Optimal. Leaf size=22 \[ \frac{x^2}{2}-2 x+\frac{1}{x+1}+3 \log (x+1) \]
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Rubi [A] time = 0.0100248, antiderivative size = 22, 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} \[ \frac{x^2}{2}-2 x+\frac{1}{x+1}+3 \log (x+1) \]
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+x-\frac{1}{(1+x)^2}+\frac{3}{1+x}\right ) \, dx\\ &=-2 x+\frac{x^2}{2}+\frac{1}{1+x}+3 \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.008752, size = 26, normalized size = 1.18 \[ \frac{1}{2} (x+1)^2-3 (x+1)+\frac{1}{x+1}+3 \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 21, normalized size = 1. \begin{align*} -2\,x+{\frac{{x}^{2}}{2}}+ \left ( 1+x \right ) ^{-1}+3\,\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957131, size = 27, normalized size = 1.23 \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 = 2.24834, size = 81, normalized size = 3.68 \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 = 0.079324, size = 19, normalized size = 0.86 \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 = 1.12364, size = 28, normalized size = 1.27 \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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