Optimal. Leaf size=25 \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]
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Rubi [A] time = 0.041823, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {1594, 1628} \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rubi steps
\begin{align*} \int \frac{-1+2 x+x^2}{-2 x+3 x^2+2 x^3} \, dx &=\int \frac{-1+2 x+x^2}{x \left (-2+3 x+2 x^2\right )} \, dx\\ &=\int \left (\frac{1}{2 x}-\frac{1}{10 (2+x)}+\frac{1}{5 (-1+2 x)}\right ) \, dx\\ &=\frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0058306, size = 25, normalized size = 1. \[ \frac{1}{10} \log (1-2 x)+\frac{\log (x)}{2}-\frac{1}{10} \log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 20, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 2+x \right ) }{10}}+{\frac{\ln \left ( x \right ) }{2}}+{\frac{\ln \left ( 2\,x-1 \right ) }{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.918427, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{10} \, \log \left (2 \, x - 1\right ) - \frac{1}{10} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13412, size = 68, normalized size = 2.72 \begin{align*} \frac{1}{10} \, \log \left (2 \, x - 1\right ) - \frac{1}{10} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122663, size = 19, normalized size = 0.76 \begin{align*} \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (x - \frac{1}{2} \right )}}{10} - \frac{\log{\left (x + 2 \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04813, size = 30, normalized size = 1.2 \begin{align*} \frac{1}{10} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) - \frac{1}{10} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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