Integrand size = 32, antiderivative size = 283 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {x \left (A c d+a C d+a B e+(B c d+A c e+a C e) x^2\right )}{6 a c \left (a-c x^4\right )^{3/2}}+\frac {x \left (5 A c d-a (C d+B e)+3 (B c d+A c e-a C e) x^2\right )}{12 a^2 c \sqrt {a-c x^4}}-\frac {(B c d+A c e-a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{4 a^{5/4} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (3 \sqrt {a} \left (B d+\left (A-\frac {a C}{c}\right ) e\right )+\frac {5 A c d-a (C d+B e)}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{12 a^{7/4} c^{3/4} \sqrt {a-c x^4}} \] Output:
1/6*x*(A*c*d+C*a*d+B*a*e+(A*c*e+B*c*d+C*a*e)*x^2)/a/c/(-c*x^4+a)^(3/2)+1/1 2*x*(5*A*c*d-a*(B*e+C*d)+3*(A*c*e+B*c*d-C*a*e)*x^2)/a^2/c/(-c*x^4+a)^(1/2) -1/4*(A*c*e+B*c*d-C*a*e)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/ a^(5/4)/c^(7/4)/(-c*x^4+a)^(1/2)+1/12*(3*a^(1/2)*(B*d+(A-a*C/c)*e)+(5*A*c* d-a*(B*e+C*d))/c^(1/2))*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a ^(7/4)/c^(3/4)/(-c*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.67 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {-5 A c^2 d x^5+a c x \left (7 A d+(C d+B e) x^4\right )+a^2 x \left (B e+C \left (d+4 e x^2\right )\right )-(-5 A c d+a C d+a B e) x \left (a-c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )-4 (B c d+A c e-a C e) x^3 \left (-a+c x^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{12 a^2 c \left (a-c x^4\right )^{3/2}} \] Input:
Integrate[((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x]
Output:
(-5*A*c^2*d*x^5 + a*c*x*(7*A*d + (C*d + B*e)*x^4) + a^2*x*(B*e + C*(d + 4* e*x^2)) - (-5*A*c*d + a*C*d + a*B*e)*x*(a - c*x^4)*Sqrt[1 - (c*x^4)/a]*Hyp ergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] - 4*(B*c*d + A*c*e - a*C*e)*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*(a - c*x^4)^(3/2))
Time = 0.82 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.59, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^2 (a C e+A c e+B c d)+a B e+a C d+A c d}{c \left (a-c x^4\right )^{5/2}}+\frac {B e+C d+C e x^2}{c \sqrt {a-c x^4} \left (c x^4-a\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (\frac {5 \sqrt {c} (a B e+a C d+A c d)}{\sqrt {a}}+3 (a C e+A c e+B c d)\right )}{12 a^{5/4} c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (a C e+A c e+B c d)}{4 a^{5/4} c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right ) \left (\sqrt {a} C e+\sqrt {c} (B e+C d)\right )}{2 a^{3/4} c^{7/4} \sqrt {a-c x^4}}+\frac {x \left (3 x^2 (a C e+A c e+B c d)+5 (a B e+a C d+A c d)\right )}{12 a^2 c \sqrt {a-c x^4}}+\frac {x \left (x^2 (a C e+A c e+B c d)+a B e+a C d+A c d\right )}{6 a c \left (a-c x^4\right )^{3/2}}+\frac {C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}-\frac {x \left (B e+C d+C e x^2\right )}{2 a c \sqrt {a-c x^4}}\) |
Input:
Int[((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x]
Output:
(x*(A*c*d + a*C*d + a*B*e + (B*c*d + A*c*e + a*C*e)*x^2))/(6*a*c*(a - c*x^ 4)^(3/2)) - (x*(C*d + B*e + C*e*x^2))/(2*a*c*Sqrt[a - c*x^4]) + (x*(5*(A*c *d + a*C*d + a*B*e) + 3*(B*c*d + A*c*e + a*C*e)*x^2))/(12*a^2*c*Sqrt[a - c *x^4]) + (C*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], - 1])/(2*a^(1/4)*c^(7/4)*Sqrt[a - c*x^4]) - ((B*c*d + A*c*e + a*C*e)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(4*a^(5/4)*c^(7/4 )*Sqrt[a - c*x^4]) - ((Sqrt[a]*C*e + Sqrt[c]*(C*d + B*e))*Sqrt[1 - (c*x^4) /a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(2*a^(3/4)*c^(7/4)*Sqrt[a - c*x^4]) + (((5*Sqrt[c]*(A*c*d + a*C*d + a*B*e))/Sqrt[a] + 3*(B*c*d + A*c *e + a*C*e))*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1 ])/(12*a^(5/4)*c^(7/4)*Sqrt[a - c*x^4])
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 = 0.83 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\left (\frac {\left (A c e +B c d +C a e \right ) x^{3}}{6 a \,c^{3}}+\frac {\left (A c d +B a e +C a d \right ) x}{6 a \,c^{3}}\right ) \sqrt {-c \,x^{4}+a}}{\left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {2 c \left (\frac {\left (A c e +B c d -C a e \right ) x^{3}}{8 a^{2} c^{2}}+\frac {\left (5 A c d -B a e -C a d \right ) x}{24 a^{2} c^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (5 A c d -B a e -C a d \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 a^{2} c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (A c e +B c d -C a e \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(328\) |
| default | \(A d \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {5 x}{12 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {5 \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 a^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A e +B d \right ) \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 a \,c^{2} \left (x^{4}-\frac {a}{c}\right )^{2}}+\frac {x^{3}}{4 a^{2} \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 a^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+\left (B e +C d \right ) \left (\frac {x \sqrt {-c \,x^{4}+a}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x}{12 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{12 c a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e C \left (\frac {x^{3} \sqrt {-c \,x^{4}+a}}{6 c^{3} \left (x^{4}-\frac {a}{c}\right )^{2}}-\frac {x^{3}}{4 c a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{4 c^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(555\) |
Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/6/a/c^3*(A*c*e+B*c*d+C*a*e)*x^3+1/6/a/c^3*(A*c*d+B*a*e+C*a*d)*x)*(-c*x^ 4+a)^(1/2)/(x^4-a/c)^2+2*c*(1/8*(A*c*e+B*c*d-C*a*e)/a^2/c^2*x^3+1/24/a^2/c ^2*(5*A*c*d-B*a*e-C*a*d)*x)/(-(x^4-a/c)*c)^(1/2)+1/12/a^2/c*(5*A*c*d-B*a*e -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)+ 1/4*(A*c*e+B*c*d-C*a*e)/a^(3/2)/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)*(Ellipt icF(x*(c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(c^(1/2)/a^(1/2))^(1/2),I))
Time = 0.10 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.49 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (B c^{3} d - {\left (C a c^{2} - A c^{3}\right )} e\right )} x^{8} + B a^{2} c d - 2 \, {\left (B a c^{2} d - {\left (C a^{2} c - A a c^{2}\right )} e\right )} x^{4} - {\left (C a^{3} - A a^{2} c\right )} e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left ({\left ({\left (C a c^{2} - {\left (5 \, A + 3 \, B\right )} c^{3}\right )} d + {\left ({\left (B + 3 \, C\right )} a c^{2} - 3 \, A c^{3}\right )} e\right )} x^{8} - 2 \, {\left ({\left (C a^{2} c - {\left (5 \, A + 3 \, B\right )} a c^{2}\right )} d + {\left ({\left (B + 3 \, C\right )} a^{2} c - 3 \, A a c^{2}\right )} e\right )} x^{4} + {\left (C a^{3} - {\left (5 \, A + 3 \, B\right )} a^{2} c\right )} d + {\left ({\left (B + 3 \, C\right )} a^{3} - 3 \, A a^{2} c\right )} e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (3 \, {\left (B c^{3} d - {\left (C a c^{2} - A c^{3}\right )} e\right )} x^{7} - {\left (B a c^{2} e + {\left (C a c^{2} - 5 \, A c^{3}\right )} d\right )} x^{5} - {\left (5 \, B a c^{2} d - {\left (C a^{2} c - 5 \, A a c^{2}\right )} e\right )} x^{3} - {\left (B a^{2} c e + {\left (C a^{2} c + 7 \, A a c^{2}\right )} d\right )} x\right )} \sqrt {-c x^{4} + a}}{12 \, {\left (a^{2} c^{4} x^{8} - 2 \, a^{3} c^{3} x^{4} + a^{4} c^{2}\right )}} \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="fricas" )
Output:
-1/12*(3*((B*c^3*d - (C*a*c^2 - A*c^3)*e)*x^8 + B*a^2*c*d - 2*(B*a*c^2*d - (C*a^2*c - A*a*c^2)*e)*x^4 - (C*a^3 - A*a^2*c)*e)*sqrt(a)*(c/a)^(3/4)*ell iptic_e(arcsin(x*(c/a)^(1/4)), -1) + (((C*a*c^2 - (5*A + 3*B)*c^3)*d + ((B + 3*C)*a*c^2 - 3*A*c^3)*e)*x^8 - 2*((C*a^2*c - (5*A + 3*B)*a*c^2)*d + ((B + 3*C)*a^2*c - 3*A*a*c^2)*e)*x^4 + (C*a^3 - (5*A + 3*B)*a^2*c)*d + ((B + 3*C)*a^3 - 3*A*a^2*c)*e)*sqrt(a)*(c/a)^(3/4)*elliptic_f(arcsin(x*(c/a)^(1/ 4)), -1) + (3*(B*c^3*d - (C*a*c^2 - A*c^3)*e)*x^7 - (B*a*c^2*e + (C*a*c^2 - 5*A*c^3)*d)*x^5 - (5*B*a*c^2*d - (C*a^2*c - 5*A*a*c^2)*e)*x^3 - (B*a^2*c *e + (C*a^2*c + 7*A*a*c^2)*d)*x)*sqrt(-c*x^4 + a))/(a^2*c^4*x^8 - 2*a^3*c^ 3*x^4 + a^4*c^2)
Time = 66.99 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {A d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {A e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {C d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {C e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)*(C*x**4+B*x**2+A)/(-c*x**4+a)**(5/2),x)
Output:
A*d*x*gamma(1/4)*hyper((1/4, 5/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4* a**(5/2)*gamma(5/4)) + A*e*x**3*gamma(3/4)*hyper((3/4, 5/2), (7/4,), c*x** 4*exp_polar(2*I*pi)/a)/(4*a**(5/2)*gamma(7/4)) + B*d*x**3*gamma(3/4)*hyper ((3/4, 5/2), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a**(5/2)*gamma(7/4)) + B*e*x**5*gamma(5/4)*hyper((5/4, 5/2), (9/4,), c*x**4*exp_polar(2*I*pi)/a) /(4*a**(5/2)*gamma(9/4)) + C*d*x**5*gamma(5/4)*hyper((5/4, 5/2), (9/4,), c *x**4*exp_polar(2*I*pi)/a)/(4*a**(5/2)*gamma(9/4)) + C*e*x**7*gamma(7/4)*h yper((7/4, 5/2), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a**(5/2)*gamma(11 /4))
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)/(-c*x^4 + a)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (-c x^{4} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x^2+d)*(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)/(-c*x^4 + a)^(5/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{5/2}} \,d x \] Input:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2),x)
Output:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(5/2), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{5/2}} \, dx=\frac {3 \sqrt {-c \,x^{4}+a}\, b e x +3 \sqrt {-c \,x^{4}+a}\, c d x +5 \sqrt {-c \,x^{4}+a}\, c e \,x^{3}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a^{3} b e +12 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a^{3} c d +6 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a^{2} b c e \,x^{4}-24 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a^{2} c^{2} d \,x^{4}-3 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a b \,c^{2} e \,x^{8}+12 \left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a \,c^{3} d \,x^{8}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a^{2} b c d -30 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) a b \,c^{2} d \,x^{4}+15 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c^{3} x^{12}+3 a \,c^{2} x^{8}-3 a^{2} c \,x^{4}+a^{3}}d x \right ) b \,c^{3} d \,x^{8}}{15 c \left (c^{2} x^{8}-2 a c \,x^{4}+a^{2}\right )} \] Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(5/2),x)
Output:
(3*sqrt(a - c*x**4)*b*e*x + 3*sqrt(a - c*x**4)*c*d*x + 5*sqrt(a - c*x**4)* c*e*x**3 - 3*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*e + 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*d + 6*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*e*x**4 - 24*int(sqr t(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**2*d*x**4 - 3*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**2*e*x**8 + 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*c**3*d*x**8 + 15*int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*a**2* b*c*d - 30*int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x* *8 - c**3*x**12),x)*a*b*c**2*d*x**4 + 15*int((sqrt(a - c*x**4)*x**2)/(a**3 - 3*a**2*c*x**4 + 3*a*c**2*x**8 - c**3*x**12),x)*b*c**3*d*x**8)/(15*c*(a* *2 - 2*a*c*x**4 + c**2*x**8))