\[ \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx \]
Optimal antiderivative \[ -\frac {b \,d^{2} n \,x^{\frac {2}{3}}}{2 e^{2}}+\frac {b d n \,x^{\frac {4}{3}}}{4 e}-\frac {b n \,x^{2}}{6}+\frac {b \,d^{3} n \ln \left (d +e \,x^{\frac {2}{3}}\right )}{2 e^{3}}+\frac {x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}{2} \]
command
integrate(x*(a+b*ln(c*(d+e*x**(2/3))**n)),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a x^{2}}{2} + b \left (- \frac {e n \left (- \frac {3 d^{3} \left (\begin {cases} \frac {x^{\frac {2}{3}}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{\frac {2}{3}} \right )}}{e} & \text {otherwise} \end {cases}\right )}{2 e^{3}} + \frac {3 d^{2} x^{\frac {2}{3}}}{2 e^{3}} - \frac {3 d x^{\frac {4}{3}}}{4 e^{2}} + \frac {x^{2}}{2 e}\right )}{3} + \frac {x^{2} \log {\left (c \left (d + e x^{\frac {2}{3}}\right )^{n} \right )}}{2}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________