\[ \int x^{-1-15 n} \left (a+b x^n\right )^8 \, dx \]
Optimal antiderivative \[ -\frac {a^{8} x^{-15 n}}{15 n}-\frac {4 a^{7} b \,x^{-14 n}}{7 n}-\frac {28 a^{6} b^{2} x^{-13 n}}{13 n}-\frac {14 a^{5} b^{3} x^{-12 n}}{3 n}-\frac {70 a^{4} b^{4} x^{-11 n}}{11 n}-\frac {28 a^{3} b^{5} x^{-10 n}}{5 n}-\frac {28 a^{2} b^{6} x^{-9 n}}{9 n}-\frac {a \,b^{7} x^{-8 n}}{n}-\frac {b^{8} x^{-7 n}}{7 n} \]
command
integrate(x**(-1-15*n)*(a+b*x**n)**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {a^{8} x^{- 15 n}}{15 n} - \frac {4 a^{7} b x^{- 14 n}}{7 n} - \frac {28 a^{6} b^{2} x^{- 13 n}}{13 n} - \frac {14 a^{5} b^{3} x^{- 12 n}}{3 n} - \frac {70 a^{4} b^{4} x^{- 11 n}}{11 n} - \frac {28 a^{3} b^{5} x^{- 10 n}}{5 n} - \frac {28 a^{2} b^{6} x^{- 9 n}}{9 n} - \frac {a b^{7} x^{- 8 n}}{n} - \frac {b^{8} x^{- 7 n}}{7 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________