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

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 = 0.0010932, 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, Rules used = {26, 32} \[ \frac{2}{3} (x+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[1 - x],x]

[Out]

(2*(1 + x)^(3/2))/3

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{1-x}} \, dx &=\int \sqrt{1+x} \, dx\\ &=\frac{2}{3} (1+x)^{3/2}\\ \end{align*}

Mathematica [B]  time = 0.0216299, 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. \begin{align*}{\frac{2+2\,x}{3}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{1-x}}}} \end{align*}

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 = 0.986706, size = 9, normalized size = 0.82 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^(1/2),x, algorithm="maxima")

[Out]

2/3*(x + 1)^(3/2)

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Fricas [B]  time = 1.46537, size = 68, normalized size = 6.18 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}{\left (x + 1\right )} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^(1/2),x, algorithm="fricas")

[Out]

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

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

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(1 - x), x)

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Giac [A]  time = 1.10436, size = 18, normalized size = 1.64 \begin{align*} \frac{2}{3} \,{\left (x + 1\right )}^{\frac{3}{2}} - \frac{4}{3} \, \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1-x)^(1/2),x, algorithm="giac")

[Out]

2/3*(x + 1)^(3/2) - 4/3*sqrt(2)