Integrand size = 21, antiderivative size = 84 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=-\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^2} \]
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Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {269, 46, 2393, 2341, 2338, 2354, 2438} \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{e^2}-\frac {a+b \log (c x)}{e x}+\frac {b d \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{e^2}-\frac {b}{e x} \]
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Rule 46
Rule 269
Rule 2338
Rule 2341
Rule 2354
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log (c x)}{e x^2}-\frac {d (a+b \log (c x))}{e^2 x}+\frac {d^2 (a+b \log (c x))}{e^2 (e+d x)}\right ) \, dx \\ & = -\frac {d \int \frac {a+b \log (c x)}{x} \, dx}{e^2}+\frac {d^2 \int \frac {a+b \log (c x)}{e+d x} \, dx}{e^2}+\frac {\int \frac {a+b \log (c x)}{x^2} \, dx}{e} \\ & = -\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}-\frac {(b d) \int \frac {\log \left (1+\frac {d x}{e}\right )}{x} \, dx}{e^2} \\ & = -\frac {b}{e x}-\frac {a+b \log (c x)}{e x}-\frac {d (a+b \log (c x))^2}{2 b e^2}+\frac {d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )}{e^2}+\frac {b d \text {Li}_2\left (-\frac {d x}{e}\right )}{e^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=-\frac {\frac {2 b e}{x}+\frac {2 e (a+b \log (c x))}{x}+\frac {d (a+b \log (c x))^2}{b}-2 d (a+b \log (c x)) \log \left (1+\frac {d x}{e}\right )-2 b d \operatorname {PolyLog}\left (2,-\frac {d x}{e}\right )}{2 e^2} \]
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Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {a d \ln \left (d x +e \right )}{e^{2}}-\frac {a}{e x}-\frac {a d \ln \left (x \right )}{e^{2}}+\frac {b d \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{e^{2}}+\frac {b d \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{e^{2}}-\frac {b \ln \left (x c \right )}{e x}-\frac {b}{e x}-\frac {b d \ln \left (x c \right )^{2}}{2 e^{2}}\) | \(115\) |
parts | \(a \left (\frac {d \ln \left (d x +e \right )}{e^{2}}-\frac {1}{e x}-\frac {d \ln \left (x \right )}{e^{2}}\right )+\frac {b d \operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{e^{2}}+\frac {b d \ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{e^{2}}-\frac {b \ln \left (x c \right )}{e x}-\frac {b}{e x}-\frac {b d \ln \left (x c \right )^{2}}{2 e^{2}}\) | \(115\) |
derivativedivides | \(c^{2} \left (a \left (-\frac {1}{e \,c^{2} x}-\frac {d \ln \left (x c \right )}{e^{2} c^{2}}+\frac {d \ln \left (c d x +c e \right )}{e^{2} c^{2}}\right )+b \left (-\frac {d \ln \left (x c \right )^{2}}{2 e^{2} c^{2}}+\frac {-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}}{e c}+\frac {d^{2} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{e^{2} c^{2}}\right )\right )\) | \(151\) |
default | \(c^{2} \left (a \left (-\frac {1}{e \,c^{2} x}-\frac {d \ln \left (x c \right )}{e^{2} c^{2}}+\frac {d \ln \left (c d x +c e \right )}{e^{2} c^{2}}\right )+b \left (-\frac {d \ln \left (x c \right )^{2}}{2 e^{2} c^{2}}+\frac {-\frac {\ln \left (x c \right )}{x c}-\frac {1}{x c}}{e c}+\frac {d^{2} \left (\frac {\operatorname {dilog}\left (\frac {c d x +c e}{e c}\right )}{d}+\frac {\ln \left (x c \right ) \ln \left (\frac {c d x +c e}{e c}\right )}{d}\right )}{e^{2} c^{2}}\right )\right )\) | \(151\) |
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\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int { \frac {b \log \left (c x\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{3}} \,d x } \]
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Time = 37.73 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.45 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\frac {a d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} - \frac {a d \log {\left (x \right )}}{e^{2}} - \frac {a}{e x} - \frac {b d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (e \right )} \log {\left (x \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (e \right )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (e \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (e \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{e^{2}} + \frac {b d^{2} \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{e^{2}} + \frac {b d \log {\left (x \right )}^{2}}{2 e^{2}} - \frac {b d \log {\left (x \right )} \log {\left (c x \right )}}{e^{2}} - \frac {b \log {\left (c x \right )}}{e x} - \frac {b}{e x} \]
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Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b d}{e^{2}} + \frac {{\left (b d \log \left (c\right ) + a d\right )} \log \left (d x + e\right )}{e^{2}} - \frac {b d x \log \left (x\right )^{2} + 2 \, {\left (e \log \left (c\right ) + e\right )} b + 2 \, a e + 2 \, {\left (b e + {\left (b d \log \left (c\right ) + a d\right )} x\right )} \log \left (x\right )}{2 \, e^{2} x} \]
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\[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int { \frac {b \log \left (c x\right ) + a}{{\left (d + \frac {e}{x}\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log (c x)}{\left (d+\frac {e}{x}\right ) x^3} \, dx=\int \frac {a+b\,\ln \left (c\,x\right )}{x^3\,\left (d+\frac {e}{x}\right )} \,d x \]
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