3.1111 \(\int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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Rubi [A]  time = 0.0156638, antiderivative size = 17, 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 \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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Rubi in Sympy [A]  time = 2.37912, size = 12, normalized size = 0.71 \[ \frac{\sqrt{x + 1}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1)/sqrt(-x + 1)

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Mathematica [A]  time = 0.0129875, size = 19, normalized size = 1.12 \[ \frac{\sqrt{1-x^2}}{1-x} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[1 - x^2]/(1 - x)

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Maple [A]  time = 0.006, size = 14, 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)^(3/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.499, size = 22, normalized size = 1.29 \[ -\frac{\sqrt{-x^{2} + 1}}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 0.204786, size = 28, normalized size = 1.65 \[ \frac{2 \, x}{x + \sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.61606, size = 31, normalized size = 1.82 \[ \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{i}{\sqrt{1 - \frac{2}{x + 1}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((1/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 = 0.207985, size = 26, normalized size = 1.53 \[ -\frac{\sqrt{x + 1} \sqrt{-x + 1}}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(x + 1)*sqrt(-x + 1)/(x - 1)