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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.02, 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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fricas [B]  time = 0.41, size = 26, normalized size = 2.36 \[ -\frac {2 \, \sqrt {-x^{2} + 1} {\left (x + 1\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}} \]

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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giac [A]  time = 0.33, size = 13, normalized size = 1.18 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, \sqrt {2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x + 1)^(3/2) - 4/3*sqrt(2)

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maple [B]  time = 0.00, size = 22, normalized size = 2.00 \[ \frac {2 \left (x +1\right ) \sqrt {-x^{2}+1}}{3 \sqrt {-x +1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.92, size = 7, normalized size = 0.64 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \]

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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mupad [B]  time = 3.49, size = 22, normalized size = 2.00 \[ \frac {\left (\frac {2\,x}{3}+\frac {2}{3}\right )\,\sqrt {1-x^2}}{\sqrt {1-x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(1 - x)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {1 - x}}\, dx \]

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(1 - x), x)

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