Integrand size = 21, antiderivative size = 146 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d}-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}+\frac {e p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d^2} \]
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Time = 0.12 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {46, 2463, 2442, 36, 29, 31, 2441, 2352, 2440, 2438} \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {\log \left (c (a+b x)^p\right )}{d x}+\frac {e p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right )}{d^2}+\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c (a+b x)^p\right )}{d x^2}-\frac {e \log \left (c (a+b x)^p\right )}{d^2 x}+\frac {e^2 \log \left (c (a+b x)^p\right )}{d^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{d^2} \\ & = -\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}+\frac {(b p) \int \frac {1}{x (a+b x)} \, dx}{d}+\frac {(b e p) \int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx}{d^2}-\frac {(b e p) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d^2} \\ & = -\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d^2}+\frac {(b p) \int \frac {1}{x} \, dx}{a d}-\frac {\left (b^2 p\right ) \int \frac {1}{a+b x} \, dx}{a d}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d^2} \\ & = \frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d}-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}+\frac {e p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x}{a}\right )}{d^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\frac {b p \log (x)}{a d}-\frac {b p \log (a+b x)}{a d}-\frac {\log \left (c (a+b x)^p\right )}{d x}-\frac {e \log \left (-\frac {b x}{a}\right ) \log \left (c (a+b x)^p\right )}{d^2}+\frac {e \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {a+b x}{a}\right )}{d^2}+\frac {e p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d^2} \]
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Time = 0.91 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.38
method | result | size |
parts | \(\frac {\ln \left (c \left (b x +a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (c \left (b x +a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-p b \left (\frac {e \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}\right )}{d^{2}}+\frac {\ln \left (b x +a \right )}{d a}-\frac {\ln \left (x \right )}{d a}-\frac {e \operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d^{2} b}-\frac {e \ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d^{2} b}\right )\) | \(201\) |
risch | \(\frac {\ln \left (\left (b x +a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{d^{2}}-\frac {b p \ln \left (b x +a \right )}{a d}+\frac {b p \ln \left (x \right )}{a d}+\frac {p e \operatorname {dilog}\left (\frac {b x +a}{a}\right )}{d^{2}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {b x +a}{a}\right )}{d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\) | \(322\) |
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\[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=b p {\left (\frac {{\left (\log \left (\frac {b x}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x}{a}\right )\right )} e}{b d^{2}} - \frac {{\left (\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )\right )} e}{b d^{2}} - \frac {\log \left (b x + a\right )}{a d} + \frac {\log \left (x\right )}{a d}\right )} + {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \left (x\right )}{d^{2}} - \frac {1}{d x}\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
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\[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c (a+b x)^p\right )}{x^2 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
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