Optimal. Leaf size=27 \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
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Rubi [A] time = 0.0121766, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {701, 632, 31} \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
Antiderivative was successfully verified.
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Rule 701
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3}{2-3 x+x^2} \, dx &=\int \left (3+x-\frac{6-7 x}{2-3 x+x^2}\right ) \, dx\\ &=3 x+\frac{x^2}{2}-\int \frac{6-7 x}{2-3 x+x^2} \, dx\\ &=3 x+\frac{x^2}{2}+8 \int \frac{1}{-2+x} \, dx-\int \frac{1}{-1+x} \, dx\\ &=3 x+\frac{x^2}{2}-\log (1-x)+8 \log (2-x)\\ \end{align*}
Mathematica [A] time = 0.0035987, size = 27, normalized size = 1. \[ \frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+3\,x-\ln \left ( -1+x \right ) +8\,\ln \left ( -2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974202, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2626, size = 58, normalized size = 2.15 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.095846, size = 19, normalized size = 0.7 \begin{align*} \frac{x^{2}}{2} + 3 x + 8 \log{\left (x - 2 \right )} - \log{\left (x - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09396, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left ({\left | x - 1 \right |}\right ) + 8 \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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