Optimal. Leaf size=63 \[ -\frac {e \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac {a x}{d}+\frac {b x \log (c x)}{d}-\frac {b e \text {Li}_2\left (-\frac {d x}{e}\right )}{d^2}-\frac {b x}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {193, 43, 2330, 2295, 2317, 2391} \[ -\frac {b e \text {PolyLog}\left (2,-\frac {d x}{e}\right )}{d^2}-\frac {e \log \left (\frac {d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac {a x}{d}+\frac {b x \log (c x)}{d}-\frac {b x}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 193
Rule 2295
Rule 2317
Rule 2330
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log (c x)}{d+\frac {e}{x}} \, dx &=\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*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 1.02 \[ -\frac {e \log \left (\frac {d x+e}{e}\right ) (a+b \log (c x))}{d^2}+\frac {a x}{d}+\frac {b x \log (c x)}{d}-\frac {b e \text {Li}_2\left (-\frac {d x}{e}\right )}{d^2}-\frac {b x}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left (c x\right ) + a x}{d x + e}, x\right ) \]
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 \[ \int \frac {b \log \left (c x\right ) + a}{d + \frac {e}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 91, normalized size = 1.44 \[ \frac {b x \ln \left (c x \right )}{d}-\frac {b e \ln \left (c x \right ) \ln \left (\frac {c d x +c e}{c e}\right )}{d^{2}}+\frac {a x}{d}-\frac {a e \ln \left (c d x +c e \right )}{d^{2}}-\frac {b x}{d}-\frac {b e \dilog \left (\frac {c d x +c e}{c e}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 69, normalized size = 1.10 \[ -\frac {{\left (\log \left (\frac {d x}{e} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {d x}{e}\right )\right )} b e}{d^{2}} + \frac {b x \log \relax (x) + {\left (b {\left (\log \relax (c) - 1\right )} + a\right )} x}{d} - \frac {{\left (b e \log \relax (c) + a e\right )} \log \left (d x + e\right )}{d^{2}} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x\right )}{d+\frac {e}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 71.22, size = 138, normalized size = 2.19 \[ - \frac {a e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right )}{d} + \frac {a x}{d} + \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\begin {cases} \log {\relax (e )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (e )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (e )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (e )} - \operatorname {Li}_{2}\left (\frac {d x e^{i \pi }}{e}\right ) & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right )}{d} - \frac {b e \left (\begin {cases} \frac {x}{e} & \text {for}\: d = 0 \\\frac {\log {\left (d x + e \right )}}{d} & \text {otherwise} \end {cases}\right ) \log {\left (c x \right )}}{d} + \frac {b x \log {\left (c x \right )}}{d} - \frac {b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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