\[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx \]
Optimal antiderivative \[ -\frac {x^{-3 n}}{6 n}+\frac {b \,x^{-2 n}}{8 n}-\frac {b^{2} x^{-n}}{8 n}-\frac {b^{3} \ln \left (x \right )}{16}+\frac {b^{3} \ln \left (2+b \,x^{n}\right )}{16 n} \]
command
integrate(x**(-1-3*n)/(2+b*x**n),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\log {\left (x \right )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {\log {\left (x \right )}}{b + 2} & \text {for}\: n = 0 \\- \frac {x^{- 3 n}}{6 n} & \text {for}\: b = 0 \\- \frac {b^{3} \log {\left (x^{n} \right )}}{16 n} + \frac {b^{3} \log {\left (x^{n} + \frac {2}{b} \right )}}{16 n} - \frac {b^{2} x^{- n}}{8 n} + \frac {b x^{- 2 n}}{8 n} - \frac {x^{- 3 n}}{6 n} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________