3.982 \(\int \frac{x}{(x (2+x))^{3/2}} \, dx\)

Optimal. Leaf size=13 \[ \frac{x}{\sqrt{x^2+2 x}} \]

[Out]

x/Sqrt[2*x + x^2]

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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.

[In]

Int[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[2*x + x^2]

Rule 1980

Int[(u_)^(p_.)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[(c*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{c, m, p}, x] &&
GeneralizedBinomialQ[u, x] &&  !GeneralizedBinomialMatchQ[u, x]

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

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.

[In]

Integrate[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[x*(2 + x)]

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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.

[In]

int(x/(x*(2+x))^(3/2),x)

[Out]

x^2*(2+x)/(x*(2+x))^(3/2)

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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.

[In]

integrate(x/(x*(2+x))^(3/2),x, algorithm="maxima")

[Out]

x/sqrt(x^2 + 2*x)

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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.

[In]

integrate(x/(x*(2+x))^(3/2),x, algorithm="fricas")

[Out]

(x + sqrt(x^2 + 2*x) + 2)/(x + 2)

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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.

[In]

integrate(x/(x*(2+x))**(3/2),x)

[Out]

Integral(x/(x*(x + 2))**(3/2), x)

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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.

[In]

integrate(x/(x*(2+x))^(3/2),x, algorithm="giac")

[Out]

2/(x - sqrt((x + 2)*x) + 2)