3.2252 \(\int \frac{1}{(1+\sqrt{x})^2 \sqrt{x}} \, dx\)

Optimal. Leaf size=11 \[ -\frac{2}{\sqrt{x}+1} \]

[Out]

-2/(1 + Sqrt[x])

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Rubi [A]  time = 0.0020008, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{2}{\sqrt{x}+1} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])^2*Sqrt[x]),x]

[Out]

-2/(1 + Sqrt[x])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{1}{\left (1+\sqrt{x}\right )^2 \sqrt{x}} \, dx &=-\frac{2}{1+\sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0017484, size = 11, normalized size = 1. \[ -\frac{2}{\sqrt{x}+1} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + Sqrt[x])^2*Sqrt[x]),x]

[Out]

-2/(1 + Sqrt[x])

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Maple [A]  time = 0.003, size = 10, normalized size = 0.9 \begin{align*} -2\, \left ( \sqrt{x}+1 \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x^(1/2)+1)^2,x)

[Out]

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

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Maxima [A]  time = 0.965887, size = 12, normalized size = 1.09 \begin{align*} -\frac{2}{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^2,x, algorithm="maxima")

[Out]

-2/(sqrt(x) + 1)

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Fricas [A]  time = 1.19421, size = 35, normalized size = 3.18 \begin{align*} -\frac{2 \,{\left (\sqrt{x} - 1\right )}}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^2,x, algorithm="fricas")

[Out]

-2*(sqrt(x) - 1)/(x - 1)

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Sympy [A]  time = 0.346045, size = 8, normalized size = 0.73 \begin{align*} - \frac{2}{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(1+x**(1/2))**2,x)

[Out]

-2/(sqrt(x) + 1)

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Giac [A]  time = 1.08511, size = 12, normalized size = 1.09 \begin{align*} -\frac{2}{\sqrt{x} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^2,x, algorithm="giac")

[Out]

-2/(sqrt(x) + 1)