Optimal. Leaf size=308 \[ -\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \log (x) \log \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )}{2 a^3}-\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \text {Li}_2\left (-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{2 a^3} \]
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Rubi [A]
time = 0.36, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2404, 2341,
2338, 2354, 2438} \begin {gather*} -\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 a^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 a^3}+\frac {\log ^2(x) \left (b^2-a c\right )}{2 a^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 a^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 a^3}+\frac {b}{a^2 x}+\frac {b \log (x)}{a^2 x}-\frac {1}{4 a x^2}-\frac {\log (x)}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2338
Rule 2341
Rule 2354
Rule 2404
Rule 2438
Rubi steps
\begin {align*} \int \frac {\log (x)}{x^3 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {\log (x)}{a x^3}-\frac {b \log (x)}{a^2 x^2}+\frac {\left (b^2-a c\right ) \log (x)}{a^3 x}+\frac {\left (-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x\right ) \log (x)}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (-b \left (b^2-2 a c\right )-c \left (b^2-a c\right ) x\right ) \log (x)}{a+b x+c x^2} \, dx}{a^3}+\frac {\int \frac {\log (x)}{x^3} \, dx}{a}-\frac {b \int \frac {\log (x)}{x^2} \, dx}{a^2}+\frac {\left (b^2-a c\right ) \int \frac {\log (x)}{x} \, dx}{a^3}\\ &=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^3}+\frac {\int \left (\frac {\left (-\frac {b c \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-c \left (b^2-a c\right )\right ) \log (x)}{b-\sqrt {b^2-4 a c}+2 c x}+\frac {\left (\frac {b c \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-c \left (b^2-a c\right )\right ) \log (x)}{b+\sqrt {b^2-4 a c}+2 c x}\right ) \, dx}{a^3}\\ &=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^3}-\frac {\left (c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b+\sqrt {b^2-4 a c}+2 c x} \, dx}{a^3}-\frac {\left (c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {\log (x)}{b-\sqrt {b^2-4 a c}+2 c x} \, dx}{a^3}\\ &=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^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 a^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 a^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 a^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 a^3}\\ &=-\frac {1}{4 a x^2}+\frac {b}{a^2 x}-\frac {\log (x)}{2 a x^2}+\frac {b \log (x)}{a^2 x}+\frac {\left (b^2-a c\right ) \log ^2(x)}{2 a^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 a^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 a^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 a^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 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 311, normalized size = 1.01 \begin {gather*} -\frac {\frac {a^2}{x^2}-\frac {4 a b}{x}+\frac {2 a^2 \log (x)}{x^2}-\frac {4 a b \log (x)}{x}-2 \left (b^2-a c\right ) \log ^2(x)+2 \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 (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )+2 \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 (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )+2 \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 \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 )}{4 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(781\) vs.
\(2(278)=556\).
time = 0.96, size = 782, normalized size = 2.54
method | result | size |
default | \(\frac {-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}}{a}-\frac {\left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right ) b}{a^{2}}+\frac {\left (-c a +b^{2}\right ) \ln \left (x \right )^{2}}{2 a^{3}}+\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}\right )}{2 \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}+3 \dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) a b c -\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}+\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, a c -\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) \sqrt {-4 c a +b^{2}}\, b^{2}-3 \dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) a b c +\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 \sqrt {-4 c a +b^{2}}}}{a^{3}}\) | \(782\) |
risch | \(-\frac {\ln \left (x \right )}{2 x^{2} a}-\frac {1}{4 x^{2} a}+\frac {b \ln \left (x \right )}{a^{2} x}+\frac {b}{a^{2} x}-\frac {\ln \left (x \right )^{2} c}{2 a^{2}}+\frac {\ln \left (x \right )^{2} b^{2}}{2 a^{3}}+\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {3 \ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}-\frac {\ln \left (x \right ) \ln \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}-\frac {3 \ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\ln \left (x \right ) \ln \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}+\frac {3 \dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}-\frac {\dilog \left (\frac {-2 c x +\sqrt {-4 c a +b^{2}}-b}{-b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) c}{2 a^{2}}-\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{2}}{2 a^{3}}-\frac {3 \dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b c}{2 a^{2} \sqrt {-4 c a +b^{2}}}+\frac {\dilog \left (\frac {b +2 c x +\sqrt {-4 c a +b^{2}}}{b +\sqrt {-4 c a +b^{2}}}\right ) b^{3}}{2 a^{3} \sqrt {-4 c a +b^{2}}}\) | \(816\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (x\right )}{x^3\,\left (c\,x^2+b\,x+a\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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