Optimal. Leaf size=23 \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]
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Rubi [A] time = 0.0367433, antiderivative size = 23, 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} \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rubi steps
\begin{align*} \int \frac{-1-3 x+x^2}{-2 x+x^2+x^3} \, dx &=\int \frac{-1-3 x+x^2}{x \left (-2+x+x^2\right )} \, dx\\ &=\int \left (\frac{1}{1-x}+\frac{1}{2 x}+\frac{3}{2 (2+x)}\right ) \, dx\\ &=-\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0058489, size = 23, normalized size = 1. \[ -\log (1-x)+\frac{\log (x)}{2}+\frac{3}{2} \log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 18, normalized size = 0.8 \begin{align*} -\ln \left ( x-1 \right ) +{\frac{\ln \left ( x \right ) }{2}}+{\frac{3\,\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 = 0.970412, size = 23, normalized size = 1. \begin{align*} \frac{3}{2} \, \log \left (x + 2\right ) - \log \left (x - 1\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 = 1.34757, size = 57, normalized size = 2.48 \begin{align*} \frac{3}{2} \, \log \left (x + 2\right ) - \log \left (x - 1\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 = 0.123064, size = 17, normalized size = 0.74 \begin{align*} \frac{\log{\left (x \right )}}{2} - \log{\left (x - 1 \right )} + \frac{3 \log{\left (x + 2 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1124, size = 27, normalized size = 1.17 \begin{align*} \frac{3}{2} \, \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x - 1 \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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