Optimal. Leaf size=43 \[ \frac{a \text{PolyLog}(2,a+b x)}{b^2}-\frac{(-a-b x+1) \log (-a-b x+1)}{b^2}-\frac{x}{b} \]
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Rubi [A] time = 0.0725418, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {43, 2416, 2389, 2295, 2393, 2391} \[ \frac{a \text{PolyLog}(2,a+b x)}{b^2}-\frac{(-a-b x+1) \log (-a-b x+1)}{b^2}-\frac{x}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \log (1-a-b x)}{a+b x} \, dx &=\int \left (\frac{\log (1-a-b x)}{b}-\frac{a \log (1-a-b x)}{b (a+b x)}\right ) \, dx\\ &=\frac{\int \log (1-a-b x) \, dx}{b}-\frac{a \int \frac{\log (1-a-b x)}{a+b x} \, dx}{b}\\ &=-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a-b x)}{b^2}-\frac{a \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,a+b x\right )}{b^2}\\ &=-\frac{x}{b}-\frac{(1-a-b x) \log (1-a-b x)}{b^2}+\frac{a \text{Li}_2(a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0177899, size = 35, normalized size = 0.81 \[ \frac{a \text{PolyLog}(2,a+b x)+(a+b x-1) \log (-a-b x+1)-b x}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 77, normalized size = 1.8 \begin{align*}{\frac{\ln \left ( -bx-a+1 \right ) x}{b}}+{\frac{a{\it dilog} \left ( -bx-a+1 \right ) }{{b}^{2}}}+{\frac{\ln \left ( -bx-a+1 \right ) a}{{b}^{2}}}-{\frac{x}{b}}-{\frac{\ln \left ( -bx-a+1 \right ) }{{b}^{2}}}-{\frac{a}{{b}^{2}}}+{b}^{-2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03289, size = 111, normalized size = 2.58 \begin{align*} b{\left (\frac{{\left (\log \left (b x + a\right ) \log \left (-b x - a + 1\right ) +{\rm Li}_2\left (b x + a\right )\right )} a}{b^{3}} - \frac{x}{b^{2}} + \frac{{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} +{\left (\frac{x}{b} - \frac{a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (-b x - a + 1\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 (-b x - a + 1\right )}{b x + a}, 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{x \log{\left (- a - b x + 1 \right )}}{a + b 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{x \log \left (-b x - a + 1\right )}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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