\[ \int \frac {a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {b e n}{2 d \,x^{\frac {2}{3}}}+\frac {b \,e^{2} n}{d^{2} x^{\frac {1}{3}}}-\frac {b \,e^{3} n \ln \left (d +e \,x^{\frac {1}{3}}\right )}{d^{3}}+\frac {-a -b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )}{x}+\frac {b \,e^{3} n \ln \left (x \right )}{3 d^{3}} \]
command
integrate((a+b*ln(c*(d+e*x**(1/3))**n))/x**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {6 a d^{4} x^{\frac {2}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 a d^{3} e x}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d^{4} x^{\frac {2}{3}} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {3 b d^{3} e n x}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d^{3} e x \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {3 b d^{2} e^{2} n x^{\frac {4}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {2 b d e^{3} n x^{\frac {5}{3}} \log {\left (x \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {6 b d e^{3} n x^{\frac {5}{3}}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b d e^{3} x^{\frac {5}{3}} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} + \frac {2 b e^{4} n x^{2} \log {\left (x \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} - \frac {6 b e^{4} x^{2} \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}}{6 d^{4} x^{\frac {5}{3}} + 6 d^{3} e x^{2}} & \text {for}\: d \neq 0 \\- \frac {a}{x} - \frac {b n}{3 x} - \frac {b \log {\left (c \left (e \sqrt [3]{x}\right )^{n} \right )}}{x} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________