Optimal. Leaf size=36 \[ \frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2370, 2354,
2438} \begin {gather*} \frac {b \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d}+\frac {\log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2354
Rule 2370
Rule 2438
Rubi steps
\begin {align*} \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx &=\int \frac {a+b \log (c x)}{e+d x} \, dx\\ &=\frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d}-\frac {b \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d}\\ &=\frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b \text {Li}_2\left (-\frac {d x}{e}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.94 \begin {gather*} \frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )+b \text {Li}_2\left (-\frac {d x}{e}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 62, normalized size = 1.72
method | result | size |
risch | \(\frac {a \ln \left (d x +e \right )}{d}+\frac {b \dilog \left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(59\) |
derivativedivides | \(\frac {a \ln \left (c d x +c e \right )}{d}+\frac {b \dilog \left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(62\) |
default | \(\frac {a \ln \left (c d x +c e \right )}{d}+\frac {b \dilog \left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 42, normalized size = 1.17 \begin {gather*} \frac {{\left (\log \left (d x e^{\left (-1\right )} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-d x e^{\left (-1\right )}\right )\right )} b}{d} + \frac {{\left (b \log \left (c\right ) + a\right )} \log \left (d x + e\right )}{d} \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 {a + b \log {\left (c x \right )}}{d x + e}\, 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 = -0.03 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x\right )}{x\,\left (d+\frac {e}{x}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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