Optimal. Leaf size=286 \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^3}+\frac{\log (x) \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c^3}+\frac{\log (x) \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c^3}+\frac{b x}{c^2}-\frac{b x \log (x)}{c^2}-\frac{x^2}{4 c}+\frac{x^2 \log (x)}{2 c} \]
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Rubi [A] time = 0.443137, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {2357, 2295, 2304, 2317, 2391} \[ \frac{\left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{2 c^3}+\frac{\left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{2 c^3}+\frac{\log (x) \left (-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{b-\sqrt{b^2-4 a c}}+1\right )}{2 c^3}+\frac{\log (x) \left (\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}-a c+b^2\right ) \log \left (\frac{2 c x}{\sqrt{b^2-4 a c}+b}+1\right )}{2 c^3}+\frac{b x}{c^2}-\frac{b x \log (x)}{c^2}-\frac{x^2}{4 c}+\frac{x^2 \log (x)}{2 c} \]
Antiderivative was successfully verified.
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Rule 2357
Rule 2295
Rule 2304
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \log (x)}{a+b x+c x^2} \, dx &=\int \left (-\frac{b \log (x)}{c^2}+\frac{x \log (x)}{c}+\frac{\left (a b+\left (b^2-a c\right ) x\right ) \log (x)}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (a b+\left (b^2-a c\right ) x\right ) \log (x)}{a+b x+c x^2} \, dx}{c^2}-\frac{b \int \log (x) \, dx}{c^2}+\frac{\int x \log (x) \, dx}{c}\\ &=\frac{b x}{c^2}-\frac{x^2}{4 c}-\frac{b x \log (x)}{c^2}+\frac{x^2 \log (x)}{2 c}+\frac{\int \left (\frac{\left (b^2-a c+\frac{b \left (-b^2+3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{\left (b^2-a c-\frac{b \left (-b^2+3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \log (x)}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{c^2}\\ &=\frac{b x}{c^2}-\frac{x^2}{4 c}-\frac{b x \log (x)}{c^2}+\frac{x^2 \log (x)}{2 c}+\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{c^2}+\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{\log (x)}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{c^2}\\ &=\frac{b x}{c^2}-\frac{x^2}{4 c}-\frac{b x \log (x)}{c^2}+\frac{x^2 \log (x)}{2 c}+\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\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^3}+\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\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^3}-\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\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^3}-\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\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^3}\\ &=\frac{b x}{c^2}-\frac{x^2}{4 c}-\frac{b x \log (x)}{c^2}+\frac{x^2 \log (x)}{2 c}+\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\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^3}+\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\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^3}+\frac{\left (b^2-a c-\frac{b \left (b^2-3 a c\right )}{\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^3}+\frac{\left (b^2-a c+\frac{b \left (b^2-3 a c\right )}{\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^3}\\ \end{align*}
Mathematica [A] time = 0.650866, size = 464, normalized size = 1.62 \[ \frac{2 \left (b^2-a c\right ) \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \text{PolyLog}\left (2,\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+\frac{4 a b c \text{PolyLog}\left (2,\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )}{\sqrt{b^2-4 a c}}+2 \left (b^2-a c\right ) \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )-\frac{4 a b c \text{PolyLog}\left (2,-\frac{2 c x}{\sqrt{b^2-4 a c}+b}\right )}{\sqrt{b^2-4 a c}}+2 \log (x) \left (b^2-a c\right ) \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )+\frac{4 a b c \log (x) \log \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}+2 \log (x) \left (b^2-a c\right ) \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )-\frac{4 a b c \log (x) \log \left (\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}+b}\right )}{\sqrt{b^2-4 a c}}+4 b c x-4 b c x \log (x)-c^2 x^2+2 c^2 x^2 \log (x)}{4 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 791, normalized size = 2.8 \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^{3} \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^{3} \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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