Optimal. Leaf size=81 \[ -\frac{a \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\frac{a \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{x}{b} \]
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Rubi [A] time = 0.0994792, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {43, 2416, 2389, 2295, 2394, 2393, 2391} \[ -\frac{a \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b^2}-\frac{a \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \log (c+d x)}{a+b x} \, dx &=\int \left (\frac{\log (c+d x)}{b}-\frac{a \log (c+d x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \log (c+d x) \, dx}{b}-\frac{a \int \frac{\log (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{b d}+\frac{(a d) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2}\\ &=-\frac{x}{b}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}+\frac{a \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^2}\\ &=-\frac{x}{b}+\frac{(c+d x) \log (c+d x)}{b d}-\frac{a \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b^2}-\frac{a \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0291286, size = 73, normalized size = 0.9 \[ \frac{-a d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (-a d \log \left (\frac{d (a+b x)}{a d-b c}\right )+b c+b d x\right )-b d x}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 114, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) x}{b}}+{\frac{\ln \left ( dx+c \right ) c}{db}}-{\frac{x}{b}}-{\frac{c}{db}}-{\frac{a}{{b}^{2}}{\it dilog} \left ({\frac{ \left ( dx+c \right ) b+ad-bc}{ad-bc}} \right ) }-{\frac{a\ln \left ( dx+c \right ) }{{b}^{2}}\ln \left ({\frac{ \left ( dx+c \right ) b+ad-bc}{ad-bc}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01102, size = 150, normalized size = 1.85 \begin{align*} d{\left (\frac{{\left (\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 )\right )} a}{b^{2} d} - \frac{x}{b d} + \frac{c \log \left (d x + c\right )}{b d^{2}}\right )} +{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (d x + c\right ) \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{x \log \left (d x + c\right )}{b x + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{x \log \left (d x + c\right )}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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