Optimal. Leaf size=105 \[ -\frac{\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d}+\frac{\text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{d}+\frac{\log \left (\frac{a}{x}+b\right ) \log (c+d x)}{d}-\frac{\log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d}+\frac{\log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d} \]
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Rubi [A] time = 0.167946, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2465, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d}+\frac{\text{PolyLog}\left (2,\frac{d x}{c}+1\right )}{d}+\frac{\log \left (\frac{a}{x}+b\right ) \log (c+d x)}{d}-\frac{\log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d}+\frac{\log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2465
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{a+b x}{x}\right )}{c+d x} \, dx &=\int \frac{\log \left (b+\frac{a}{x}\right )}{c+d x} \, dx\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{a \int \frac{\log (c+d x)}{\left (b+\frac{a}{x}\right ) x^2} \, dx}{d}\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{a \int \left (\frac{\log (c+d x)}{a x}-\frac{b \log (c+d x)}{a (a+b x)}\right ) \, dx}{d}\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{\int \frac{\log (c+d x)}{x} \, dx}{d}-\frac{b \int \frac{\log (c+d x)}{a+b x} \, dx}{d}\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{\log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d}-\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\int \frac{\log \left (-\frac{d x}{c}\right )}{c+d x} \, dx+\int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{\log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d}-\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}+\frac{\text{Li}_2\left (1+\frac{d x}{c}\right )}{d}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\log \left (b+\frac{a}{x}\right ) \log (c+d x)}{d}+\frac{\log \left (-\frac{d x}{c}\right ) \log (c+d x)}{d}-\frac{\log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{\text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d}+\frac{\text{Li}_2\left (1+\frac{d x}{c}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0286185, size = 80, normalized size = 0.76 \[ \frac{-\text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\text{PolyLog}\left (2,\frac{d x}{c}+1\right )+\log (c+d x) \left (-\log \left (\frac{d (a+b x)}{a d-b c}\right )+\log \left (\frac{a}{x}+b\right )+\log \left (-\frac{d x}{c}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.411, size = 114, normalized size = 1.1 \begin{align*} -{\frac{1}{d}\ln \left ( b+{\frac{a}{x}} \right ) \ln \left ( -{\frac{a}{bx}} \right ) }-{\frac{1}{d}{\it dilog} \left ( -{\frac{a}{bx}} \right ) }+{\frac{1}{d}{\it dilog} \left ({\frac{1}{ad-bc} \left ( c \left ( b+{\frac{a}{x}} \right ) +ad-bc \right ) } \right ) }+{\frac{1}{d}\ln \left ( b+{\frac{a}{x}} \right ) \ln \left ({\frac{1}{ad-bc} \left ( c \left ( b+{\frac{a}{x}} \right ) +ad-bc \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0664, size = 167, normalized size = 1.59 \begin{align*} -\frac{{\left (\log \left (b x + a\right ) - \log \left (x\right )\right )} \log \left (d x + c\right )}{d} + \frac{\log \left (d x + c\right ) \log \left (\frac{b x + a}{x}\right )}{d} - \frac{\log \left (\frac{d x}{c} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{d x}{c}\right )}{d} + \frac{\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\frac{b x + a}{x}\right )}{d x + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\frac{a}{x} + b \right )}}{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{b x + a}{x}\right )}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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