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