3.9.61 \(\int \frac {\sqrt {1-x^2}}{\sqrt {1-x}} \, dx\) [861]

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

[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} \begin {gather*} \frac {2}{3} (x+1)^{3/2} \end {gather*}

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] Leaf count is larger than twice the leaf count of optimal. \(24\) vs. \(2(11)=22\).
time = 0.03, size = 24, normalized size = 2.18 \begin {gather*} \frac {2 \left (1-x^2\right )^{3/2}}{3 (1-x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1 - x^2)^(3/2))/(3*(1 - x)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs. \(2(7)=14\).
time = 0.59, size = 27, normalized size = 2.45

method	result	size
gosper	\(\frac {2 \left (1+x \right ) \sqrt {-x^{2}+1}}{3 \sqrt {1-x}}\)	\(22\)
default	\(-\frac {2 \sqrt {-x^{2}+1}\, \sqrt {1-x}\, \left (1+x \right )}{3 \left (-1+x \right )}\)	\(27\)
risch	\(-\frac {2 \sqrt {\frac {\left (1-x \right ) \left (-x^{2}+1\right )}{\left (-1+x \right )^{2}}}\, \left (-1+x \right ) \left (1+x \right )^{\frac {3}{2}}}{3 \sqrt {1-x}\, \sqrt {-x^{2}+1}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/(1-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 7, normalized size = 0.64 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (7) = 14\).
time = 0.33, size = 26, normalized size = 2.36 \begin {gather*} -\frac {2 \, \sqrt {-x^{2} + 1} {\left (x + 1\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {1 - x}}\, dx \end {gather*}

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 = 2.37, size = 13, normalized size = 1.18 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, \sqrt {2} \end {gather*}

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

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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