3.8.51 \(\int \frac {\sqrt {1+\frac {1}{x}}}{(1+x)^2} \, dx\) [751]

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

[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, 267} \begin {gather*} \frac {2}{\sqrt {\frac {1}{x}+1}} \end {gather*}

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 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*} \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 = 0.02, size = 19, normalized size = 1.73 \begin {gather*} \frac {2 x \sqrt {\frac {1+x}{x}}}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[(1 + x)/x])/(1 + x)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(9)=18\).
time = 0.39, size = 32, normalized size = 2.91

method	result	size
gosper	\(\frac {2 x \sqrt {\frac {1+x}{x}}}{1+x}\)	\(18\)
risch	\(\frac {2 x \sqrt {\frac {1+x}{x}}}{1+x}\)	\(18\)
trager	\(\frac {2 x \sqrt {-\frac {-1-x}{x}}}{1+x}\)	\(21\)
default	\(\frac {2 \sqrt {x^{2}+x}\, x \sqrt {\frac {1+x}{x}}}{\left (1+x \right ) \sqrt {x \left (1+x \right )}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.36, size = 17, normalized size = 1.55 \begin {gather*} \frac {2 \, x \sqrt {\frac {x + 1}{x}}}{x + 1} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 + \frac {1}{x}}}{\left (x + 1\right )^{2}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (9) = 18\).
time = 4.82, size = 23, normalized size = 2.09 \begin {gather*} \frac {2 \, \mathrm {sgn}\left (x\right )}{x - \sqrt {x^{2} + x} + 1} - 2 \, \mathrm {sgn}\left (x\right ) \end {gather*}

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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Mupad [B]
time = 3.13, size = 15, normalized size = 1.36 \begin {gather*} \frac {2\,x\,\sqrt {\frac {1}{x}+1}}{x+1} \end {gather*}

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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