3.593 \(\int \frac{x}{(1+x) \sqrt{-1+\frac{2}{1+x}}} \, dx\)

Optimal. Leaf size=18 \[ -(x+1) \sqrt{\frac{2}{x+1}-1} \]

[Out]

-((1 + x)*Sqrt[-1 + 2/(1 + x)])

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Rubi [A]  time = 0.115905, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -(x+1) \sqrt{\frac{2}{x+1}-1} \]

Antiderivative was successfully verified.

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

[Out]

-((1 + x)*Sqrt[-1 + 2/(1 + x)])

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Rubi in Sympy [A]  time = 6.16638, size = 14, normalized size = 0.78 \[ - \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1 + 2/(x + 1))*(x + 1)

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Mathematica [A]  time = 0.0154088, size = 17, normalized size = 0.94 \[ \frac{x-1}{\sqrt{\frac{2}{x+1}-1}} \]

Antiderivative was successfully verified.

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

[Out]

(-1 + x)/Sqrt[-1 + 2/(1 + x)]

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Maple [A]  time = 0.004, size = 17, normalized size = 0.9 \[{(-1+x){\frac{1}{\sqrt{-{\frac{-1+x}{1+x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.774482, size = 22, normalized size = 1.22 \[ \frac{\sqrt{x + 1}{\left (x - 1\right )}}{\sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1)*(x - 1)/sqrt(-x + 1)

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Fricas [A]  time = 0.268601, size = 23, normalized size = 1.28 \[ -{\left (x + 1\right )} \sqrt{-\frac{x - 1}{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x + 1)*sqrt(-(x - 1)/(x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(sqrt(-(x - 1)/(x + 1))*(x + 1)), x)

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GIAC/XCAS [A]  time = 0.281223, size = 39, normalized size = 2.17 \[ -\frac{2}{\sqrt{-\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{-\frac{x - 1}{x + 1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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