32.2 Problem number 343

\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx \]

Optimal antiderivative \[ -\frac {b}{e x}+\frac {-a -b \ln \! \left (c x \right )}{e x}-\frac {d \left (a +b \ln \! \left (c x \right )\right )^{2}}{2 b \,e^{2}}+\frac {d \left (a +b \ln \! \left (c x \right )\right ) \ln \! \left (1+\frac {d x}{e}\right )}{e^{2}}+\frac {b d \polylog \! \left (2, -\frac {d x}{e}\right )}{e^{2}} \]

command

integrate((a+b*ln(c*x))/(d+e/x)/x**3,x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \text {Timed out} \]

Sympy 1.8 under Python 3.8.8 output

\[ \frac {a d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {a d \log {\left (x \right )}}{e^{2}} - \frac {a}{e x} - \frac {b d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} \log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{e^{2}} + \frac {b d \log {\left (x \right )}^{2}}{2 e^{2}} - \frac {b d \log {\left (x \right )} \log {\left (c x \right )}}{e^{2}} - \frac {b \log {\left (c x \right )}}{e x} - \frac {b}{e x} \]