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