3.576 \(\int \sqrt{\frac{x}{1+x}} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]]

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Rubi [A]  time = 0.0167364, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \sqrt{x} \sqrt{x+1}-\sinh ^{-1}\left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x/(1 + x)],x]

[Out]

Sqrt[x]*Sqrt[1 + x] - ArcSinh[Sqrt[x]]

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Rubi in Sympy [A]  time = 1.97574, size = 24, normalized size = 1.09 \[ \frac{\sqrt{\frac{x}{x + 1}}}{- \frac{x}{x + 1} + 1} - \operatorname{atanh}{\left (\sqrt{\frac{x}{x + 1}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x/(x + 1))/(-x/(x + 1) + 1) - atanh(sqrt(x/(x + 1)))

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Mathematica [A]  time = 0.00576897, size = 42, normalized size = 1.91 \[ \frac{\sqrt{\frac{x}{x+1}} \left (\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x/(1 + x)],x]

[Out]

(Sqrt[x/(1 + x)]*(Sqrt[x]*(1 + x) - Sqrt[1 + x]*ArcSinh[Sqrt[x]]))/Sqrt[x]

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Maple [B]  time = 0.005, size = 45, normalized size = 2.1 \[{\frac{1+x}{2}\sqrt{{\frac{x}{1+x}}} \left ( 2\,\sqrt{{x}^{2}+x}-\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x/(1+x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.713359, size = 69, normalized size = 3.14 \[ -\frac{\sqrt{\frac{x}{x + 1}}}{\frac{x}{x + 1} - 1} - \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x/(x + 1))/(x/(x + 1) - 1) - 1/2*log(sqrt(x/(x + 1)) + 1) + 1/2*log(sqrt(x
/(x + 1)) - 1)

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Fricas [A]  time = 0.272926, size = 57, normalized size = 2.59 \[{\left (x + 1\right )} \sqrt{\frac{x}{x + 1}} - \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{x}{x + 1}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x + 1)*sqrt(x/(x + 1)) - 1/2*log(sqrt(x/(x + 1)) + 1) + 1/2*log(sqrt(x/(x + 1))
- 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x}{x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x/(1+x))**(1/2),x)

[Out]

Integral(sqrt(x/(x + 1)), x)

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GIAC/XCAS [A]  time = 0.272301, size = 47, normalized size = 2.14 \[ \frac{1}{2} \,{\rm ln}\left ({\left | -2 \, x + 2 \, \sqrt{x^{2} + x} - 1 \right |}\right ){\rm sign}\left (x + 1\right ) + \sqrt{x^{2} + x}{\rm sign}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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