\[ \int x^{3/2} \left (a+b x^2\right )^p \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{1+p} \hypergeom \left (\left [1, \frac {9}{4}+p \right ], \left [\frac {9}{4}\right ], -\frac {b \,x^{2}}{a}\right )}{5 a} \]
command
integrate(x**(3/2)*(b*x**2+a)**p,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a^{p} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________