3.1552 \(\int \frac{1}{\sqrt{x} \sqrt{-3+2 x}} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-3}}{\sqrt{3}}\right ) \]

[Out]

Sqrt[2]*ArcSinh[Sqrt[-3 + 2*x]/Sqrt[3]]

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Rubi [A]  time = 0.0040966, antiderivative size = 22, 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, 215} \[ \sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-3}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[-3 + 2*x]),x]

[Out]

Sqrt[2]*ArcSinh[Sqrt[-3 + 2*x]/Sqrt[3]]

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 215

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} \sqrt{-3+2 x}} \, dx &=\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+x^2}} \, dx,x,\sqrt{-3+2 x}\right )\\ &=\sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{-3+2 x}}{\sqrt{3}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0083228, size = 31, normalized size = 1.41 \[ \frac{\sqrt{4 x-6} \sin ^{-1}\left (\sqrt{1-\frac{2 x}{3}}\right )}{\sqrt{3-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-6 + 4*x]*ArcSin[Sqrt[1 - (2*x)/3]])/Sqrt[3 - 2*x]

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Maple [B]  time = 0.005, size = 48, normalized size = 2.2 \begin{align*}{\frac{\sqrt{2}}{2}\sqrt{x \left ( -3+2\,x \right ) }\ln \left ({\frac{\sqrt{2}}{2} \left ( -{\frac{3}{2}}+2\,x \right ) }+\sqrt{2\,{x}^{2}-3\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-3+2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.46487, size = 55, normalized size = 2.5 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{\sqrt{2 \, x - 3}}{\sqrt{x}}}{\sqrt{2} + \frac{\sqrt{2 \, x - 3}}{\sqrt{x}}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9871, size = 82, normalized size = 3.73 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{2 \, x - 3} \sqrt{x} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-2*sqrt(2)*sqrt(2*x - 3)*sqrt(x) - 4*x + 3)

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Sympy [A]  time = 1.05414, size = 44, normalized size = 2. \begin{align*} \begin{cases} \sqrt{2} \operatorname{acosh}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{for}\: \frac{2 \left |{x}\right |}{3} > 1 \\- \sqrt{2} i \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/x**(1/2)/(-3+2*x)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.06913, size = 31, normalized size = 1.41 \begin{align*} -\sqrt{2} \log \left (\sqrt{2} \sqrt{x} - \sqrt{2 \, x - 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*log(sqrt(2)*sqrt(x) - sqrt(2*x - 3))