Optimal. Leaf size=17 \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]
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Rubi [A] time = 0.003884, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {636} \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{x}{\left (1+x+x^2\right )^{3/2}} \, dx &=-\frac{2 (2+x)}{3 \sqrt{1+x+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0338795, size = 17, normalized size = 1. \[ -\frac{2 (x+2)}{3 \sqrt{x^2+x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 14, normalized size = 0.8 \begin{align*} -{\frac{4+2\,x}{3}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958795, size = 30, normalized size = 1.76 \begin{align*} -\frac{2 \, x}{3 \, \sqrt{x^{2} + x + 1}} - \frac{4}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05286, size = 84, normalized size = 4.94 \begin{align*} -\frac{2 \,{\left (x^{2} + \sqrt{x^{2} + x + 1}{\left (x + 2\right )} + x + 1\right )}}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06772, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \,{\left (x + 2\right )}}{3 \, \sqrt{x^{2} + x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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