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

Optimal. Leaf size=32 \[ \sqrt{-\frac{x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0308614, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \sqrt{-\frac{x}{x+1}} (x+1)-\tan ^{-1}\left (\sqrt{-\frac{x}{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-(x/(1 + x))],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.0556, size = 27, normalized size = 0.84 \[ \frac{\sqrt{- \frac{x}{x + 1}}}{- \frac{x}{x + 1} + 1} - \operatorname{atan}{\left (\sqrt{- \frac{x}{x + 1}} \right )} \]

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) - atan(sqrt(-x/(x + 1)))

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Mathematica [A]  time = 0.0283316, size = 43, normalized size = 1.34 \[ \frac{\sqrt{-\frac{x}{x+1}} \left (\sqrt{x} (x+1)-\sqrt{x+1} \sinh ^{-1}\left (\sqrt{x}\right )\right )}{\sqrt{x}} \]

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 [A]  time = 0.005, size = 46, normalized size = 1.4 \[{\frac{1+x}{2}\sqrt{-{\frac{x}{1+x}}} \left ( 2\,\sqrt{{x}^{2}+x}-\ln \left ({\frac{1}{2}}+x+\sqrt{{x}^{2}+x} \right ) \right ){\frac{1}{\sqrt{x \left ( 1+x \right ) }}}} \]

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 = 0.798798, size = 50, normalized size = 1.56 \[ -\frac{\sqrt{-\frac{x}{x + 1}}}{\frac{x}{x + 1} - 1} - \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]

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) - arctan(sqrt(-x/(x + 1)))

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Fricas [A]  time = 0.274071, size = 38, normalized size = 1.19 \[{\left (x + 1\right )} \sqrt{-\frac{x}{x + 1}} - \arctan \left (\sqrt{-\frac{x}{x + 1}}\right ) \]

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)) - arctan(sqrt(-x/(x + 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 = 0.274055, size = 49, normalized size = 1.53 \[ \frac{1}{4} \, \pi{\rm sign}\left (x + 1\right ) + \frac{1}{2} \, \arcsin \left (2 \, x + 1\right ){\rm sign}\left (x + 1\right ) + \sqrt{-x^{2} - x}{\rm sign}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*pi*sign(x + 1) + 1/2*arcsin(2*x + 1)*sign(x + 1) + sqrt(-x^2 - x)*sign(x + 1
)