Optimal. Leaf size=26 \[ \frac{x^2}{2}+2 x+\frac{1}{1-x}+3 \log (1-x) \]
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Rubi [A] time = 0.0101849, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {27, 43} \[ \frac{x^2}{2}+2 x+\frac{1}{1-x}+3 \log (1-x) \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{1-2 x+x^2} \, dx &=\int \frac{x^3}{(-1+x)^2} \, dx\\ &=\int \left (2+\frac{1}{(-1+x)^2}+\frac{3}{-1+x}+x\right ) \, dx\\ &=\frac{1}{1-x}+2 x+\frac{x^2}{2}+3 \log (1-x)\\ \end{align*}
Mathematica [A] time = 0.0074819, size = 25, normalized size = 0.96 \[ \frac{1}{2} \left (x^2+4 x-\frac{2}{x-1}+6 \log (x-1)-5\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 23, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+2\,x+3\,\ln \left ( -1+x \right ) - \left ( -1+x \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02233, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x - \frac{1}{x - 1} + 3 \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1213, size = 81, normalized size = 3.12 \begin{align*} \frac{x^{3} + 3 \, x^{2} + 6 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 2}{2 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079259, size = 19, normalized size = 0.73 \begin{align*} \frac{x^{2}}{2} + 2 x + 3 \log{\left (x - 1 \right )} - \frac{1}{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09125, size = 31, normalized size = 1.19 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x - \frac{1}{x - 1} + 3 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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