Integrand size = 21, antiderivative size = 36 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d}+\frac {b \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d} \]
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Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2370, 2354, 2438} \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {\log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d}+\frac {b \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d} \]
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Rule 2354
Rule 2370
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {(a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )+b \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.64
method | result | size |
risch | \(\frac {a \ln \left (d x +e \right )}{d}+\frac {b \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(59\) |
parts | \(\frac {a \ln \left (d x +e \right )}{d}+\frac {b \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(59\) |
derivativedivides | \(\frac {a \ln \left (c d x +c e \right )}{d}+\frac {b \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(62\) |
default | \(\frac {a \ln \left (c d x +c e \right )}{d}+\frac {b \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {b \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\) | \(62\) |
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\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\int { \frac {b \log \left (c x\right ) + a}{{\left (d + \frac {e}{x}\right )} x} \,d x } \]
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\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\int \frac {a + b \log {\left (c x \right )}}{d x + e}\, dx \]
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none
Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b}{d} + \frac {{\left (b \log \left (c\right ) + a\right )} \log \left (d x + e\right )}{d} \]
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\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\int { \frac {b \log \left (c x\right ) + a}{{\left (d + \frac {e}{x}\right )} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x} \, dx=\int \frac {a+b\,\ln \left (c\,x\right )}{x\,\left (d+\frac {e}{x}\right )} \,d x \]
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