\[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx \]
Optimal antiderivative \[ \frac {a^{8} x^{8 n}}{8 n}+\frac {8 a^{7} b \,x^{9 n}}{9 n}+\frac {14 a^{6} b^{2} x^{10 n}}{5 n}+\frac {56 a^{5} b^{3} x^{11 n}}{11 n}+\frac {35 a^{4} b^{4} x^{12 n}}{6 n}+\frac {56 a^{3} b^{5} x^{13 n}}{13 n}+\frac {2 a^{2} b^{6} x^{14 n}}{n}+\frac {8 a \,b^{7} x^{15 n}}{15 n}+\frac {b^{8} x^{16 n}}{16 n} \]
command
integrate(x**(-1+8*n)*(a+b*x**n)**8,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {a^{8} x^{8 n}}{8 n} + \frac {8 a^{7} b x^{9 n}}{9 n} + \frac {14 a^{6} b^{2} x^{10 n}}{5 n} + \frac {56 a^{5} b^{3} x^{11 n}}{11 n} + \frac {35 a^{4} b^{4} x^{12 n}}{6 n} + \frac {56 a^{3} b^{5} x^{13 n}}{13 n} + \frac {2 a^{2} b^{6} x^{14 n}}{n} + \frac {8 a b^{7} x^{15 n}}{15 n} + \frac {b^{8} x^{16 n}}{16 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________