Optimal. Leaf size=193 \[ \frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c}+\frac{\log (x) \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c}+\frac{\log (x) \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c} \]
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Rubi [A] time = 0.180522, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2357, 2317, 2391} \[ \frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c}+\frac{\log (x) \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c}+\frac{\log (x) \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 2357
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x \log (x)}{a+b x+c x^2} \, dx &=\int \left (\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx\\ &=\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b-\sqrt{b^2-4 a c}+2 c x} \, dx+\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b+\sqrt{b^2-4 a c}+2 c x} \, dx\\ &=\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 c}-\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{x} \, dx}{2 c}-\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx}{2 c}\\ &=\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \log (x) \log \left (1+\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{Li}_2\left (-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c}+\frac{\left (1+\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{Li}_2\left (-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.131616, size = 210, normalized size = 1.09 \[ \frac{\left (\sqrt{b^2-4 a c}-b\right ) \text{PolyLog}\left (2,\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+\left (\sqrt{b^2-4 a c}+b\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )+\log (x) \left (\left (\sqrt{b^2-4 a c}-b\right ) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )+\left (\sqrt{b^2-4 a c}+b\right ) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )\right )}{2 c \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 361, normalized size = 1.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2\,c} \left ( \ln \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) \sqrt{-4\,ac+{b}^{2}}-\ln \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) b+\ln \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) \sqrt{-4\,ac+{b}^{2}}+\ln \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) b \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}+{\frac{1}{2\,c}{\it dilog} \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) }+{\frac{b}{2\,c}{\it dilog} \left ({ \left ( 2\,cx+\sqrt{-4\,ac+{b}^{2}}+b \right ) \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}+{\frac{1}{2\,c}{\it dilog} \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ) }-{\frac{b}{2\,c}{\it dilog} \left ({ \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (x\right )}{c x^{2} + 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 (x\right )}{c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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