\[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx \]
Optimal antiderivative \[ -\frac {2}{3 n \ln \left (a \,x^{n}\right )^{\frac {3}{2}}} \]
command
integrate(1/x/ln(a*x**n)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {2}{3 n \log {\left (a x^{n} \right )}^{\frac {3}{2}}} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{\log {\left (a \right )}^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________