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 = 0.0161486, 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 \[ \sqrt{2} \sinh ^{-1}\left (\frac{\sqrt{2 x-3}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.62344, size = 20, normalized size = 0.91 \[ \sqrt{2} \operatorname{asinh}{\left (\frac{\sqrt{3} \sqrt{2 x - 3}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(1/2)/(-3+2*x)**(1/2),x)

[Out]

sqrt(2)*asinh(sqrt(3)*sqrt(2*x - 3)/3)

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Mathematica [A]  time = 0.0153173, size = 24, normalized size = 1.09 \[ \sqrt{2} \log \left (2 \sqrt{x}+\sqrt{4 x-6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[2]*Log[2*Sqrt[x] + Sqrt[-6 + 4*x]]

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Maple [B]  time = 0.009, size = 48, normalized size = 2.2 \[{\frac{\sqrt{2}}{2}\sqrt{x \left ( -3+2\,x \right ) }\ln \left ({\frac{\sqrt{2}}{2} \left ( -{\frac{3}{2}}+2\,x \right ) }+\sqrt{2\,{x}^{2}-3\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-3+2\,x}}}} \]

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 [A]  time = 1.49366, size = 57, normalized size = 2.59 \[ -\frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \,{\left (\sqrt{2} - \frac{\sqrt{2 \, x - 3}}{\sqrt{x}}\right )}}{2 \, \sqrt{2} + \frac{2 \, \sqrt{2 \, x - 3}}{\sqrt{x}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 3)*sqrt(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212996, size = 35, normalized size = 1.59 \[ \frac{1}{2} \, \sqrt{2} \log \left (-2 \, \sqrt{2} \sqrt{2 \, x - 3} \sqrt{x} - 4 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 3)*sqrt(x)),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 = 1.70016, size = 44, normalized size = 2. \[ \begin{cases} \sqrt{2} \operatorname{acosh}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{for}\: \frac{2 \left |{x}\right |}{3} > 1 \\- \sqrt{2} i \operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{otherwise} \end{cases} \]

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(s
qrt(6)*sqrt(x)/3), True))

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GIAC/XCAS [A]  time = 0.210884, size = 31, normalized size = 1.41 \[ -\sqrt{2}{\rm ln}\left (\sqrt{2} \sqrt{x} - \sqrt{2 \, x - 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(2*x - 3)*sqrt(x)),x, algorithm="giac")

[Out]

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