Optimal. Leaf size=58 \[ \frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e} \]
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Rubi [A] time = 0.05, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2394, 2393, 2391} \[ \frac {p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e}+\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rubi steps
\begin {align*} \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx &=\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {(b p) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e}\\ &=\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e}\\ &=\frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 57, normalized size = 0.98 \[ \frac {\log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e}+\frac {p \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )}{e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 242, normalized size = 4.17 \[ -\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{2 e}+\frac {i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e}-\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 e}-\frac {p \ln \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right ) \ln \left (e x +d \right )}{e}-\frac {p \dilog \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right )}{e}+\frac {\ln \relax (c ) \ln \left (e x +d \right )}{e}+\frac {\ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 118, normalized size = 2.03 \[ \frac {b p {\left (\frac {\log \left (b x + a\right ) \log \left (e x + d\right )}{b} - \frac {\log \left (e x + d\right ) \log \left (-\frac {b e x + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b e x + b d}{b d - a e}\right )}{b}\right )}}{e} - \frac {p \log \left (b x + a\right ) \log \left (e x + d\right )}{e} + \frac {\log \left ({\left (b x + a\right )}^{p} c\right ) \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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