3.23.52 \(\int \frac {1}{(1+\sqrt {x})^2 \sqrt {x}} \, dx\) [2252]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 = {267} \begin {gather*} -\frac {2}{\sqrt {x}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])^2*Sqrt[x]),x]

[Out]

-2/(1 + Sqrt[x])

Rule 267

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*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} -\frac {2}{1+\sqrt {x}} \end {gather*}

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 = 0.20, size = 10, normalized size = 0.91

method	result	size
derivativedivides	\(-\frac {2}{\sqrt {x}+1}\)	\(10\)
default	\(-\frac {2}{\sqrt {x}+1}\)	\(10\)
meijerg	\(\frac {2 \sqrt {x}}{\sqrt {x}+1}\)	\(13\)
trager	\(-\frac {2 \left (x -2\right )}{x -1}-\frac {2 \sqrt {x}}{x -1}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x^(1/2)+1)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 9, normalized size = 0.82 \begin {gather*} -\frac {2}{\sqrt {x} + 1} \end {gather*}

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 = 0.37, size = 12, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left (\sqrt {x} - 1\right )}}{x - 1} \end {gather*}

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 = 0.14, size = 8, normalized size = 0.73 \begin {gather*} - \frac {2}{\sqrt {x} + 1} \end {gather*}

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 = 1.67, size = 9, normalized size = 0.82 \begin {gather*} -\frac {2}{\sqrt {x} + 1} \end {gather*}

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)

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Mupad [B]
time = 0.03, size = 9, normalized size = 0.82 \begin {gather*} -\frac {2}{\sqrt {x}+1} \end {gather*}

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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