Integrand size = 23, antiderivative size = 159 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {b}{a x}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{d} \]
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Time = 0.16 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2516, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}-\frac {\log \left (-\frac {b}{a x}\right ) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {b}{a x}+1\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d} \]
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Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2504
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d x}-\frac {e \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x}\right )}{d}-\frac {(b p) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {(b p) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{d}+\frac {(b p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x}\right )}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \int \frac {\log (d+e x)}{x} \, dx}{d}+\frac {(a p) \int \frac {\log (d+e x)}{b+a x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {(e p) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{d}-\frac {(e p) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{d} \\ & = -\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \text {Li}_2\left (1+\frac {b}{a x}\right )}{d}+\frac {p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{d}-\frac {p \text {Li}_2\left (1+\frac {e x}{d}\right )}{d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log \left (-\frac {b}{a x}\right )}{d}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{d}-\frac {p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {a+\frac {b}{x}}{a}\right )}{d}-\frac {p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{d} \]
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Time = 0.92 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.31
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (x \right )}{d}+p b \left (\frac {\ln \left (x \right )^{2}}{2 d b}-\frac {\operatorname {dilog}\left (\frac {a x +b}{b}\right )}{d b}-\frac {\ln \left (x \right ) \ln \left (\frac {a x +b}{b}\right )}{d b}-\frac {\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d b}-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )}{d b}+\frac {\operatorname {dilog}\left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{d b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{d b}\right )\) | \(209\) |
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{x \left (d + e x\right )}\, dx \]
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Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.13 \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=-\frac {1}{2} \, b p {\left (\frac {2 \, \log \left (e x + d\right ) \log \left (x\right ) - \log \left (x\right )^{2}}{b d} + \frac {2 \, {\left (\log \left (\frac {a x}{b} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {a x}{b}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )}}{b d} - \frac {2 \, {\left (\log \left (e x + d\right ) \log \left (-\frac {a e x + a d}{a d - b e} + 1\right ) + {\rm Li}_2\left (\frac {a e x + a d}{a d - b e}\right )\right )}}{b d}\right )} - {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \left (x\right )}{d}\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right ) \]
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\[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int { \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{{\left (e x + d\right )} x} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{x\,\left (d+e\,x\right )} \,d x \]
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