Optimal. Leaf size=13 \[ \frac{x}{\sqrt{x^2+2 x}} \]
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Rubi [A] time = 0.0126261, antiderivative size = 13, normalized size of antiderivative = 1., 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 1980
Rule 636
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*}
Mathematica [A] time = 0.0033524, size = 11, normalized size = 0.85 \[ \frac{x}{\sqrt{x (x+2)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 15, normalized size = 1.2 \begin{align*}{{x}^{2} \left ( 2+x \right ) \left ( x \left ( 2+x \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08519, size = 15, normalized size = 1.15 \begin{align*} \frac{x}{\sqrt{x^{2} + 2 \, x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4272, size = 47, normalized size = 3.62 \begin{align*} \frac{x + \sqrt{x^{2} + 2 \, x} + 2}{x + 2} \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 \left (x + 2\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15854, size = 22, normalized size = 1.69 \begin{align*} \frac{2}{x - \sqrt{{\left (x + 2\right )} x} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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