Optimal. Leaf size=12 \[ x+x^2+\log \left (1+x+x^2\right ) \]
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Rubi [A]
time = 0.01, 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 = {1671, 642}
\begin {gather*} x^2+\log \left (x^2+x+1\right )+x \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1671
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+x^2+\log \left (1+x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 13, normalized size = 1.08
method | result | size |
default | \(x +x^{2}+\ln \left (x^{2}+x +1\right )\) | \(13\) |
norman | \(x +x^{2}+\ln \left (x^{2}+x +1\right )\) | \(13\) |
risch | \(x +x^{2}+\ln \left (x^{2}+x +1\right )\) | \(13\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, 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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Fricas [A]
time = 0.38, 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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Sympy [A]
time = 0.02, 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 = 3.68, 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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