Integrand size = 19, antiderivative size = 91 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2} \]
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Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {45, 2463, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\frac {p x}{e} \]
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c (a+b x)^p\right )}{e}-\frac {d \log \left (c (a+b x)^p\right )}{e (d+e x)}\right ) \, dx \\ & = \frac {\int \log \left (c (a+b x)^p\right ) \, dx}{e}-\frac {d \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e} \\ & = -\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e}+\frac {(b d p) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^2} \\ & = -\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}+\frac {(d p) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^2} \\ & = -\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}-\frac {d p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.87 \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {-b e p x+\log \left (c (a+b x)^p\right ) \left (a e+b e x-b d \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )-b d p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{b e^2} \]
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Time = 1.71 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.71
method | result | size |
parts | \(\frac {x \ln \left (c \left (b x +a \right )^{p}\right )}{e}-\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p b \left (\frac {e x +d}{e b}-\frac {a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b^{2}}-\frac {d \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{b}\right )}{e}\right )}{e}\) | \(156\) |
risch | \(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x}{e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p x}{e}-\frac {p d}{e^{2}}+\frac {p a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b e}+\frac {p d \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\frac {p d \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )\) | \(270\) |
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\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]
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\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int { \frac {x \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\int \frac {x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]
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