Optimal. Leaf size=13 \[ \frac {x}{\sqrt {x^2+2 x}} \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1980, 636} \[ \frac {x}{\sqrt {x^2+2 x}} \]
Antiderivative was successfully verified.
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Rule 636
Rule 1980
Rubi steps
\begin {align*} \int \frac {x}{(x (2+x))^{3/2}} \, dx &=\int \frac {x}{\left (2 x+x^2\right )^{3/2}} \, dx\\ &=\frac {x}{\sqrt {2 x+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 11, normalized size = 0.85 \[ \frac {x}{\sqrt {x (x+2)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 18, normalized size = 1.38 \[ \frac {x + \sqrt {x^{2} + 2 \, x} + 2}{x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 16, normalized size = 1.23 \[ \frac {2}{x - \sqrt {{\left (x + 2\right )} x} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 1.15 \[ \frac {\left (x +2\right ) x^{2}}{\left (\left (x +2\right ) x \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 11, normalized size = 0.85 \[ \frac {x}{\sqrt {x^{2} + 2 \, x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.52, size = 13, normalized size = 1.00 \[ \frac {\sqrt {x\,\left (x+2\right )}}{x+2} \]
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 \[ \int \frac {x}{\left (x \left (x + 2\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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