\[ \int \frac {a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{x^2} \, dx \]
Optimal antiderivative \[ \frac {b n}{3 x}-\frac {b d n}{2 e \,x^{\frac {2}{3}}}+\frac {b \,d^{2} n}{e^{2} x^{\frac {1}{3}}}-\frac {b \,d^{3} n \ln \left (d +\frac {e}{x^{\frac {1}{3}}}\right )}{e^{3}}+\frac {-a -b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )}{x} \]
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
\[ \begin {cases} - \frac {6 a d e^{3} x^{\frac {7}{3}}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {6 a e^{4} x^{2}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {6 b d^{4} x^{\frac {10}{3}} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} + \frac {6 b d^{3} e n x^{3}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {6 b d^{3} e x^{3} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} + \frac {3 b d^{2} e^{2} n x^{\frac {8}{3}}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {b d e^{3} n x^{\frac {7}{3}}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {6 b d e^{3} x^{\frac {7}{3}} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} + \frac {2 b e^{4} n x^{2}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} - \frac {6 b e^{4} x^{2} \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}}{6 d e^{3} x^{\frac {10}{3}} + 6 e^{4} x^{3}} & \text {for}\: e \neq 0 \\- \frac {a + b \log {\left (c d^{n} \right )}}{x} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________