\[ \int \frac {a+b x^2}{x (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {a \arctan \! \left (\frac {\sqrt {d x -c}\, \sqrt {d x +c}}{c}\right )}{c^{3}}+\frac {-\frac {a}{c^{2}}-\frac {b}{d^{2}}}{\sqrt {d x -c}\, \sqrt {d x +c}} \]
command
integrate((b*x**2+a)/x/(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
\[ a \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & 1, 2, \frac {5}{2} \\\frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{3}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, 1 & \\\frac {3}{4}, \frac {5}{4} & 0, \frac {1}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{3}}\right ) + b \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4}, 1 & 0, 1, \frac {3}{2} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c d^{2}} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, 1 & \\- \frac {1}{4}, \frac {1}{4} & -1, - \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c d^{2}}\right ) \]