3.1120 \(\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx\)

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

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Rubi [A]  time = 0.011929, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 - x]*(1 + x)^(3/2)),x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Rubi in Sympy [A]  time = 2.43945, size = 14, normalized size = 0.78 \[ - \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1-x)**(1/2)/(1+x)**(3/2),x)

[Out]

-sqrt(-x + 1)/sqrt(x + 1)

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Mathematica [A]  time = 0.00990987, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 - x]*(1 + x)^(3/2)),x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[ -{1\sqrt{1-x}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1-x)^(1/2)/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.5101, size = 22, normalized size = 1.22 \[ -\frac{\sqrt{-x^{2} + 1}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*sqrt(-x + 1)),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 1)/(x + 1)

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Fricas [A]  time = 0.203705, size = 30, normalized size = 1.67 \[ -\frac{2 \, x}{x - \sqrt{x + 1} \sqrt{-x + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*sqrt(-x + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.89458, size = 31, normalized size = 1.72 \[ \begin{cases} - \sqrt{-1 + \frac{2}{x + 1}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- i \sqrt{1 - \frac{2}{x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1-x)**(1/2)/(1+x)**(3/2),x)

[Out]

Piecewise((-sqrt(-1 + 2/(x + 1)), 2*Abs(1/(x + 1)) > 1), (-I*sqrt(1 - 2/(x + 1))
, True))

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GIAC/XCAS [A]  time = 0.207629, size = 58, normalized size = 3.22 \[ \frac{\sqrt{2} - \sqrt{-x + 1}}{2 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x + 1)^(3/2)*sqrt(-x + 1)),x, algorithm="giac")

[Out]

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