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