Optimal. Leaf size=12 \[ x^2+\log \left (x^2+x+1\right )+x \]
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Rubi [A] time = 0.0162013, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1657, 628} \[ x^2+\log \left (x^2+x+1\right )+x \]
Antiderivative was successfully verified.
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Rule 1657
Rule 628
Rubi steps
\begin{align*} \int \frac{2+5 x+3 x^2+2 x^3}{1+x+x^2} \, dx &=\int \left (1+2 x+\frac{1+2 x}{1+x+x^2}\right ) \, dx\\ &=x+x^2+\int \frac{1+2 x}{1+x+x^2} \, dx\\ &=x+x^2+\log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0053581, size = 12, normalized size = 1. \[ x^2+\log \left (x^2+x+1\right )+x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 13, normalized size = 1.1 \begin{align*} x+{x}^{2}+\ln \left ({x}^{2}+x+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04541, size = 16, normalized size = 1.33 \begin{align*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52452, size = 38, normalized size = 3.17 \begin{align*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079665, size = 12, normalized size = 1. \begin{align*} x^{2} + x + \log{\left (x^{2} + x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12055, size = 16, normalized size = 1.33 \begin{align*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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