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

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

[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 = {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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(1 + Sqrt[x])

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fricas [A]  time = 0.96, size = 12, normalized size = 1.09 \[ -\frac {2 \, {\left (\sqrt {x} - 1\right )}}{x - 1} \]

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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giac [A]  time = 0.16, size = 9, normalized size = 0.82 \[ -\frac {2}{\sqrt {x} + 1} \]

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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maple [A]  time = 0.00, size = 10, normalized size = 0.91 \[ -\frac {2}{\sqrt {x}+1} \]

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 = 0.81, size = 9, normalized size = 0.82 \[ -\frac {2}{\sqrt {x} + 1} \]

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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mupad [B]  time = 0.03, size = 9, normalized size = 0.82 \[ -\frac {2}{\sqrt {x}+1} \]

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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sympy [A]  time = 0.36, size = 8, normalized size = 0.73 \[ - \frac {2}{\sqrt {x} + 1} \]

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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