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