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