\[ \int \frac {x \left (a+b x^2\right )}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {a}{c^{2}}+\frac {b}{d^{2}}\right ) x^{2}}{\sqrt {d x -c}\, \sqrt {d x +c}}+\frac {\left (a \,d^{2}+2 b \,c^{2}\right ) \sqrt {d x -c}\, \sqrt {d x +c}}{c^{2} d^{4}} \]
command
integrate(x*(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
\[ a \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4}, 1 & 0, 1, \frac {3}{2} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c d^{2}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, 1 & \\- \frac {1}{4}, \frac {1}{4} & -1, - \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c d^{2}}\right ) + b \left (\frac {c {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{4} & -1, 0, \frac {1}{2}, 1 \\- \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{4}} - \frac {i c {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, 1 & \\- \frac {5}{4}, - \frac {3}{4} & -2, - \frac {3}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} d^{4}}\right ) \]