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 = 0.0307892, antiderivative size = 27, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0051538, size = 27, normalized size = 1. \[ \frac{x^2}{2}-x-\log (1-x)+3 \log (x)+\log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 0.9 \begin{align*} -x+{\frac{{x}^{2}}{2}}+\ln \left ( 2+x \right ) +3\,\ln \left ( x \right ) -\ln \left ( -1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.944474, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.660478, size = 68, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x^{2} - x + \log \left (x + 2\right ) - \log \left (x - 1\right ) + 3 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.114105, size = 20, normalized size = 0.74 \begin{align*} \frac{x^{2}}{2} - x + 3 \log{\left (x \right )} - \log{\left (x - 1 \right )} + \log{\left (x + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09188, size = 35, normalized size = 1.3 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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