Integrand size = 32, antiderivative size = 223 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (A c d+a C d+a B e+(B c d+A c e+a C e) x^2\right )}{2 a c \sqrt {a-c x^4}}-\frac {(B c d+A c e+3 a C e) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {\left (\sqrt {a} \left (B d+\left (A+\frac {3 a C}{c}\right ) e\right )+\frac {A c d-a (C d+B e)}{\sqrt {c}}\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{3/4} \sqrt {a-c x^4}} \] Output:
1/2*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)^(1/2)-1/2 *(A*c*e+B*c*d+3*C*a*e)*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/a^ (1/4)/c^(7/4)/(-c*x^4+a)^(1/2)+1/2*(a^(1/2)*(B*d+(A+3*a*C/c)*e)+(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^(3/4 )/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 = 10.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 x \left (A c d+a B e+a C \left (d-2 e x^2\right )\right )-3 (-A c d+a C d+a B e) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 (B c d+A c e+3 a C e) x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{6 a c \sqrt {a-c x^4}} \] Input:
Integrate[((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x]
Output:
(3*x*(A*c*d + a*B*e + a*C*(d - 2*e*x^2)) - 3*(-(A*c*d) + a*C*d + a*B*e)*x* Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 2*(B*c*d + A*c*e + 3*a*C*e)*x^3*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/ 4, (c*x^4)/a])/(6*a*c*Sqrt[a - c*x^4])
Time = 0.64 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.73, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {x^2 (a C e+A c e+B c d)+a B e+a C d+A c d}{c \left (a-c x^4\right )^{3/2}}-\frac {B e+C d}{c \sqrt {a-c x^4}}-\frac {C e x^2}{c \sqrt {a-c x^4}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\sqrt {a} B \sqrt {c}+a C+A c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{7/4} \sqrt {a-c x^4}}+\frac {a^{3/4} C e \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{7/4} \sqrt {a-c x^4}}-\frac {a^{3/4} C e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{7/4} \sqrt {a-c x^4}}-\frac {\sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right ) (a C e+A c e+B c d)}{2 \sqrt [4]{a} c^{7/4} \sqrt {a-c x^4}}+\frac {x \left (x^2 (a C e+A c e+B c d)+a B e+a C d+A c d\right )}{2 a c \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (B e+C d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} \sqrt {a-c x^4}}\) |
Input:
Int[((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x]
Output:
(x*(A*c*d + a*C*d + a*B*e + (B*c*d + A*c*e + a*C*e)*x^2))/(2*a*c*Sqrt[a - c*x^4]) - (a^(3/4)*C*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[(c^(1/4)*x)/a^ (1/4)], -1])/(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])/(2*a^(1/4)*c^(7/4) *Sqrt[a - c*x^4]) + (a^(3/4)*C*e*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^( 1/4)*x)/a^(1/4)], -1])/(c^(7/4)*Sqrt[a - c*x^4]) + ((Sqrt[a]*B*Sqrt[c] + A *c + a*C)*(Sqrt[c]*d + Sqrt[a]*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]) - (a^(1/4)*(C* d + B*e)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/( c^(5/4)*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 = 0.87 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.26
| method | result | size |
| elliptic | \(\frac {2 c \left (\frac {\left (A c e +B c d +C a e \right ) x^{3}}{4 a \,c^{2}}+\frac {\left (A c d +B a e +C a d \right ) x}{4 c^{2} a}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {B e +C d}{c}+\frac {A c d +B a e +C a d}{2 a c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (-\frac {e C}{c}-\frac {A c e +B c d +C a e}{2 a c}\right ) \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\) | \(280\) |
| default | \(A d \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+\left (A e +B d \right ) \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )+\left (B e +C d \right ) \left (\frac {x}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e C \left (\frac {x^{3}}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {3 \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 c^{\frac {3}{2}} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )\) | \(424\) |
Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
2*c*(1/4*(A*c*e+B*c*d+C*a*e)/a/c^2*x^3+1/4*(A*c*d+B*a*e+C*a*d)/c^2/a*x)/(- (x^4-a/c)*c)^(1/2)+(-(B*e+C*d)/c+1/2*(A*c*d+B*a*e+C*a*d)/a/c)/(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*e*C-1/2*(A*c*e +B*c*d+C*a*e)/a/c)*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 = 290, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (B a c^{2} d + {\left (3 \, C a^{2} c + A a c^{2}\right )} e\right )} x^{5} - {\left (B a^{2} c d + {\left (3 \, C a^{3} + A a^{2} c\right )} e\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left ({\left ({\left (B + C\right )} a c^{2} - A c^{3}\right )} d + {\left (3 \, C a^{2} c + {\left (A + B\right )} a c^{2}\right )} e\right )} x^{5} - {\left ({\left ({\left (B + C\right )} a^{2} c - A a c^{2}\right )} d + {\left (3 \, C a^{3} + {\left (A + B\right )} a^{2} c\right )} e\right )} x\right )} \sqrt {-c} \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (2 \, C a^{2} c e x^{4} - B a^{2} c d - {\left (B a^{2} c e + {\left (C a^{2} c + A a c^{2}\right )} d\right )} x^{2} - {\left (3 \, C a^{3} + A a^{2} c\right )} e\right )} \sqrt {-c x^{4} + a}}{2 \, {\left (a^{2} c^{3} x^{5} - a^{3} c^{2} x\right )}} \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="fricas" )
Output:
1/2*(((B*a*c^2*d + (3*C*a^2*c + A*a*c^2)*e)*x^5 - (B*a^2*c*d + (3*C*a^3 + A*a^2*c)*e)*x)*sqrt(-c)*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) - ((((B + C)*a*c^2 - A*c^3)*d + (3*C*a^2*c + (A + B)*a*c^2)*e)*x^5 - (((B + C)*a^2*c - A*a*c^2)*d + (3*C*a^3 + (A + B)*a^2*c)*e)*x)*sqrt(-c)*(a/c)^( 3/4)*elliptic_f(arcsin((a/c)^(1/4)/x), -1) + (2*C*a^2*c*e*x^4 - B*a^2*c*d - (B*a^2*c*e + (C*a^2*c + A*a*c^2)*d)*x^2 - (3*C*a^3 + A*a^2*c)*e)*sqrt(-c *x^4 + a))/(a^2*c^3*x^5 - a^3*c^2*x)
Time = 11.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {A d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {A e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B e x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {C d x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {C e x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)*(C*x**4+B*x**2+A)/(-c*x**4+a)**(3/2),x)
Output:
A*d*x*gamma(1/4)*hyper((1/4, 3/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4* a**(3/2)*gamma(5/4)) + A*e*x**3*gamma(3/4)*hyper((3/4, 3/2), (7/4,), c*x** 4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + B*d*x**3*gamma(3/4)*hyper ((3/4, 3/2), (7/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(7/4)) + B*e*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), c*x**4*exp_polar(2*I*pi)/a) /(4*a**(3/2)*gamma(9/4)) + C*d*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), c *x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(9/4)) + C*e*x**7*gamma(7/4)*h yper((3/2, 7/4), (11/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/2)*gamma(11 /4))
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="maxima" )
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)/(-c*x^4 + a)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}}{{\left (-c x^{4} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)/(-c*x^4 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (C\,x^4+B\,x^2+A\right )}{{\left (a-c\,x^4\right )}^{3/2}} \,d x \] Input:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2),x)
Output:
int(((d + e*x^2)*(A + B*x^2 + C*x^4))/(a - c*x^4)^(3/2), x)
\[ \int \frac {\left (d+e x^2\right ) \left (A+B x^2+C x^4\right )}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {\sqrt {-c \,x^{4}+a}\, b e x +\sqrt {-c \,x^{4}+a}\, c d x -\sqrt {-c \,x^{4}+a}\, c e \,x^{3}-\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} b e +\left (\int \frac {\sqrt {-c \,x^{4}+a}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b c e \,x^{4}+4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a^{2} c e +\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a b c d -4 \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) a \,c^{2} e \,x^{4}-\left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) b \,c^{2} d \,x^{4}}{c \left (-c \,x^{4}+a \right )} \] Input:
int((e*x^2+d)*(C*x^4+B*x^2+A)/(-c*x^4+a)^(3/2),x)
Output:
(sqrt(a - c*x**4)*b*e*x + sqrt(a - c*x**4)*c*d*x - sqrt(a - c*x**4)*c*e*x* *3 - int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*b*e + in t(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c*e*x**4 + 4*int ((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a**2*c*e + int ((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*b*c*d - 4*in t((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c**2*e*x**4 - int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*b*c**2*d *x**4)/(c*(a - c*x**4))