Optimal. Leaf size=17 \[ \frac {\sqrt {2 x+x^2}}{2+x} \]
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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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {664}
\begin {gather*} \frac {\sqrt {x^2+2 x}}{x+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rubi steps
\begin {align*} \int \frac {1}{(2+x) \sqrt {2 x+x^2}} \, dx &=\frac {\sqrt {2 x+x^2}}{2+x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 11, normalized size = 0.65 \begin {gather*} \frac {x}{\sqrt {x (2+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 19, normalized size = 1.12
method | result | size |
risch | \(\frac {x}{\sqrt {x \left (2+x \right )}}\) | \(10\) |
gosper | \(\frac {x}{\sqrt {x^{2}+2 x}}\) | \(12\) |
trager | \(\frac {\sqrt {x^{2}+2 x}}{2+x}\) | \(16\) |
meijerg | \(\frac {\sqrt {2}\, \sqrt {x}}{2 \sqrt {\frac {x}{2}+1}}\) | \(16\) |
default | \(\frac {\sqrt {\left (2+x \right )^{2}-4-2 x}}{2+x}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 15, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x^{2} + 2 \, x}}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 18, normalized size = 1.06 \begin {gather*} \frac {x + \sqrt {x^{2} + 2 \, x} + 2}{x + 2} \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}{\sqrt {x \left (x + 2\right )} \left (x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 18, normalized size = 1.06 \begin {gather*} \frac {2}{x - \sqrt {x^{2} + 2 \, x} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 15, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x^2+2\,x}}{x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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