Optimal. Leaf size=63 \[ \frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0453695, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2394, 2393, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(b e n) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g}+\frac{b n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g}\\ \end{align*}
Mathematica [A] time = 0.0065298, size = 62, normalized size = 0.98 \[ \frac{b n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )}{g}+\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.115, size = 261, normalized size = 4.1 \begin{align*}{\frac{b\ln \left ( gx+f \right ) \ln \left ( \left ( ex+d \right ) ^{n} \right ) }{g}}-{\frac{bn}{g}{\it dilog} \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{dg-fe}} \right ) }-{\frac{bn\ln \left ( gx+f \right ) }{g}\ln \left ({\frac{ \left ( gx+f \right ) e+dg-fe}{dg-fe}} \right ) }-{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) }{g}}+{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}+{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g}}-{\frac{{\frac{i}{2}}\ln \left ( gx+f \right ) b\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{g}}+{\frac{b\ln \left ( gx+f \right ) \ln \left ( c \right ) }{g}}+{\frac{a\ln \left ( gx+f \right ) }{g}} \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*} b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g x + f}\,{d x} + \frac{a \log \left (g x + f\right )}{g} \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 ({\left (e x + d\right )}^{n} c\right ) + a}{g x + f}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}{f + g x}\, dx \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 ({\left (e x + d\right )}^{n} c\right ) + a}{g x + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]