Optimal. Leaf size=286 \[ \frac {\left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 c^3}+\frac {\log (x) \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 c^3}+\frac {\log (x) \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 c^3}+\frac {b x}{c^2}-\frac {b x \log (x)}{c^2}-\frac {x^2}{4 c}+\frac {x^2 \log (x)}{2 c} \]
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Rubi [A] time = 0.44, antiderivative size = 286, normalized size of antiderivative = 1.00, 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 2295
Rule 2304
Rule 2317
Rule 2357
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*}
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Mathematica [A] time = 0.69, size = 464, normalized size = 1.62 \[ \frac {2 \left (b^2-a c\right ) \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+\frac {4 a b c \text {Li}_2\left (\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )}{\sqrt {b^2-4 a c}}+2 \left (b^2-a c\right ) \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )-\frac {4 a b c \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \log (x) \left (b^2-a c\right ) \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )+\frac {4 a b c \log (x) \log \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+2 \log (x) \left (b^2-a c\right ) \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )-\frac {4 a b c \log (x) \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )}{\sqrt {b^2-4 a c}}+4 b c x-4 b c x \log (x)-c^2 x^2+2 c^2 x^2 \log (x)}{4 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \log \relax (x)}{c x^{2} + b x + a}, x\right ) \]
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 \[ \int \frac {x^{3} \log \relax (x)}{c x^{2} + b x + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 791, normalized size = 2.77 \[ -\frac {3 a b \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{2}}+\frac {3 a b \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{2}}+\frac {b^{3} \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{3}}-\frac {b^{3} \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{3}}+\frac {x^{2} \ln \relax (x )}{2 c}-\frac {3 a b \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{2}}+\frac {3 a b \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{2}}-\frac {a \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{2}}-\frac {a \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{2}}+\frac {b^{3} \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{3}}-\frac {b^{3} \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, c^{3}}+\frac {b^{2} \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{3}}+\frac {b^{2} \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{3}}-\frac {b x \ln \relax (x )}{c^{2}}-\frac {x^{2}}{4 c}-\frac {a \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{2}}-\frac {a \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{2}}+\frac {b^{2} \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{3}}+\frac {b^{2} \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 c^{3}}+\frac {b x}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\ln \relax (x)}{c\,x^2+b\,x+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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