Optimal. Leaf size=27 \[ \frac {x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]
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Rubi [A] time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1594, 1628} \[ \frac {x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rubi steps
\begin {align*} \int \frac {-6+2 x+x^4}{-2 x+x^2+x^3} \, dx &=\int \frac {-6+2 x+x^4}{x \left (-2+x+x^2\right )} \, dx\\ &=\int \left (-1+\frac {1}{1-x}+\frac {3}{x}+x+\frac {1}{2+x}\right ) \, dx\\ &=-x+\frac {x^2}{2}-\log (1-x)+3 \log (x)+\log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 23, normalized size = 0.85 \[ \frac {1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 26, normalized size = 0.96 \[ \frac {1}{2} \, x^{2} - x + \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x - 1 \right |}\right ) + 3 \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.89 \[ \frac {x^{2}}{2}-x +3 \ln \relax (x )-\ln \left (x -1\right )+\ln \left (x +2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 23, normalized size = 0.85 \[ \frac {1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 30, normalized size = 1.11 \[ 3\,\ln \relax (x)-x+\frac {x^2}{2}+\mathrm {atan}\left (\frac {192{}\mathrm {i}}{7\,\left (28\,x-40\right )}+\frac {9}{7}{}\mathrm {i}\right )\,2{}\mathrm {i} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.74 \[ \frac {x^{2}}{2} - x + 3 \log {\relax (x )} - \log {\left (x - 1 \right )} + \log {\left (x + 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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