\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {b n}{2 d^{2} x^{2}}+\frac {a +b \ln \left (c \,x^{n}\right )}{2 d \,x^{2} \left (e \,x^{2}+d \right )}+\frac {-4 a +b n -4 b \ln \left (c \,x^{n}\right )}{4 d^{2} x^{2}}+\frac {e \ln \left (1+\frac {d}{e \,x^{2}}\right ) \left (4 a -b n +4 b \ln \left (c \,x^{n}\right )\right )}{4 d^{3}}-\frac {b e n \polylog \left (2, -\frac {d}{e \,x^{2}}\right )}{2 d^{3}} \]
command
integrate((a+b*ln(c*x**n))/x**3/(e*x**2+d)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {a e^{2} \left (\begin {cases} \frac {x^{2}}{2 d^{2}} & \text {for}\: e = 0 \\- \frac {1}{2 d e + 2 e^{2} x^{2}} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a}{2 d^{2} x^{2}} - \frac {2 a e \log {\left (x \right )}}{d^{3}} + \frac {a e \log {\left (d + e x^{2} \right )}}{d^{3}} - \frac {b e^{2} n \left (\begin {cases} \frac {x^{2}}{2 d^{2}} & \text {for}\: e = 0 \\- \frac {\log {\left (x \right )}}{d e} + \frac {\log {\left (\frac {d}{e} + x^{2} \right )}}{2 d e} & \text {otherwise} \end {cases}\right )}{2 d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x^{2}}{d^{2}} & \text {for}\: e = 0 \\- \frac {1}{d e + e^{2} x^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 d^{2}} - \frac {b n}{4 d^{2} x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 d^{2} x^{2}} - \frac {b e^{2} n \left (\begin {cases} \frac {x^{2}}{2 d} & \text {for}\: e = 0 \\\frac {\begin {cases} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} - \frac {\operatorname {Li}_{2}\left (\frac {e x^{2} e^{i \pi }}{d}\right )}{2} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{3}} + \frac {b e^{2} \left (\begin {cases} \frac {x^{2}}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x^{2} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{3}} + \frac {b e n \log {\left (x^{2} \right )}^{2}}{4 d^{3}} - \frac {b e \log {\left (x^{2} \right )} \log {\left (c x^{n} \right )}}{d^{3}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________