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 = 0.03, 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 = {2441, 2440,
2438} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2441
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 \text {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 \begin {gather*} \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 {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 = 0.41, size = 242, normalized size = 4.17
method | result | size |
risch | \(\frac {\ln \left (e x +d \right ) \ln \left (\left (b x +a \right )^{p}\right )}{e}-\frac {p \dilog \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +a e -b d}{a e -b d}\right )}{e}+\frac {i \ln \left (e x +d \right ) \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 e}-\frac {i \ln \left (e x +d \right ) \pi \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 e}-\frac {i \ln \left (e x +d \right ) \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2 e}+\frac {i \ln \left (e x +d \right ) \pi \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 e}+\frac {\ln \left (e x +d \right ) \ln \left (c \right )}{e}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs.
\(2 (59) = 118\).
time = 0.28, size = 123, normalized size = 2.12 \begin {gather*} b p {\left (\frac {\log \left (b x + a\right ) \log \left (x e + d\right )}{b} - \frac {\log \left (x e + d\right ) \log \left (-\frac {b x e + b d}{b d - a e} + 1\right ) + {\rm Li}_2\left (\frac {b x e + b d}{b d - a e}\right )}{b}\right )} e^{\left (-1\right )} - p e^{\left (-1\right )} \log \left (b x + a\right ) \log \left (x e + d\right ) + e^{\left (-1\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \log \left (x e + d\right ) \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 {\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 = -0.02 \begin {gather*} \int \frac {\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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