3.1556 \(\int \frac{1}{\sqrt{3-2 x} \sqrt{x}} \, dx\)

Optimal. Leaf size=20 \[ \sqrt{2} \sin ^{-1}\left (\sqrt{\frac{2}{3}} \sqrt{x}\right ) \]

[Out]

Sqrt[2]*ArcSin[Sqrt[2/3]*Sqrt[x]]

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Rubi [A]  time = 0.0067617, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {54, 216} \[ \sqrt{2} \sin ^{-1}\left (\sqrt{\frac{2}{3}} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - 2*x]*Sqrt[x]),x]

[Out]

Sqrt[2]*ArcSin[Sqrt[2/3]*Sqrt[x]]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 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{1}{\sqrt{3-2 x} \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-2 x^2}} \, dx,x,\sqrt{x}\right )\\ &=\sqrt{2} \sin ^{-1}\left (\sqrt{\frac{2}{3}} \sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0043195, size = 20, normalized size = 1. \[ \sqrt{2} \sin ^{-1}\left (\sqrt{\frac{2}{3}} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[3 - 2*x]*Sqrt[x]),x]

[Out]

Sqrt[2]*ArcSin[Sqrt[2/3]*Sqrt[x]]

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Maple [B]  time = 0.005, size = 31, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2}}{2}\sqrt{ \left ( 3-2\,x \right ) x}\arcsin \left ({\frac{4\,x}{3}}-1 \right ){\frac{1}{\sqrt{3-2\,x}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3-2*x)^(1/2)/x^(1/2),x)

[Out]

1/2*((3-2*x)*x)^(1/2)/(3-2*x)^(1/2)/x^(1/2)*2^(1/2)*arcsin(4/3*x-1)

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Maxima [A]  time = 1.43391, size = 28, normalized size = 1.4 \begin{align*} -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x + 3}}{2 \, \sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.98882, size = 72, normalized size = 3.6 \begin{align*} -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x + 3}}{2 \, \sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(1/2)/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.0215, size = 44, normalized size = 2.2 \begin{align*} \begin{cases} - \sqrt{2} i \operatorname{acosh}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{for}\: \frac{2 \left |{x}\right |}{3} > 1 \\\sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)**(1/2)/x**(1/2),x)

[Out]

Piecewise((-sqrt(2)*I*acosh(sqrt(6)*sqrt(x)/3), 2*Abs(x)/3 > 1), (sqrt(2)*asin(sqrt(6)*sqrt(x)/3), True))

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Giac [A]  time = 1.06279, size = 18, normalized size = 0.9 \begin{align*} \sqrt{2} \arcsin \left (\frac{1}{3} \, \sqrt{6} \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3-2*x)^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

sqrt(2)*arcsin(1/3*sqrt(6)*sqrt(x))