3.647 \(\int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=27 \[ -\sqrt{1-x} \sqrt{x}-\frac{1}{2} \sin ^{-1}(1-2 x) \]

[Out]

-(Sqrt[1 - x]*Sqrt[x]) - ArcSin[1 - 2*x]/2

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Rubi [A]  time = 0.005159, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {50, 53, 619, 216} \[ -\sqrt{1-x} \sqrt{x}-\frac{1}{2} \sin ^{-1}(1-2 x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/Sqrt[1 - x],x]

[Out]

-(Sqrt[1 - x]*Sqrt[x]) - ArcSin[1 - 2*x]/2

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{\sqrt{1-x}} \, dx &=-\sqrt{1-x} \sqrt{x}+\frac{1}{2} \int \frac{1}{\sqrt{1-x} \sqrt{x}} \, dx\\ &=-\sqrt{1-x} \sqrt{x}+\frac{1}{2} \int \frac{1}{\sqrt{x-x^2}} \, dx\\ &=-\sqrt{1-x} \sqrt{x}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,1-2 x\right )\\ &=-\sqrt{1-x} \sqrt{x}-\frac{1}{2} \sin ^{-1}(1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0080975, size = 25, normalized size = 0.93 \[ -\sqrt{-(x-1) x}-\sin ^{-1}\left (\sqrt{1-x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/Sqrt[1 - x],x]

[Out]

-Sqrt[-((-1 + x)*x)] - ArcSin[Sqrt[1 - x]]

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Maple [A]  time = 0.005, size = 41, normalized size = 1.5 \begin{align*} -\sqrt{1-x}\sqrt{x}+{\frac{\arcsin \left ( 2\,x-1 \right ) }{2}\sqrt{x \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.59106, size = 50, normalized size = 1.85 \begin{align*} \frac{\sqrt{-x + 1}}{\sqrt{x}{\left (\frac{x - 1}{x} - 1\right )}} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45219, size = 73, normalized size = 2.7 \begin{align*} -\sqrt{x} \sqrt{-x + 1} - \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.75505, size = 54, normalized size = 2. \begin{align*} \begin{cases} - i \sqrt{x} \sqrt{x - 1} - i \operatorname{acosh}{\left (\sqrt{x} \right )} & \text{for}\: \left |{x}\right | > 1 \\\frac{x^{\frac{3}{2}}}{\sqrt{1 - x}} - \frac{\sqrt{x}}{\sqrt{1 - x}} + \operatorname{asin}{\left (\sqrt{x} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(1-x)**(1/2),x)

[Out]

Piecewise((-I*sqrt(x)*sqrt(x - 1) - I*acosh(sqrt(x)), Abs(x) > 1), (x**(3/2)/sqrt(1 - x) - sqrt(x)/sqrt(1 - x)
+ asin(sqrt(x)), True))

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Giac [A]  time = 1.0529, size = 23, normalized size = 0.85 \begin{align*} -\sqrt{x} \sqrt{-x + 1} + \arcsin \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x)*sqrt(-x + 1) + arcsin(sqrt(x))