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