\[ \int x^{-2+n} (a+b x)^{-n} \, dx \]
Optimal antiderivative \[ -\frac {x^{-1+n} \left (b x +a \right )^{1-n}}{a \left (1-n \right )} \]
command
integrate(x**(-2+n)/((b*x+a)**n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {1}{b x} & \text {for}\: a = 0 \wedge n = 1 \\- \frac {x^{n} \left (b x\right )^{- n}}{x} & \text {for}\: a = 0 \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: n = 1 \\\frac {a x^{n}}{a n x \left (a + b x\right )^{n} - a x \left (a + b x\right )^{n}} + \frac {b x x^{n}}{a n x \left (a + b x\right )^{n} - a x \left (a + b x\right )^{n}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________