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