Optimal. Leaf size=12 \[ x^2+\log \left (x^2+x+1\right )+x \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} x^2+\log \left (x^2+x+1\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 1657
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*}
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Mathematica [A] time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} x^2+\log \left (x^2+x+1\right )+x \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+5 x+3 x^2+2 x^3}{1+x+x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.06, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.08 \begin {gather*} x^{2}+x +\ln \left (x^{2}+x +1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 1.00 \begin {gather*} x+\ln \left (x^2+x+1\right )+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 12, normalized size = 1.00 \begin {gather*} x^{2} + x + \log {\left (x^{2} + x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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