Optimal. Leaf size=43 \[ \frac {a \text {Li}_2(a+b x)}{b^2}-\frac {(-a-b x+1) \log (-a-b x+1)}{b^2}-\frac {x}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2416
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*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.81 \[ \frac {a \text {Li}_2(a+b x)+(a+b x-1) \log (-a-b x+1)-b x}{b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left (-b x - a + 1\right )}{b x + a}, x\right ) \]
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 \[ \int \frac {x \log \left (-b x - a + 1\right )}{b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 77, normalized size = 1.79 \[ \frac {x \ln \left (-b x -a +1\right )}{b}+\frac {a \dilog \left (-b x -a +1\right )}{b^{2}}+\frac {a \ln \left (-b x -a +1\right )}{b^{2}}-\frac {x}{b}-\frac {a}{b^{2}}-\frac {\ln \left (-b x -a +1\right )}{b^{2}}+\frac {1}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 82, normalized size = 1.91 \[ b {\left (\frac {{\left (\log \left (b x + a\right ) \log \left (-b x - a + 1\right ) + {\rm Li}_2\left (b x + a\right )\right )} a}{b^{3}} - \frac {x}{b^{2}} + \frac {{\left (a - 1\right )} \log \left (b x + a - 1\right )}{b^{3}}\right )} + {\left (\frac {x}{b} - \frac {a \log \left (b x + a\right )}{b^{2}}\right )} \log \left (-b x - a + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 59, normalized size = 1.37 \[ -\frac {\ln \left (1-b\,x-a\right )+b\,\left (x-x\,\ln \left (1-b\,x-a\right )\right )-a\,{\mathrm {Li}}_{\mathrm {2}}\left (1-b\,x-a\right )-a\,\ln \left (1-b\,x-a\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log {\left (- a - b x + 1 \right )}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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