Optimal. Leaf size=30 \[ \text {Li}_2\left (-\frac {c x}{b}\right )+\log (x) \log \left (\frac {c x}{b}+1\right )+\frac {\log ^2(x)}{2} \]
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Rubi [A] time = 0.10, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2357, 2301, 2317, 2391} \[ \text {PolyLog}\left (2,-\frac {c x}{b}\right )+\log (x) \log \left (\frac {c x}{b}+1\right )+\frac {\log ^2(x)}{2} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2317
Rule 2357
Rule 2391
Rubi steps
\begin {align*} \int \frac {(b+2 c x) \log (x)}{x (b+c x)} \, dx &=\int \left (\frac {\log (x)}{x}+\frac {c \log (x)}{b+c x}\right ) \, dx\\ &=c \int \frac {\log (x)}{b+c x} \, dx+\int \frac {\log (x)}{x} \, dx\\ &=\frac {\log ^2(x)}{2}+\log (x) \log \left (1+\frac {c x}{b}\right )-\int \frac {\log \left (1+\frac {c x}{b}\right )}{x} \, dx\\ &=\frac {\log ^2(x)}{2}+\log (x) \log \left (1+\frac {c x}{b}\right )+\text {Li}_2\left (-\frac {c x}{b}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.03 \[ \text {Li}_2\left (-\frac {c x}{b}\right )+\log (x) \log \left (\frac {b+c x}{b}\right )+\frac {\log ^2(x)}{2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (2 \, c x + b\right )} \log \relax (x)}{c x^{2} + b x}, x\right ) \]
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 \[ \int \frac {{\left (2 \, c x + b\right )} \log \relax (x)}{{\left (c x + b\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 31, normalized size = 1.03 \[ \frac {\ln \relax (x )^{2}}{2}+\ln \relax (x ) \ln \left (\frac {c x +b}{b}\right )+\dilog \left (\frac {c x +b}{b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 49, normalized size = 1.63 \[ {\left (\log \left (c x + b\right ) + \log \relax (x)\right )} \log \relax (x) - \log \left (c x + b\right ) \log \relax (x) + \log \left (\frac {c x}{b} + 1\right ) \log \relax (x) - \frac {1}{2} \, \log \relax (x)^{2} + {\rm Li}_2\left (-\frac {c x}{b}\right ) \]
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 \[ \int \frac {\ln \relax (x)\,\left (b+2\,c\,x\right )}{x\,\left (b+c\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 117.58, size = 192, normalized size = 6.40 \[ b \left (\begin {cases} - \frac {1}{c x} & \text {for}\: b = 0 \\\frac {\begin {cases} \log {\relax (c )} \log {\relax (x )} + \operatorname {Li}_{2}\left (\frac {b e^{i \pi }}{c x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (c )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {b e^{i \pi }}{c x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (c )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (c )} + \operatorname {Li}_{2}\left (\frac {b e^{i \pi }}{c x}\right ) & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) - b \left (\begin {cases} \frac {1}{c x} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {b}{x} + c \right )}}{b} & \text {otherwise} \end {cases}\right ) \log {\relax (x )} - 2 c \left (\begin {cases} \frac {x}{b} & \text {for}\: c = 0 \\\frac {\begin {cases} \log {\relax (b )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {c x e^{i \pi }}{b}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (b )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {c x e^{i \pi }}{b}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (b )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (b )} - \operatorname {Li}_{2}\left (\frac {c x e^{i \pi }}{b}\right ) & \text {otherwise} \end {cases}}{c} & \text {otherwise} \end {cases}\right ) + 2 c \left (\begin {cases} \frac {x}{b} & \text {for}\: c = 0 \\\frac {\log {\left (b + c x \right )}}{c} & \text {otherwise} \end {cases}\right ) \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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