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

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

[Out]

2/Sqrt[1 + x^(-1)]

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Rubi [A]  time = 0.0034446, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {25, 261} \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^(-1)]/(1 + x)^2,x]

[Out]

2/Sqrt[1 + x^(-1)]

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

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

Mathematica [A]  time = 0.0052393, size = 11, normalized size = 1. \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/Sqrt[1 + x^(-1)]

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Maple [A]  time = 0.003, size = 18, normalized size = 1.6 \begin{align*} 2\,{\frac{x}{1+x}\sqrt{{\frac{1+x}{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{1}{x} + 1}}{{\left (x + 1\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(1/x + 1)/(x + 1)^2, x)

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Fricas [A]  time = 1.62278, size = 39, normalized size = 3.55 \begin{align*} \frac{2 \, x \sqrt{\frac{x + 1}{x}}}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*sqrt((x + 1)/x)/(x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 + \frac{1}{x}}}{\left (x + 1\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(1 + 1/x)/(x + 1)**2, x)

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Giac [B]  time = 1.10869, size = 31, normalized size = 2.82 \begin{align*} \frac{2 \, \mathrm{sgn}\left (x\right )}{x - \sqrt{x^{2} + x} + 1} - 2 \, \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sgn(x)/(x - sqrt(x^2 + x) + 1) - 2*sgn(x)