3.2253 \(\int \frac{1}{(1+\sqrt{x})^3 \sqrt{x}} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

[Out]

-(1 + Sqrt[x])^(-2)

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Rubi [A]  time = 0.0022699, 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} \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + Sqrt[x])^3*Sqrt[x]),x]

[Out]

-(1 + Sqrt[x])^(-2)

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 )^3 \sqrt{x}} \, dx &=-\frac{1}{\left (1+\sqrt{x}\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0018168, size = 11, normalized size = 1. \[ -\frac{1}{\left (\sqrt{x}+1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + Sqrt[x])^3*Sqrt[x]),x]

[Out]

-(1 + Sqrt[x])^(-2)

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Maple [A]  time = 0.001, size = 10, normalized size = 0.9 \begin{align*} - \left ( \sqrt{x}+1 \right ) ^{-2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x^(1/2)+1)^3,x)

[Out]

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

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Maxima [A]  time = 0.976081, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{{\left (\sqrt{x} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="maxima")

[Out]

-1/(sqrt(x) + 1)^2

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Fricas [B]  time = 1.30151, size = 51, normalized size = 4.64 \begin{align*} -\frac{x - 2 \, \sqrt{x} + 1}{x^{2} - 2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="fricas")

[Out]

-(x - 2*sqrt(x) + 1)/(x^2 - 2*x + 1)

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Sympy [A]  time = 0.438852, size = 12, normalized size = 1.09 \begin{align*} - \frac{1}{2 \sqrt{x} + x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(1+x**(1/2))**3,x)

[Out]

-1/(2*sqrt(x) + x + 1)

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Giac [A]  time = 1.10769, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{{\left (\sqrt{x} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(1+x^(1/2))^3,x, algorithm="giac")

[Out]

-1/(sqrt(x) + 1)^2