3.705 \(\int \frac{\sqrt{1-x^2}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=11 \[ \frac{2}{3} (x+1)^{3/2} \]

[Out]

(2*(1 + x)^(3/2))/3

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Rubi [A]  time = 0.00532804, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2]/Sqrt[1 - x],x]

[Out]

(2*(1 + x)^(3/2))/3

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Rubi in Sympy [A]  time = 1.40369, size = 8, normalized size = 0.73 \[ \frac{2 \left (x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x + 1)**(3/2)/3

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Mathematica [B]  time = 0.0120282, size = 27, normalized size = 2.45 \[ \frac{2 (x+1) \sqrt{1-x^2}}{3 \sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^2]/Sqrt[1 - x],x]

[Out]

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

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Maple [B]  time = 0.003, size = 22, normalized size = 2. \[{\frac{2+2\,x}{3}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{1-x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(1+x)*(-x^2+1)^(1/2)/(1-x)^(1/2)

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Maxima [A]  time = 0.69087, size = 9, normalized size = 0.82 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x + 1)^(3/2)

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Fricas [A]  time = 0.263995, size = 39, normalized size = 3.55 \[ -\frac{2 \,{\left (x^{3} + x^{2} - x - 1\right )}}{3 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x^3 + x^2 - x - 1)/(sqrt(-x^2 + 1)*sqrt(-x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.264834, size = 18, normalized size = 1.64 \[ \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x + 1)^(3/2) - 4/3*sqrt(2)