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

Optimal. Leaf size=20 \[ -\sqrt{\frac{1-x}{x+1}} (x+1) \]

[Out]

-(Sqrt[(1 - x)/(1 + x)]*(1 + x))

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Rubi [A]  time = 0.0885777, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\sqrt{\frac{1-x}{x+1}} (x+1) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(1 - x)/(1 + x)]*(1 + x))

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Rubi in Sympy [A]  time = 3.94377, size = 20, normalized size = 1. \[ - \frac{2 \sqrt{\frac{- x + 1}{x + 1}}}{\frac{- x + 1}{x + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0197909, size = 19, normalized size = 0.95 \[ \frac{x-1}{\sqrt{\frac{1-x}{x+1}}} \]

Antiderivative was successfully verified.

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

[Out]

(-1 + x)/Sqrt[(1 - x)/(1 + x)]

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Maple [A]  time = 0.005, 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-x)/(1+x))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.716299, size = 36, normalized size = 1.8 \[ \frac{2 \, \sqrt{-\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272183, size = 23, normalized size = 1.15 \[ -{\left (x + 1\right )} \sqrt{-\frac{x - 1}{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280352, size = 39, normalized size = 1.95 \[ -\frac{2}{\sqrt{-\frac{x - 1}{x + 1}} + \frac{1}{\sqrt{-\frac{x - 1}{x + 1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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