Optimal. Leaf size=33 \[ -\frac {(x+1) (2 x+1)}{6 \left (x^2+x+1\right )^2}-\frac {1}{6 \left (x^2+x+1\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {818, 629} \begin {gather*} -\frac {(x+1) (2 x+1)}{6 \left (x^2+x+1\right )^2}-\frac {1}{6 \left (x^2+x+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rule 818
Rubi steps
\begin {align*} \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx &=-\frac {(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6} \int \frac {-1-2 x}{\left (1+x+x^2\right )^2} \, dx\\ &=-\frac {(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6 \left (1+x+x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.67 \begin {gather*} -\frac {3 x^2+4 x+2}{6 \left (x^2+x+1\right )^2} \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 {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 32, normalized size = 0.97 \begin {gather*} -\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 20, normalized size = 0.61 \begin {gather*} -\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{2} + x + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 20, normalized size = 0.61 \begin {gather*} \frac {-\frac {1}{2} x^{2}-\frac {2}{3} x -\frac {1}{3}}{\left (x^{2}+x +1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 32, normalized size = 0.97 \begin {gather*} -\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 0.61 \begin {gather*} -\frac {3\,x^2+4\,x+2}{6\,{\left (x^2+x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 31, normalized size = 0.94 \begin {gather*} \frac {- 3 x^{2} - 4 x - 2}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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