Optimal. Leaf size=152 \[ -\frac{a^2 \text{PolyLog}(2,c (a+b x))}{2 b^2}+\frac{1}{2} x^2 \text{PolyLog}(2,c (a+b x))-\frac{(1-a c)^2 \log (-a c-b c x+1)}{4 b^2 c^2}+\frac{a (-a c-b c x+1) \log (-a c-b c x+1)}{2 b^2 c}+\frac{1}{4} x^2 \log (-a c-b c x+1)-\frac{x (1-a c)}{4 b c}+\frac{a x}{2 b}-\frac{x^2}{8} \]
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Rubi [A] time = 0.169219, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {6598, 43, 2416, 2389, 2295, 2395, 2393, 2391} \[ -\frac{a^2 \text{PolyLog}(2,c (a+b x))}{2 b^2}+\frac{1}{2} x^2 \text{PolyLog}(2,c (a+b x))-\frac{(1-a c)^2 \log (-a c-b c x+1)}{4 b^2 c^2}+\frac{a (-a c-b c x+1) \log (-a c-b c x+1)}{2 b^2 c}+\frac{1}{4} x^2 \log (-a c-b c x+1)-\frac{x (1-a c)}{4 b c}+\frac{a x}{2 b}-\frac{x^2}{8} \]
Antiderivative was successfully verified.
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Rule 6598
Rule 43
Rule 2416
Rule 2389
Rule 2295
Rule 2395
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int x \text{Li}_2(c (a+b x)) \, dx &=\frac{1}{2} x^2 \text{Li}_2(c (a+b x))+\frac{1}{2} b \int \frac{x^2 \log (1-a c-b c x)}{a+b x} \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2(c (a+b x))+\frac{1}{2} b \int \left (-\frac{a \log (1-a c-b c x)}{b^2}+\frac{x \log (1-a c-b c x)}{b}+\frac{a^2 \log (1-a c-b c x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \text{Li}_2(c (a+b x))+\frac{1}{2} \int x \log (1-a c-b c x) \, dx-\frac{a \int \log (1-a c-b c x) \, dx}{2 b}+\frac{a^2 \int \frac{\log (1-a c-b c x)}{a+b x} \, dx}{2 b}\\ &=\frac{1}{4} x^2 \log (1-a c-b c x)+\frac{1}{2} x^2 \text{Li}_2(c (a+b x))+\frac{a^2 \operatorname{Subst}\left (\int \frac{\log (1-c x)}{x} \, dx,x,a+b x\right )}{2 b^2}+\frac{a \operatorname{Subst}(\int \log (x) \, dx,x,1-a c-b c x)}{2 b^2 c}+\frac{1}{4} (b c) \int \frac{x^2}{1-a c-b c x} \, dx\\ &=\frac{a x}{2 b}+\frac{1}{4} x^2 \log (1-a c-b c x)+\frac{a (1-a c-b c x) \log (1-a c-b c x)}{2 b^2 c}-\frac{a^2 \text{Li}_2(c (a+b x))}{2 b^2}+\frac{1}{2} x^2 \text{Li}_2(c (a+b x))+\frac{1}{4} (b c) \int \left (\frac{-1+a c}{b^2 c^2}-\frac{x}{b c}-\frac{(-1+a c)^2}{b^2 c^2 (-1+a c+b c x)}\right ) \, dx\\ &=\frac{a x}{2 b}-\frac{(1-a c) x}{4 b c}-\frac{x^2}{8}-\frac{(1-a c)^2 \log (1-a c-b c x)}{4 b^2 c^2}+\frac{1}{4} x^2 \log (1-a c-b c x)+\frac{a (1-a c-b c x) \log (1-a c-b c x)}{2 b^2 c}-\frac{a^2 \text{Li}_2(c (a+b x))}{2 b^2}+\frac{1}{2} x^2 \text{Li}_2(c (a+b x))\\ \end{align*}
Mathematica [A] time = 0.100904, size = 96, normalized size = 0.63 \[ \frac{-4 c^2 \left (a^2-b^2 x^2\right ) \text{PolyLog}(2,c (a+b x))+\left (-6 a^2 c^2-4 a c (b c x-2)+2 b^2 c^2 x^2-2\right ) \log (-a c-b c x+1)-b c x (-6 a c+b c x+2)}{8 b^2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 177, normalized size = 1.2 \begin{align*} -{\frac{{\it polylog} \left ( 2,xbc+ac \right ){a}^{2}}{2\,{b}^{2}}}+{\frac{{\it polylog} \left ( 2,xbc+ac \right ){x}^{2}}{2}}-{\frac{\ln \left ( -xbc-ac+1 \right ) xa}{2\,b}}-{\frac{3\,\ln \left ( -xbc-ac+1 \right ){a}^{2}}{4\,{b}^{2}}}+{\frac{3\,ax}{4\,b}}+{\frac{7\,{a}^{2}}{8\,{b}^{2}}}+{\frac{\ln \left ( -xbc-ac+1 \right ) a}{{b}^{2}c}}-{\frac{5\,a}{4\,{b}^{2}c}}+{\frac{{x}^{2}\ln \left ( -xbc-ac+1 \right ) }{4}}-{\frac{\ln \left ( -xbc-ac+1 \right ) }{4\,{b}^{2}{c}^{2}}}-{\frac{{x}^{2}}{8}}-{\frac{x}{4\,bc}}+{\frac{3}{8\,{b}^{2}{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992845, size = 196, normalized size = 1.29 \begin{align*} \frac{{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) +{\rm Li}_2\left (-b c x - a c + 1\right )\right )} a^{2}}{2 \, b^{2}} + \frac{4 \, b^{2} c^{2} x^{2}{\rm Li}_2\left (b c x + a c\right ) - b^{2} c^{2} x^{2} + 2 \,{\left (3 \, a b c^{2} - b c\right )} x + 2 \,{\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x - 3 \, a^{2} c^{2} + 4 \, a c - 1\right )} \log \left (-b c x - a c + 1\right )}{8 \, b^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.348, size = 242, normalized size = 1.59 \begin{align*} -\frac{b^{2} c^{2} x^{2} - 2 \,{\left (3 \, a b c^{2} - b c\right )} x - 4 \,{\left (b^{2} c^{2} x^{2} - a^{2} c^{2}\right )}{\rm Li}_2\left (b c x + a c\right ) - 2 \,{\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x - 3 \, a^{2} c^{2} + 4 \, a c - 1\right )} \log \left (-b c x - a c + 1\right )}{8 \, b^{2} c^{2}} \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 x{\rm Li}_2\left ({\left (b x + a\right )} c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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