\[ \int x^{-1+n} (a+b x)^{-1-n} \, dx \]
Optimal antiderivative \[ \frac {x^{n} \left (b x +a \right )^{-n}}{a n} \]
command
integrate(x**(-1+n)*(b*x+a)**(-1-n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {x^{n} \left (b x\right )^{- n}}{b x} & \text {for}\: a = 0 \\\frac {0^{- n - 1} x^{n}}{n} & \text {for}\: a = - b x \\\frac {x^{n} \left (0^{\frac {1}{n}}\right )^{- n - 1}}{n} & \text {for}\: a = 0^{\frac {1}{n}} - b x \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: n = 0 \\\frac {a^{2} x^{n}}{a^{3} n \left (a + b x\right )^{n} + 2 a^{2} b n x \left (a + b x\right )^{n} + a b^{2} n x^{2} \left (a + b x\right )^{n}} + \frac {a b x x^{n}}{a^{3} n \left (a + b x\right )^{n} + 2 a^{2} b n x \left (a + b x\right )^{n} + a b^{2} n x^{2} \left (a + b x\right )^{n}} + \frac {b x x^{n}}{a^{2} n \left (a + b x\right )^{n} + a b n x \left (a + b x\right )^{n}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________