3.1557 \(\int \frac{1}{\sqrt{3-2 x} \sqrt{x}} \, dx\)

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 = 0.0194492, 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 \[ \sqrt{2} \sin ^{-1}\left (\sqrt{\frac{2}{3}} \sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.56503, size = 17, normalized size = 0.85 \[ \sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*asin(sqrt(6)*sqrt(x)/3)

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Mathematica [A]  time = 0.00934318, 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 = 0.008, size = 31, normalized size = 1.6 \[{\frac{\sqrt{2}}{2}\sqrt{ \left ( 3-2\,x \right ) x}\arcsin \left ({\frac{4\,x}{3}}-1 \right ){\frac{1}{\sqrt{3-2\,x}}}{\frac{1}{\sqrt{x}}}} \]

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 = 1.50453, size = 28, normalized size = 1.4 \[ -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x + 3}}{2 \, \sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208743, size = 28, normalized size = 1.4 \[ -\sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-2 \, x + 3}}{2 \, \sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.70845, size = 44, normalized size = 2.2 \[ \begin{cases} - \sqrt{2} i \operatorname{acosh}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{for}\: \frac{2 \left |{x}\right |}{3} > 1 \\\sqrt{2} \operatorname{asin}{\left (\frac{\sqrt{6} \sqrt{x}}{3} \right )} & \text{otherwise} \end{cases} \]

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(s
qrt(6)*sqrt(x)/3), True))

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GIAC/XCAS [A]  time = 0.217587, size = 18, normalized size = 0.9 \[ \sqrt{2} \arcsin \left (\frac{1}{3} \, \sqrt{6} \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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