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

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

[Out]

2/Sqrt[1 + x^(-1)]

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Rubi [A]  time = 0.0135279, 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 \[ \frac{2}{\sqrt{\frac{1}{x}+1}} \]

Antiderivative was successfully verified.

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

[Out]

2/Sqrt[1 + x^(-1)]

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Rubi in Sympy [A]  time = 1.34967, size = 8, normalized size = 0.73 \[ \frac{2}{\sqrt{1 + \frac{1}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+1/x)**(1/2)/(1+x)**2,x)

[Out]

2/sqrt(1 + 1/x)

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Mathematica [A]  time = 0.0147826, size = 17, normalized size = 1.55 \[ \frac{2 \sqrt{\frac{1}{x}+1} x}{x+1} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 + x^(-1)]*x)/(1 + x)

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Maple [A]  time = 0.005, size = 18, normalized size = 1.6 \[ 2\,{\frac{x}{1+x}\sqrt{{\frac{1+x}{x}}}} \]

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 [A]  time = 0.88832, size = 15, normalized size = 1.36 \[ \frac{2}{\sqrt{\frac{x + 1}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x + 1)/(x + 1)^2,x, algorithm="maxima")

[Out]

2/sqrt((x + 1)/x)

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Fricas [A]  time = 0.28111, size = 15, normalized size = 1.36 \[ \frac{2}{\sqrt{\frac{x + 1}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x + 1)/(x + 1)^2,x, algorithm="fricas")

[Out]

2/sqrt((x + 1)/x)

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

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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GIAC/XCAS [A]  time = 0.269729, size = 31, normalized size = 2.82 \[ \frac{2 \,{\rm sign}\left (x\right )}{x - \sqrt{x^{2} + x} + 1} - 2 \,{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x + 1)/(x + 1)^2,x, algorithm="giac")

[Out]

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