Optimal. Leaf size=91 \[ -\frac {p x}{e}+\frac {(a+b x) \log \left (c (a+b x)^p\right )}{b e}-\frac {d \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^2}-\frac {d p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {45, 2463, 2436,
2332, 2441, 2440, 2438} \begin {gather*} -\frac {d p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^2}-\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 {p x}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rubi steps
\begin {align*} \int \frac {x \log \left (c (a+b x)^p\right )}{d+e x} \, dx &=\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*}
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Mathematica [A]
time = 0.03, size = 79, normalized size = 0.87 \begin {gather*} \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 \text {Li}_2\left (\frac {e (a+b x)}{-b d+a e}\right )}{b e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.49, size = 427, normalized size = 4.69
method | result | size |
risch | \(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x}{e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d \ln \left (e x +d \right )}{e^{2}}-\frac {p x}{e}-\frac {p d}{e^{2}}+\frac {p a \ln \left (\left (e x +d \right ) b +a e -b d \right )}{b e}+\frac {p d \dilog \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\frac {p d \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} d \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} x}{2 e}-\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) d \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) x}{2 e}-\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} d \ln \left (e x +d \right )}{2 e^{2}}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} x}{2 e}+\frac {i \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) d \ln \left (e x +d \right )}{2 e^{2}}+\frac {i \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) x}{2 e}+\frac {\ln \left (c \right ) x}{e}-\frac {\ln \left (c \right ) d \ln \left (e x +d \right )}{e^{2}}\) | \(427\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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