3.10.82 \(\int \frac {x}{(x (2+x))^{3/2}} \, dx\) [982]

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

[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 = {1976, 650} \begin {gather*} \frac {x}{\sqrt {x^2+2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[2*x + x^2]

Rule 650

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 1976

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Int[u*(a*c*e + (b*c
+ a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, n, p}, 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 = 0.01, size = 11, normalized size = 0.85 \begin {gather*} \frac {x}{\sqrt {x (2+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(x*(2 + x))^(3/2),x]

[Out]

x/Sqrt[x*(2 + x)]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(11)=22\).
time = 0.22, size = 29, normalized size = 2.23

method	result	size
risch	\(\frac {x}{\sqrt {x \left (x +2\right )}}\)	\(10\)
gosper	\(\frac {x^{2} \left (x +2\right )}{\left (x \left (x +2\right )\right )^{\frac {3}{2}}}\)	\(15\)
trager	\(\frac {\sqrt {x^{2}+2 x}}{x +2}\)	\(16\)
meijerg	\(\frac {\sqrt {2}\, \sqrt {x}}{2 \sqrt {1+\frac {x}{2}}}\)	\(16\)
default	\(-\frac {1}{\sqrt {x^{2}+2 x}}+\frac {2 x +2}{2 \sqrt {x^{2}+2 x}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x*(x+2))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, 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]

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

[Out]

x/sqrt(x^2 + 2*x)

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Fricas [A]
time = 0.36, 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]

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 = 0.00, 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]

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

[Out]

Integral(x/(x*(x + 2))**(3/2), x)

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Giac [A]
time = 4.30, size = 18, normalized size = 1.38 \begin {gather*} \frac {2}{x - \sqrt {x^{2} + 2 \, x} + 2} \end {gather*}

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 + 2*x) + 2)

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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]

int(x/(x*(x + 2))^(3/2),x)

[Out]

(x*(x + 2))^(1/2)/(x + 2)

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