Optimal. Leaf size=41 \[ \frac {x^2}{2}+\frac {3}{2} \log \left (x^2+x+1\right )-2 x+\frac {11 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1657, 634, 618, 204, 628} \[ \frac {x^2}{2}+\frac {3}{2} \log \left (x^2+x+1\right )-2 x+\frac {11 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rubi steps
\begin {align*} \int \frac {5+2 x-x^2+x^3}{1+x+x^2} \, dx &=\int \left (-2+x+\frac {7+3 x}{1+x+x^2}\right ) \, dx\\ &=-2 x+\frac {x^2}{2}+\int \frac {7+3 x}{1+x+x^2} \, dx\\ &=-2 x+\frac {x^2}{2}+\frac {3}{2} \int \frac {1+2 x}{1+x+x^2} \, dx+\frac {11}{2} \int \frac {1}{1+x+x^2} \, dx\\ &=-2 x+\frac {x^2}{2}+\frac {3}{2} \log \left (1+x+x^2\right )-11 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=-2 x+\frac {x^2}{2}+\frac {11 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.00 \[ \frac {x^2}{2}+\frac {3}{2} \log \left (x^2+x+1\right )-2 x+\frac {11 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 34, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + \frac {11}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, x + \frac {3}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 34, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + \frac {11}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, x + \frac {3}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 35, normalized size = 0.85 \[ \frac {x^{2}}{2}-2 x +\frac {11 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{3}+\frac {3 \ln \left (x^{2}+x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.18, size = 34, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} + \frac {11}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 2 \, x + \frac {3}{2} \, \log \left (x^{2} + x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.88 \[ \frac {3\,\ln \left (x^2+x+1\right )}{2}-2\,x+\frac {11\,\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{3}+\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 1.12 \[ \frac {x^{2}}{2} - 2 x + \frac {3 \log {\left (x^{2} + x + 1 \right )}}{2} + \frac {11 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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