Integrand size = 18, antiderivative size = 63 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\frac {a x}{d}-\frac {b x}{d}+\frac {b x \log (c x)}{d}-\frac {e (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {199, 45, 2367, 2332, 2354, 2438} \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=-\frac {e \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac {a x}{d}+\frac {b x \log (c x)}{d}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2}-\frac {b x}{d} \]
[In]
[Out]
Rule 45
Rule 199
Rule 2332
Rule 2354
Rule 2367
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log (c x)}{d}-\frac {e (a+b \log (c x))}{d (e+d x)}\right ) \, dx \\ & = \frac {\int (a+b \log (c x)) \, dx}{d}-\frac {e \int \frac {a+b \log (c x)}{e+d x} \, dx}{d} \\ & = \frac {a x}{d}-\frac {e (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^2}+\frac {b \int \log (c x) \, dx}{d}+\frac {(b e) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{d^2} \\ & = \frac {a x}{d}-\frac {b x}{d}+\frac {b x \log (c x)}{d}-\frac {e (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{d^2}-\frac {b e \text {Li}_2\left (-\frac {d x}{e}\right )}{d^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\frac {a x}{d}-\frac {b x}{d}+\frac {b x \log (c x)}{d}-\frac {e (a+b \log (c x)) \log \left (\frac {e+d x}{e}\right )}{d^2}-\frac {b e \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.40
method | result | size |
risch | \(\frac {a x}{d}-\frac {a e \ln \left (d x +e \right )}{d^{2}}+\frac {b x \ln \left (x c \right )}{d}-\frac {b x}{d}-\frac {b e \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{2}}-\frac {b e \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{2}}\) | \(88\) |
parts | \(\frac {a x}{d}-\frac {a e \ln \left (d x +e \right )}{d^{2}}+\frac {b x \ln \left (x c \right )}{d}-\frac {b x}{d}-\frac {b e \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d^{2}}-\frac {b e \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d^{2}}\) | \(88\) |
derivativedivides | \(\frac {\frac {a x c}{d}-\frac {a e c \ln \left (c d x +c e \right )}{d^{2}}+b \left (\frac {x c \ln \left (x c \right )-x c}{d}-\frac {e c \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d}\right )}{c}\) | \(101\) |
default | \(\frac {\frac {a x c}{d}-\frac {a e c \ln \left (c d x +c e \right )}{d^{2}}+b \left (\frac {x c \ln \left (x c \right )-x c}{d}-\frac {e c \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{d}\right )}{c}\) | \(101\) |
[In]
[Out]
\[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int { \frac {b \log \left (c x\right ) + a}{d + \frac {e}{x}} \,d x } \]
[In]
[Out]
Time = 43.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.48 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=- \frac {a e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d} + \frac {a x}{d} + \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d} - \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d} + \frac {b x \log {\left (c x \right )}}{d} - \frac {b x}{d} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=-\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e}{d^{2}} + \frac {b x \log \left (x\right ) + {\left (b {\left (\log \left (c\right ) - 1\right )} + a\right )} x}{d} - \frac {{\left (b e \log \left (c\right ) + a e\right )} \log \left (d x + e\right )}{d^{2}} \]
[In]
[Out]
\[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int { \frac {b \log \left (c x\right ) + a}{d + \frac {e}{x}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx=\int \frac {a+b\,\ln \left (c\,x\right )}{d+\frac {e}{x}} \,d x \]
[In]
[Out]