Optimal. Leaf size=13 \[ \frac {x}{\sqrt {x^2+2 x}} \]
________________________________________________________________________________________
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} \begin {gather*} \frac {x}{\sqrt {x^2+2 x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} \frac {x}{\sqrt {x (x+2)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.15, size = 17, normalized size = 1.31 \begin {gather*} \frac {\sqrt {x^2+2 x}}{x+2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 18, normalized size = 1.38 \begin {gather*} \frac {x + \sqrt {x^{2} + 2 \, x} + 2}{x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.33, size = 16, normalized size = 1.23 \begin {gather*} \frac {2}{x - \sqrt {{\left (x + 2\right )} x} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 15, normalized size = 1.15 \begin {gather*} \frac {\left (x +2\right ) x^{2}}{\left (\left (x +2\right ) x \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 11, normalized size = 0.85 \begin {gather*} \frac {x}{\sqrt {x^{2} + 2 \, x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.52, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x\,\left (x+2\right )}}{x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (x \left (x + 2\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________