Optimal. Leaf size=234 \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^2}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^2}-\frac{\log (x) \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c^2}-\frac{\log (x) \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c^2}-\frac{x}{c}+\frac{x \log (x)}{c} \]
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Rubi [A] time = 0.355269, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2357, 2295, 2317, 2391} \[ -\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^2}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^2}-\frac{\log (x) \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c^2}-\frac{\log (x) \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c^2}-\frac{x}{c}+\frac{x \log (x)}{c} \]
Antiderivative was successfully verified.
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Rule 2357
Rule 2295
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \log (x)}{a+b x+c x^2} \, dx &=\int \left (\frac{\log (x)}{c}-\frac{(a+b x) \log (x)}{c \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\int \log (x) \, dx}{c}-\frac{\int \frac{(a+b x) \log (x)}{a+b x+c x^2} \, dx}{c}\\ &=-\frac{x}{c}+\frac{x \log (x)}{c}-\frac{\int \left (\frac{\left (b+\frac{-b^2+2 a c}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (b-\frac{-b^2+2 a c}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{c}\\ &=-\frac{x}{c}+\frac{x \log (x)}{c}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{c}-\frac{\left (b+\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{c}\\ &=-\frac{x}{c}+\frac{x \log (x)}{c}-\frac{\left (b-\frac{b^2-2 a c}{\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^2}-\frac{\left (b+\frac{b^2-2 a c}{\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^2}+\frac{\left (b-\frac{b^2-2 a c}{\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^2}+\frac{\left (b+\frac{b^2-2 a c}{\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^2}\\ &=-\frac{x}{c}+\frac{x \log (x)}{c}-\frac{\left (b-\frac{b^2-2 a c}{\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^2}-\frac{\left (b+\frac{b^2-2 a c}{\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^2}-\frac{\left (b-\frac{b^2-2 a c}{\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^2}-\frac{\left (b+\frac{b^2-2 a c}{\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^2}\\ \end{align*}
Mathematica [A] time = 0.255355, size = 434, normalized size = 1.85 \[ -\frac{b \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^2}-\frac{b \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^2}-\frac{a \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{a \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{c \sqrt{b^2-4 a c}}-\frac{b \log (x) \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^2}-\frac{b \log (x) \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^2}-\frac{a \log (x) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}}+\frac{a \log (x) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )}{c \sqrt{b^2-4 a c}}-\frac{x}{c}+\frac{x \log (x)}{c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 593, normalized size = 2.5 \begin{align*} \text{result too large to display} \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^{2} \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^{2} \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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