Optimal. Leaf size=18 \[ x-12 \log (2-x)+17 \log (3-x) \]
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Rubi [A] time = 0.017693, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1657, 632, 31} \[ x-12 \log (2-x)+17 \log (3-x) \]
Antiderivative was successfully verified.
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Rule 1657
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{8+x^2}{6-5 x+x^2} \, dx &=\int \left (1+\frac{2+5 x}{6-5 x+x^2}\right ) \, dx\\ &=x+\int \frac{2+5 x}{6-5 x+x^2} \, dx\\ &=x-12 \int \frac{1}{-2+x} \, dx+17 \int \frac{1}{-3+x} \, dx\\ &=x-12 \log (2-x)+17 \log (3-x)\\ \end{align*}
Mathematica [A] time = 0.0043541, size = 18, normalized size = 1. \[ x-12 \log (2-x)+17 \log (3-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 15, normalized size = 0.8 \begin{align*} x+17\,\ln \left ( -3+x \right ) -12\,\ln \left ( -2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991097, size = 19, normalized size = 1.06 \begin{align*} x - 12 \, \log \left (x - 2\right ) + 17 \, \log \left (x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65817, size = 47, normalized size = 2.61 \begin{align*} x - 12 \, \log \left (x - 2\right ) + 17 \, \log \left (x - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.099877, size = 14, normalized size = 0.78 \begin{align*} x + 17 \log{\left (x - 3 \right )} - 12 \log{\left (x - 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27402, size = 22, normalized size = 1.22 \begin{align*} x - 12 \, \log \left ({\left | x - 2 \right |}\right ) + 17 \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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