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/(1+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 = 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*}

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

Antiderivative was successfully verified.

[In]

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

[Out]

2/Sqrt[1 + x^(-1)]

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fricas [A]  time = 0.39, size = 17, normalized size = 1.55 \[ \frac {2 \, x \sqrt {\frac {x + 1}{x}}}{x + 1} \]

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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giac [B]  time = 0.29, size = 23, normalized size = 2.09 \[ \frac {2 \, \mathrm {sgn}\relax (x)}{x - \sqrt {x^{2} + x} + 1} - 2 \, \mathrm {sgn}\relax (x) \]

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)

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maple [A]  time = 0.01, size = 18, normalized size = 1.64 \[ \frac {2 \sqrt {\frac {x +1}{x}}\, x}{x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{x} + 1}}{{\left (x + 1\right )}^{2}}\,{d x} \]

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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mupad [B]  time = 3.13, size = 15, normalized size = 1.36 \[ \frac {2\,x\,\sqrt {\frac {1}{x}+1}}{x+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/x + 1)^(1/2)/(x + 1)^2,x)

[Out]

(2*x*(1/x + 1)^(1/2))/(x + 1)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 + \frac {1}{x}}}{\left (x + 1\right )^{2}}\, dx \]

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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