Optimal. Leaf size=17 \[ -\frac {2 (2+x)}{3 \sqrt {1+x+x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {650}
\begin {gather*} -\frac {2 (x+2)}{3 \sqrt {x^2+x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
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*}
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Mathematica [A]
time = 0.08, size = 17, normalized size = 1.00 \begin {gather*} -\frac {2 (2+x)}{3 \sqrt {1+x+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 27, normalized size = 1.59
method | result | size |
gosper | \(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\) | \(14\) |
trager | \(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\) | \(14\) |
risch | \(-\frac {2 \left (2+x \right )}{3 \sqrt {x^{2}+x +1}}\) | \(14\) |
default | \(-\frac {1}{\sqrt {x^{2}+x +1}}-\frac {1+2 x}{3 \sqrt {x^{2}+x +1}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.37, size = 22, normalized size = 1.29 \begin {gather*} -\frac {2 \, x}{3 \, \sqrt {x^{2} + x + 1}} - \frac {4}{3 \, \sqrt {x^{2} + x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (13) = 26\).
time = 0.46, size = 28, normalized size = 1.65 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (x^{2} + x + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.76, size = 13, normalized size = 0.76 \begin {gather*} -\frac {2 \, {\left (x + 2\right )}}{3 \, \sqrt {x^{2} + x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} -\frac {2\,x+4}{3\,\sqrt {x^2+x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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