Optimal. Leaf size=113 \[ -\frac{p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{e}+\frac{p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{\log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e}-\frac{p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.146883, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2462, 260, 2416, 2394, 2315, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{e}+\frac{p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e}+\frac{\log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e}-\frac{p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d+e x} \, dx &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{(b p) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x}\right ) x^2} \, dx}{e}\\ &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{(b p) \int \left (\frac{\log (d+e x)}{b x}-\frac{a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e}\\ &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{p \int \frac{\log (d+e x)}{x} \, dx}{e}-\frac{(a p) \int \frac{\log (d+e x)}{b+a x} \, dx}{e}\\ &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e}-\frac{p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}-p \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx+p \int \frac{\log \left (\frac{e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx\\ &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e}-\frac{p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}+\frac{p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e}\\ &=\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e}-\frac{p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e}-\frac{p \text{Li}_2\left (\frac{a (d+e x)}{a d-b e}\right )}{e}+\frac{p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0174174, size = 114, normalized size = 1.01 \[ -\frac{p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{e}+\frac{p \text{PolyLog}\left (2,\frac{d+e x}{d}\right )}{e}+\frac{\log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e}-\frac{p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{e}+\frac{p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ex+d}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.04896, size = 215, normalized size = 1.9 \begin{align*} \frac{b p{\left (\frac{\log \left (e x + d\right ) \log \left (a + \frac{b}{x}\right )}{b} - \frac{\log \left (e x + d\right ) \log \left (-\frac{a e x + a d}{a d - b e} + 1\right ) +{\rm Li}_2\left (\frac{a e x + a d}{a d - b e}\right )}{b} + \frac{\log \left (e x + d\right ) \log \left (-\frac{e x + d}{d} + 1\right ) +{\rm Li}_2\left (\frac{e x + d}{d}\right )}{b}\right )}}{e} - \frac{p \log \left (e x + d\right ) \log \left (a + \frac{b}{x}\right )}{e} + \frac{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) \log \left (e x + d\right )}{e} \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{\log \left (c \left (\frac{a x + b}{x}\right )^{p}\right )}{e x + d}, 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{\log{\left (c \left (a + \frac{b}{x}\right )^{p} \right )}}{d + e 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{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]