Optimal. Leaf size=251 \[ \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a^2}+\frac {\log (x) \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a^2}-\frac {b \log ^2(x)}{2 a^2}-\frac {1}{a x}-\frac {\log (x)}{a x} \]
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Rubi [A] time = 0.39, antiderivative size = 251, 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, 2304, 2301, 2317, 2391} \[ \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {PolyLog}\left (2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2}+\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \text {PolyLog}\left (2,-\frac {2 c x}{\sqrt {b^2-4 a c}+b}\right )}{2 a^2}+\frac {\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )}{2 a^2}+\frac {\log (x) \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \log \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )}{2 a^2}-\frac {b \log ^2(x)}{2 a^2}-\frac {1}{a x}-\frac {\log (x)}{a x} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2304
Rule 2317
Rule 2357
Rule 2391
Rubi steps
\begin {align*} \int \frac {\log (x)}{x^2 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {\log (x)}{a x^2}-\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c+b c x\right ) \log (x)}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (b^2-a c+b c x\right ) \log (x)}{a+b x+c x^2} \, dx}{a^2}+\frac {\int \frac {\log (x)}{x^2} \, dx}{a}-\frac {b \int \frac {\log (x)}{x} \, dx}{a^2}\\ &=-\frac {1}{a x}-\frac {\log (x)}{a x}-\frac {b \log ^2(x)}{2 a^2}+\frac {\int \left (\frac {\left (b c+\frac {c \left (b^2-2 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 c-\frac {c \left (b^2-2 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a^2}\\ &=-\frac {1}{a x}-\frac {\log (x)}{a x}-\frac {b \log ^2(x)}{2 a^2}+\frac {\left (c \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{a^2}+\frac {\left (c \left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{a^2}\\ &=-\frac {1}{a x}-\frac {\log (x)}{a x}-\frac {b \log ^2(x)}{2 a^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 a^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 a^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 a^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 a^2}\\ &=-\frac {1}{a x}-\frac {\log (x)}{a x}-\frac {b \log ^2(x)}{2 a^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 a^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 a^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 a^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 a^2}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 255, normalized size = 1.02 \[ \frac {\left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \text {Li}_2\left (\frac {2 c x}{\sqrt {b^2-4 a c}-b}\right )+\left (\frac {2 a c-b^2}{\sqrt {b^2-4 a c}}+b\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )+\log (x) \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {-\sqrt {b^2-4 a c}+b+2 c x}{b-\sqrt {b^2-4 a c}}\right )+\log (x) \left (\frac {2 a c-b^2}{\sqrt {b^2-4 a c}}+b\right ) \log \left (\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}+b}\right )-\frac {2 a}{x}-\frac {2 a \log (x)}{x}-b \log ^2(x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \relax (x)}{c x^{4} + b x^{3} + a x^{2}}, x\right ) \]
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 \[ \int \frac {\log \relax (x)}{{\left (c x^{2} + b x + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 608, normalized size = 2.42 \[ \frac {c \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {-4 a c +b^{2}}\, a}-\frac {c \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {-4 a c +b^{2}}\, a}-\frac {b^{2} \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a^{2}}+\frac {b^{2} \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a^{2}}+\frac {c \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {-4 a c +b^{2}}\, a}-\frac {c \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{\sqrt {-4 a c +b^{2}}\, a}-\frac {b^{2} \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a^{2}}+\frac {b^{2} \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 \sqrt {-4 a c +b^{2}}\, a^{2}}-\frac {b \ln \relax (x )^{2}}{2 a^{2}}+\frac {b \ln \relax (x ) \ln \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2}}+\frac {b \ln \relax (x ) \ln \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2}}+\frac {b \dilog \left (\frac {2 c x +b +\sqrt {-4 a c +b^{2}}}{b +\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2}}+\frac {b \dilog \left (\frac {-2 c x -b +\sqrt {-4 a c +b^{2}}}{-b +\sqrt {-4 a c +b^{2}}}\right )}{2 a^{2}}-\frac {\ln \relax (x )}{a x}-\frac {1}{a x} \]
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 \[ \int \frac {\ln \relax (x)}{x^2\,\left (c\,x^2+b\,x+a\right )} \,d x \]
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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