Optimal. Leaf size=44 \[ \frac{b n \text{PolyLog}\left (2,-\frac{e}{d x}\right )}{e}-\frac{\log \left (\frac{e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0660136, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2337, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{e}{d x}\right )}{e}-\frac{\log \left (\frac{e}{d x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{\left (d+\frac{e}{x}\right ) x^2} \, dx &=-\frac{\log \left (1+\frac{e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{(b n) \int \frac{\log \left (1+\frac{e}{d x}\right )}{x} \, dx}{e}\\ &=-\frac{\log \left (1+\frac{e}{d x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{b n \text{Li}_2\left (-\frac{e}{d x}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0349509, size = 63, normalized size = 1.43 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{d x}{e}+1\right )\right )}{2 b e n}-\frac{b n \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.134, size = 336, normalized size = 7.6 \begin{align*}{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( x \right ) }{e}}-{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( dx+e \right ) }{e}}-{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}}{2\,e}}+{\frac{bn\ln \left ( dx+e \right ) }{e}\ln \left ( -{\frac{dx}{e}} \right ) }+{\frac{bn}{e}{\it dilog} \left ( -{\frac{dx}{e}} \right ) }-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( dx+e \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( dx+e \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( x \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( dx+e \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( x \right ) }{e}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{e}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( dx+e \right ) }{e}}+{\frac{b\ln \left ( c \right ) \ln \left ( x \right ) }{e}}-{\frac{b\ln \left ( c \right ) \ln \left ( dx+e \right ) }{e}}+{\frac{a\ln \left ( x \right ) }{e}}-{\frac{a\ln \left ( dx+e \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{\log \left (d x + e\right )}{e} - \frac{\log \left (x\right )}{e}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{d x^{2} + e x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) + a}{d x^{2} + e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 11.2964, size = 156, normalized size = 3.55 \begin{align*} \frac{2 a d \left (\begin{cases} - \frac{x}{e} - \frac{1}{2 d} & \text{for}\: d = 0 \\\frac{\log{\left (2 d x \right )}}{2 d} & \text{otherwise} \end{cases}\right )}{e} - \frac{2 a d \left (\begin{cases} \frac{x}{e} + \frac{1}{2 d} & \text{for}\: d = 0 \\\frac{\log{\left (2 d x + 2 e \right )}}{2 d} & \text{otherwise} \end{cases}\right )}{e} + b n \left (\begin{cases} - \frac{1}{d x} & \text{for}\: e = 0 \\\frac{\begin{cases} \log{\left (d \right )} \log{\left (x \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (d \right )} \log{\left (\frac{1}{x} \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\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 (d \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (d \right )} + \operatorname{Li}_{2}\left (\frac{e e^{i \pi }}{d x}\right ) & \text{otherwise} \end{cases}}{e} & \text{otherwise} \end{cases}\right ) - b \left (\begin{cases} \frac{1}{d x} & \text{for}\: e = 0 \\\frac{\log{\left (d + \frac{e}{x} \right )}}{e} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (d + \frac{e}{x}\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]