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

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

[Out]

x/(x^2+2*x)^(1/2)

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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.

[In]

Int[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[2*x + x^2]

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]

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]

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.

[In]

Integrate[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[x*(2 + x)]

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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.

[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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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.

[In]

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

[Out]

2/(x - sqrt((x + 2)*x) + 2)

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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.

[In]

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

[Out]

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

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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.

[In]

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

[Out]

x/sqrt(x^2 + 2*x)

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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.

[In]

int(x/(x*(x + 2))^(3/2),x)

[Out]

(x*(x + 2))^(1/2)/(x + 2)

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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.

[In]

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

[Out]

Integral(x/(x*(x + 2))**(3/2), x)

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