(7*d^2 + 3*e^2*x^4)) + a^2*c*(C*(d^3 + 12*d^2*e*x^2 + 21*d*e^2*x^4 - 12*e^3*x^6) + e*(A*e*(3*d + 4*e*x^2) + B*(3*d^2 + 12*d*e *x^2 + 7*e^2*x^4)))) + (A*c*d*(5*c*d^2 - 3*a*e^2) + a*(5*a*e^2*(3*C*d + B* e) - c*d^2*(C*d + 3*B*e)))*x*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*Hypergeometri c2F1[1/4, 1/2, 5/4, (c*x^4)/a] - 4*(A*c*e*(-3*c*d^2 + a*e^2) + B*c*d*(-(c* d^2) + 3*a*e^2) + a*C*e*(3*c*d^2 + 7*a*e^2))*x^3*(a - c*x^4)*Sqrt[1 - (c*x ^4)/a]*Hypergeometric2F1[3/4, 5/2, 7/4, (c*x^4)/a])/(12*a^2*c^2*(a - c*x^4 )^(3/2))
Time = 1.90 (sec) , antiderivative size = 954, normalized size of antiderivative = 1.84, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^2 \left (e (a C+A c) \left (a e^2+3 c d^2\right )+B c d \left (3 a e^2+c d^2\right )\right )+A c d \left (3 a e^2+c d^2\right )+a \left (a e^2 (B e+3 C d)+c d^2 (3 B e+C d)\right )}{c^2 \left (a-c x^4\right )^{5/2}}+\frac {-e x^2 \left (2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )-2 a e^2 (B e+3 C d)-3 c d e (A e+B d)-c C d^3}{c^2 \left (a-c x^4\right )^{3/2}}+\frac {e^2 (B e+3 C d)}{c^2 \sqrt {a-c x^4}}+\frac {C e^3 x^2}{c^2 \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^{3/4} C \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) e^3}{c^{11/4} \sqrt {a-c x^4}}-\frac {a^{3/4} C \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) e^3}{c^{11/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} (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 ) e^2}{c^{9/4} \sqrt {a-c x^4}}+\frac {\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 ) e}{2 \sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}-\frac {\left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 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 )}{4 a^{5/4} c^{11/4} \sqrt {a-c x^4}}-\frac {\left (e \left (3 c C d^2+2 a C e^2+c e (3 B d+A e)\right )+\frac {\sqrt {c} \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}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{a} c^{11/4} \sqrt {a-c x^4}}+\frac {\left (3 \left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 a e^2\right )\right )+\frac {5 \sqrt {c} \left (A c d \left (c d^2+3 a e^2\right )+a \left (c (C d+3 B e) d^2+a e^2 (3 C d+B e)\right )\right )}{\sqrt {a}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{5/4} c^{11/4} \sqrt {a-c x^4}}-\frac {x \left (c C d^3+3 c e (B d+A e) d+e \left (3 c C d^2+2 a C e^2+c e (3 B d+A e)\right ) x^2+2 a e^2 (3 C d+B e)\right )}{2 a c^2 \sqrt {a-c x^4}}+\frac {x \left (3 \left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 a e^2\right )\right ) x^2+5 \left (A c d \left (c d^2+3 a e^2\right )+a \left (c (C d+3 B e) d^2+a e^2 (3 C d+B e)\right )\right )\right )}{12 a^2 c^2 \sqrt {a-c x^4}}+\frac {x \left (\left ((A c+a C) e \left (3 c d^2+a e^2\right )+B c d \left (c d^2+3 a e^2\right )\right ) x^2+A c d \left (c d^2+3 a e^2\right )+a \left (c (C d+3 B e) d^2+a e^2 (3 C d+B e)\right )\right )}{6 a c^2 \left (a-c x^4\right )^{3/2}}\) |
Input:
Int[((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x]
Output:
(x*(A*c*d*(c*d^2 + 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e) ) + ((A*c + a*C)*e*(3*c*d^2 + a*e^2) + B*c*d*(c*d^2 + 3*a*e^2))*x^2))/(6*a *c^2*(a - c*x^4)^(3/2)) - (x*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 2*a*e^2*(3*C *d + B*e) + e*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(3*B*d + A*e))*x^2))/(2*a*c^2*S qrt[a - c*x^4]) + (x*(5*(A*c*d*(c*d^2 + 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) + c*d^2*(C*d + 3*B*e))) + 3*((A*c + a*C)*e*(3*c*d^2 + a*e^2) + B*c*d*(c*d^ 2 + 3*a*e^2))*x^2))/(12*a^2*c^2*Sqrt[a - c*x^4]) + (a^(3/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(11/4)*Sqrt[a - c*x^4]) + (e*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(3*B*d + A*e))*Sqrt[1 - (c*x^4) /a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(1/4)*c^(11/4)*Sqrt[a - c*x^4]) - (((A*c + a*C)*e*(3*c*d^2 + a*e^2) + B*c*d*(c*d^2 + 3*a*e^2))* Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(4*a^(5/4) *c^(11/4)*Sqrt[a - c*x^4]) - (a^(3/4)*C*e^3*Sqrt[1 - (c*x^4)/a]*EllipticF[ ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(11/4)*Sqrt[a - c*x^4]) + (a^(1/4)*e^ 2*(3*C*d + B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(9/4)*Sqrt[a - c*x^4]) - ((e*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(3*B*d + A*e)) + (Sqrt[c]*(c*C*d^3 + 3*c*d*e*(B*d + A*e) + 2*a*e^2*(3*C*d + B*e)) )/Sqrt[a])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) /(2*a^(1/4)*c^(11/4)*Sqrt[a - c*x^4]) + ((3*((A*c + a*C)*e*(3*c*d^2 + a*e^ 2) + B*c*d*(c*d^2 + 3*a*e^2)) + (5*Sqrt[c]*(A*c*d*(c*d^2 + 3*a*e^2) + a...
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 = 4.58 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.15
| method | result | size |
| elliptic | \(\frac {\left (\frac {\left (A a c \,e^{3}+3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}+B \,d^{3} c^{2}+C \,a^{2} e^{3}+3 C a c \,d^{2} e \right ) x^{3}}{6 a \,c^{4}}+\frac {\left (3 A a c d \,e^{2}+A \,c^{2} d^{3}+a^{2} B \,e^{3}+3 B a c \,d^{2} e +3 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{6 a \,c^{4}}\right ) \sqrt {-c \,x^{4}+a}}{\left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {2 c \left (-\frac {\left (A a c \,e^{3}-3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}-B \,d^{3} c^{2}+3 C \,a^{2} e^{3}+3 C a c \,d^{2} e \right ) x^{3}}{8 a^{2} c^{3}}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+7 a^{2} B \,e^{3}+3 B a c \,d^{2} e +21 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{24 a^{2} c^{3}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (\frac {e^{2} \left (B e +3 C d \right )}{c^{2}}-\frac {3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+7 a^{2} B \,e^{3}+3 B a c \,d^{2} e +21 C \,a^{2} d \,e^{2}+C a c \,d^{3}}{12 a^{2} c^{2}}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (\frac {e^{3} C}{c^{2}}+\frac {A a c \,e^{3}-3 A \,d^{2} e \,c^{2}+3 B a c d \,e^{2}-B \,d^{3} c^{2}+3 C \,a^{2} e^{3}+3 C a c \,d^{2} e}{4 a^{2} c^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(596\) |
| default | \(A \,d^{3} \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \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 )}{12 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+d^{2} \left (3 A e +B d \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\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 )}{4 a^{\frac {3}{2}} \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 {a x \sqrt {-c \,x^{4}+a}}{6 c^{4} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {7 x}{12 c^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \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 )}{12 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}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x^{3}}{4 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\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 )}{4 c^{\frac {3}{2}} \sqrt {a}\, \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}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x}{12 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\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 )}{12 c a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e^{3} C \left (\frac {a \,x^{3} \sqrt {-c \,x^{4}+a}}{6 c^{4} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {3 x^{3}}{4 c^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {7 \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 )}{4 c^{\frac {5}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(865\) |
Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/6/a/c^4*(A*a*c*e^3+3*A*c^2*d^2*e+3*B*a*c*d*e^2+B*c^2*d^3+C*a^2*e^3+3*C* a*c*d^2*e)*x^3+1/6/a/c^4*(3*A*a*c*d*e^2+A*c^2*d^3+B*a^2*e^3+3*B*a*c*d^2*e+ 3*C*a^2*d*e^2+C*a*c*d^3)*x)*(-c*x^4+a)^(1/2)/(x^4-a/c)^2+2*c*(-1/8*(A*a*c* e^3-3*A*c^2*d^2*e+3*B*a*c*d*e^2-B*c^2*d^3+3*C*a^2*e^3+3*C*a*c*d^2*e)/a^2/c ^3*x^3-1/24/a^2/c^3*(3*A*a*c*d*e^2-5*A*c^2*d^3+7*B*a^2*e^3+3*B*a*c*d^2*e+2 1*C*a^2*d*e^2+C*a*c*d^3)*x)/(-(x^4-a/c)*c)^(1/2)+(e^2*(B*e+3*C*d)/c^2-1/12 /a^2/c^2*(3*A*a*c*d*e^2-5*A*c^2*d^3+7*B*a^2*e^3+3*B*a*c*d^2*e+21*C*a^2*d*e ^2+C*a*c*d^3))/(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)-(1/c^2*e^3*C+1/4*(A*a*c*e^3-3*A*c^2*d^2*e+3*B*a*c*d*e^2-B*c^2*d^3+3 *C*a^2*e^3+3*C*a*c*d^2*e)/a^2/c^2)*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))
Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (469) = 938\).
Time = 0.12 (sec) , antiderivative size = 954, normalized size of antiderivative = 1.84 \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="frica s")
Output:
1/12*(3*((B*a*c^4*d^3 - 3*B*a^2*c^3*d*e^2 - 3*(C*a^2*c^3 - A*a*c^4)*d^2*e - (7*C*a^3*c^2 + A*a^2*c^3)*e^3)*x^9 - 2*(B*a^2*c^3*d^3 - 3*B*a^3*c^2*d*e^ 2 - 3*(C*a^3*c^2 - A*a^2*c^3)*d^2*e - (7*C*a^4*c + A*a^3*c^2)*e^3)*x^5 + ( B*a^3*c^2*d^3 - 3*B*a^4*c*d*e^2 - 3*(C*a^4*c - A*a^3*c^2)*d^2*e - (7*C*a^5 + A*a^4*c)*e^3)*x)*sqrt(-c)*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - ((((3*B + C)*a*c^4 - 5*A*c^5)*d^3 - 3*(3*C*a^2*c^3 - (3*A + B)*a*c^ 4)*d^2*e - 3*((3*B + 5*C)*a^2*c^3 - A*a*c^4)*d*e^2 - (21*C*a^3*c^2 + (3*A + 5*B)*a^2*c^3)*e^3)*x^9 - 2*(((3*B + C)*a^2*c^3 - 5*A*a*c^4)*d^3 - 3*(3*C *a^3*c^2 - (3*A + B)*a^2*c^3)*d^2*e - 3*((3*B + 5*C)*a^3*c^2 - A*a^2*c^3)* d*e^2 - (21*C*a^4*c + (3*A + 5*B)*a^3*c^2)*e^3)*x^5 + (((3*B + C)*a^3*c^2 - 5*A*a^2*c^3)*d^3 - 3*(3*C*a^4*c - (3*A + B)*a^3*c^2)*d^2*e - 3*((3*B + 5 *C)*a^4*c - A*a^3*c^2)*d*e^2 - (21*C*a^5 + (3*A + 5*B)*a^4*c)*e^3)*x)*sqrt (-c)*(a/c)^(3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) - (12*C*a^3*c^2*e^3 *x^8 - 3*B*a^3*c^2*d^3 + 9*B*a^4*c*d*e^2 - (3*B*a^2*c^3*d^2*e + 7*B*a^3*c^ 2*e^3 + (C*a^2*c^3 - 5*A*a*c^4)*d^3 + 3*(7*C*a^3*c^2 + A*a^2*c^3)*d*e^2)*x ^6 + (B*a^2*c^3*d^3 - 15*B*a^3*c^2*d*e^2 - 3*(5*C*a^3*c^2 - A*a^2*c^3)*d^2 *e - 5*(7*C*a^4*c + A*a^3*c^2)*e^3)*x^4 + 9*(C*a^4*c - A*a^3*c^2)*d^2*e + 3*(7*C*a^5 + A*a^4*c)*e^3 - (3*B*a^3*c^2*d^2*e - 5*B*a^4*c*e^3 + (C*a^3*c^ 2 + 7*A*a^2*c^3)*d^3 - 3*(5*C*a^4*c - A*a^3*c^2)*d*e^2)*x^2)*sqrt(-c*x^4 + a))/(a^3*c^5*x^9 - 2*a^4*c^4*x^5 + a^5*c^3*x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {\left (d + e x^{2}\right )^{3} \left (A + B x^{2} + C x^{4}\right )}{\left (a - c x^{4}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x**2+d)**3*(C*x**4+B*x**2+A)/(-c*x**4+a)**(5/2),x)
Output:
Integral((d + e*x**2)**3*(A + B*x**2 + C*x**4)/(a - c*x**4)**(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="maxim a")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^3/(-c*x^4 + a)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{3}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="giac" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^3/(-c*x^4 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{5/2}} \,d x \] Input:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x)
Output:
int(((d + e*x^2)^3*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right )^3 \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^3*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x)
Output:
( - 15*sqrt(a - c*x**4)*a*b*e**3*x - 36*sqrt(a - c*x**4)*a*c*d*e**2*x + 40 *sqrt(a - c*x**4)*a*c*e**3*x**3 + 9*sqrt(a - c*x**4)*b*c*d**2*e*x + 15*sqr t(a - c*x**4)*b*c*d*e**2*x**3 + 15*sqrt(a - c*x**4)*b*c*e**3*x**5 + 3*sqrt (a - c*x**4)*c**2*d**3*x + 15*sqrt(a - c*x**4)*c**2*d**2*e*x**3 + 45*sqrt( a - c*x**4)*c**2*d*e**2*x**5 - 15*sqrt(a - c*x**4)*c**2*e**3*x**7 + 15*int (sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a **4*b*e**3 + 36*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**4*c*d*e**2 - 9*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c* x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**3*b*c*d**2*e - 30*int(sqrt(a - c* x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**3*b*c*e**3 *x**4 + 12*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c* *3*x**12),x)*a**3*c**2*d**3 - 72*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x** 4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**3*c**2*d*e**2*x**4 + 18*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2*b*c** 2*d**2*e*x**4 + 15*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x **8 - c**3*x**12),x)*a**2*b*c**2*e**3*x**8 - 24*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2*c**3*d**3*x**4 + 36 *int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12), x)*a**2*c**3*d*e**2*x**8 - 9*int(sqrt(a - c*x**4)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a*b*c**3*d**2*e*x**8 + 12*int(sqrt(a - c...