\[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ -\frac {76 b \,d^{2} n \left (e x +d \right )^{\frac {3}{2}}}{105 e^{4}}+\frac {64 b d n \left (e x +d \right )^{\frac {5}{2}}}{175 e^{4}}-\frac {4 b n \left (e x +d \right )^{\frac {7}{2}}}{49 e^{4}}-\frac {64 b \,d^{\frac {7}{2}} n \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{35 e^{4}}+\frac {2 d^{2} \left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{e^{4}}-\frac {6 d \left (e x +d \right )^{\frac {5}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{5 e^{4}}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{7 e^{4}}+\frac {64 b \,d^{3} n \sqrt {e x +d}}{35 e^{4}}-\frac {2 d^{3} \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{e^{4}} \]
command
integrate(x**3*(a+b*ln(c*x**n))/(e*x+d)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________