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

Optimal result

Integrand size = 15, antiderivative size = 11 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{1+\sqrt {x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{\sqrt {x}+1} \]

[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*} \text {integral}& = -\frac {2}{1+\sqrt {x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{1+\sqrt {x}} \]

[In]

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

[Out]

-2/(1 + Sqrt[x])

Maple [A] (verified)

Time = 5.90 (sec) , antiderivative size = 10, normalized size of antiderivative = 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	\(-\frac {2 \left (-2+x \right )}{-1+x}-\frac {2 \sqrt {x}}{-1+x}\)	\(22\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2 \, {\left (\sqrt {x} - 1\right )}}{x - 1} \]

[In]

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

[Out]

-2*(sqrt(x) - 1)/(x - 1)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=- \frac {2}{\sqrt {x} + 1} \]

[In]

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

[Out]

-2/(sqrt(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{\sqrt {x} + 1} \]

[In]

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

[Out]

-2/(sqrt(x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{\sqrt {x} + 1} \]

[In]

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

[Out]

-2/(sqrt(x) + 1)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (1+\sqrt {x}\right )^2 \sqrt {x}} \, dx=-\frac {2}{\sqrt {x}+1} \]

[In]

int(1/(x^(1/2)*(x^(1/2) + 1)^2),x)

[Out]

-2/(x^(1/2) + 1)