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