Optimal. Leaf size=25 \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]
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Rubi [A] time = 0.0365624, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1594, 1628} \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rubi steps
\begin{align*} \int \frac{-1+x+x^2}{-6 x+x^2+x^3} \, dx &=\int \frac{-1+x+x^2}{x \left (-6+x+x^2\right )} \, dx\\ &=\int \left (\frac{1}{2 (-2+x)}+\frac{1}{6 x}+\frac{1}{3 (3+x)}\right ) \, dx\\ &=\frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (3+x)\\ \end{align*}
Mathematica [A] time = 0.005858, size = 25, normalized size = 1. \[ \frac{1}{2} \log (2-x)+\frac{\log (x)}{6}+\frac{1}{3} \log (x+3) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 18, normalized size = 0.7 \begin{align*}{\frac{\ln \left ( x \right ) }{6}}+{\frac{\ln \left ( 3+x \right ) }{3}}+{\frac{\ln \left ( -2+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931829, size = 23, normalized size = 0.92 \begin{align*} \frac{1}{3} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x - 2\right ) + \frac{1}{6} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88495, size = 62, normalized size = 2.48 \begin{align*} \frac{1}{3} \, \log \left (x + 3\right ) + \frac{1}{2} \, \log \left (x - 2\right ) + \frac{1}{6} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.117081, size = 17, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{6} + \frac{\log{\left (x - 2 \right )}}{2} + \frac{\log{\left (x + 3 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06005, size = 27, normalized size = 1.08 \begin{align*} \frac{1}{3} \, \log \left ({\left | x + 3 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - 2 \right |}\right ) + \frac{1}{6} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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