\[ \int \frac {x^5 \left (a+b x^2+c x^4\right )}{\sqrt {d-e x} \sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \left (2 a \,e^{4}+3 b \,d^{2} e^{2}+4 c \,d^{4}\right ) \left (-e x +d \right )^{\frac {3}{2}} \left (e x +d \right )^{\frac {3}{2}}}{3 e^{10}}-\frac {\left (a \,e^{4}+3 b \,d^{2} e^{2}+6 c \,d^{4}\right ) \left (-e x +d \right )^{\frac {5}{2}} \left (e x +d \right )^{\frac {5}{2}}}{5 e^{10}}+\frac {\left (b \,e^{2}+4 c \,d^{2}\right ) \left (-e x +d \right )^{\frac {7}{2}} \left (e x +d \right )^{\frac {7}{2}}}{7 e^{10}}-\frac {c \left (-e x +d \right )^{\frac {9}{2}} \left (e x +d \right )^{\frac {9}{2}}}{9 e^{10}}-\frac {d^{4} \left (a \,e^{4}+b \,d^{2} e^{2}+c \,d^{4}\right ) \sqrt {-e x +d}\, \sqrt {e x +d}}{e^{10}} \]
command
integrate(x**5*(c*x**4+b*x**2+a)/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {i a d^{5} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {9}{4}, - \frac {7}{4} & -2, -2, - \frac {3}{2}, 1 \\- \frac {5}{2}, - \frac {9}{4}, -2, - \frac {7}{4}, - \frac {3}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {a d^{5} {G_{6, 6}^{2, 6}\left (\begin {matrix} -3, - \frac {11}{4}, - \frac {5}{2}, - \frac {9}{4}, -2, 1 & \\- \frac {11}{4}, - \frac {9}{4} & -3, - \frac {5}{2}, - \frac {5}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{6}} - \frac {i b d^{7} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {13}{4}, - \frac {11}{4} & -3, -3, - \frac {5}{2}, 1 \\- \frac {7}{2}, - \frac {13}{4}, -3, - \frac {11}{4}, - \frac {5}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {b d^{7} {G_{6, 6}^{2, 6}\left (\begin {matrix} -4, - \frac {15}{4}, - \frac {7}{2}, - \frac {13}{4}, -3, 1 & \\- \frac {15}{4}, - \frac {13}{4} & -4, - \frac {7}{2}, - \frac {7}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{8}} - \frac {i c d^{9} {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {17}{4}, - \frac {15}{4} & -4, -4, - \frac {7}{2}, 1 \\- \frac {9}{2}, - \frac {17}{4}, -4, - \frac {15}{4}, - \frac {7}{2}, 0 & \end {matrix} \middle | {\frac {d^{2}}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} - \frac {c d^{9} {G_{6, 6}^{2, 6}\left (\begin {matrix} -5, - \frac {19}{4}, - \frac {9}{2}, - \frac {17}{4}, -4, 1 & \\- \frac {19}{4}, - \frac {17}{4} & -5, - \frac {9}{2}, - \frac {9}{2}, 0 \end {matrix} \middle | {\frac {d^{2} e^{- 2 i \pi }}{e^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} e^{10}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________