\[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx \]
Optimal antiderivative \[ \frac {16 b^{2} n^{2} \left (d x \right )^{\frac {7}{2}}}{343 d}-\frac {8 b n \left (d x \right )^{\frac {7}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{49 d}+\frac {2 \left (d x \right )^{\frac {7}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{7 d} \]
command
integrate((d*x)**(5/2)*(a+b*ln(c*x**n))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {2 a^{2} x \left (d x\right )^{\frac {5}{2}}}{7} - \frac {8 a b n x \left (d x\right )^{\frac {5}{2}}}{49} + \frac {4 a b x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{7} + \frac {16 b^{2} n^{2} x \left (d x\right )^{\frac {5}{2}}}{343} - \frac {8 b^{2} n x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{49} + \frac {2 b^{2} x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}^{2}}{7} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________