Optimal. Leaf size=19 \[ \frac {2 (1+2 x)}{3 \sqrt {1+x+x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {627}
\begin {gather*} \frac {2 (2 x+1)}{3 \sqrt {x^2+x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 627
Rubi steps
\begin {align*} \int \frac {1}{\left (1+x+x^2\right )^{3/2}} \, dx &=\frac {2 (1+2 x)}{3 \sqrt {1+x+x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 19, normalized size = 1.00 \begin {gather*} \frac {2 (1+2 x)}{3 \sqrt {1+x+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 16, normalized size = 0.84
method | result | size |
gosper | \(\frac {\frac {2}{3}+\frac {4 x}{3}}{\sqrt {x^{2}+x +1}}\) | \(16\) |
default | \(\frac {\frac {2}{3}+\frac {4 x}{3}}{\sqrt {x^{2}+x +1}}\) | \(16\) |
trager | \(\frac {\frac {2}{3}+\frac {4 x}{3}}{\sqrt {x^{2}+x +1}}\) | \(16\) |
risch | \(\frac {\frac {2}{3}+\frac {4 x}{3}}{\sqrt {x^{2}+x +1}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.88, size = 22, normalized size = 1.16 \begin {gather*} \frac {4 \, x}{3 \, \sqrt {x^{2} + x + 1}} + \frac {2}{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. 34 vs.
\(2 (15) = 30\).
time = 0.40, size = 34, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (2 \, x^{2} + \sqrt {x^{2} + x + 1} {\left (2 \, x + 1\right )} + 2 \, x + 2\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 {1}{\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 = 1.14, size = 15, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (2 \, x + 1\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 = 0.03, size = 13, normalized size = 0.68 \begin {gather*} \frac {4\,\left (x+\frac {1}{2}\right )}{3\,\sqrt {x^2+x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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