Optimal. Leaf size=14 \[ x+4 \log (4-x)+\log (x+2) \]
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Rubi [A] time = 0.0165158, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1657, 632, 31} \[ x+4 \log (4-x)+\log (x+2) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{-4+3 x+x^2}{-8-2 x+x^2} \, dx &=\int \left (1+\frac{4+5 x}{-8-2 x+x^2}\right ) \, dx\\ &=x+\int \frac{4+5 x}{-8-2 x+x^2} \, dx\\ &=x+4 \int \frac{1}{-4+x} \, dx+\int \frac{1}{2+x} \, dx\\ &=x+4 \log (4-x)+\log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0051096, size = 14, normalized size = 1. \[ x+4 \log (4-x)+\log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 13, normalized size = 0.9 \begin{align*} x+\ln \left ( 2+x \right ) +4\,\ln \left ( x-4 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990675, size = 16, normalized size = 1.14 \begin{align*} x + \log \left (x + 2\right ) + 4 \, \log \left (x - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61126, size = 42, normalized size = 3. \begin{align*} x + \log \left (x + 2\right ) + 4 \, \log \left (x - 4\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.097405, size = 12, normalized size = 0.86 \begin{align*} x + 4 \log{\left (x - 4 \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 = 1.24607, size = 19, normalized size = 1.36 \begin{align*} x + \log \left ({\left | x + 2 \right |}\right ) + 4 \, \log \left ({\left | x - 4 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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