\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {a +b \ln \left (c \,x^{n}\right )}{4 d \left (e \,x^{2}+d \right )^{2}}-\frac {\ln \left (1+\frac {d}{e \,x^{2}}\right ) \left (4 a -3 b n +4 b \ln \left (c \,x^{n}\right )\right )}{8 d^{3}}+\frac {4 a -b n +4 b \ln \left (c \,x^{n}\right )}{8 d^{2} \left (e \,x^{2}+d \right )}+\frac {b n \polylog \left (2, -\frac {d}{e \,x^{2}}\right )}{4 d^{3}} \]
command
integrate((a+b*ln(c*x**n))/x/(e*x**2+d)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {a e \left (\begin {cases} \frac {x^{2}}{2 d^{3}} & \text {for}\: e = 0 \\- \frac {1}{4 e \left (d + e x^{2}\right )^{2}} & \text {otherwise} \end {cases}\right )}{d} - \frac {a e \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 \log {\left (x \right )}}{d^{3}} - \frac {a \log {\left (d + e x^{2} \right )}}{2 d^{3}} + \frac {b e^{2} n \left (\begin {cases} - \frac {1}{2 e^{3} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{4 d e^{2} + 4 e^{3} x^{2}} - \frac {\log {\left (d + e x^{2} \right )}}{4 d e^{2}} & \text {otherwise} \end {cases}\right )}{2 d^{2}} - \frac {b e^{2} \left (\begin {cases} \frac {1}{e^{3} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{2 d \left (\frac {d}{x^{2}} + e\right )^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 d^{2}} - \frac {b e n \left (\begin {cases} - \frac {1}{2 e^{2} x^{2}} & \text {for}\: d = 0 \\- \frac {\log {\left (d + e x^{2} \right )}}{2 d e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e \left (\begin {cases} \frac {1}{e^{2} x^{2}} & \text {for}\: d = 0 \\- \frac {1}{\frac {d^{2}}{x^{2}} + d e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b n \left (\begin {cases} - \frac {1}{2 e x^{2}} & \text {for}\: d = 0 \\\frac {\begin {cases} \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\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 (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + \frac {\operatorname {Li}_{2}\left (\frac {d e^{i \pi }}{e x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{2 d^{2}} - \frac {b \left (\begin {cases} \frac {1}{e x^{2}} & \text {for}\: d = 0 \\\frac {\log {\left (\frac {d}{x^{2}} + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{2 d^{2}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________