\[ \int \frac {\left (a+b x^2\right )^p}{x^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (b \,x^{2}+a \right )^{1+p} \hypergeom \left (\left [1, \frac {3}{4}+p \right ], \left [\frac {3}{4}\right ], -\frac {b \,x^{2}}{a}\right )}{a \sqrt {x}} \]
command
integrate((b*x**2+a)**p/x**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{p} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - p \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________