3.16.56 \(\int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx\) [1556]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 = {56, 222} \begin {gather*} \sqrt {2} \sin ^{-1}\left (\sqrt {\frac {2}{3}} \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[3 - 2*x]*Sqrt[x]),x]

[Out]

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

Rule 56

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 222

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 \text {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*}

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Mathematica [A]
time = 0.05, size = 38, normalized size = 1.90 \begin {gather*} -2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {3}-\sqrt {3-2 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[x])/(Sqrt[3] - Sqrt[3 - 2*x])]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.20, size = 34, normalized size = 1.70 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-I \sqrt {2} \text {ArcCosh}\left [\frac {\sqrt {6} \sqrt {x}}{3}\right ],\text {Abs}\left [x\right ]>\frac {3}{2}\right \}\right \},\sqrt {2} \text {ArcSin}\left [\frac {\sqrt {6} \sqrt {x}}{3}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{-I Sqrt[2] ArcCosh[Sqrt[6] Sqrt[x] / 3], Abs[x] > 3 / 2}}, Sqrt[2] ArcSin[Sqrt[6] Sqrt[x] / 3]]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(30\) vs. \(2(13)=26\).
time = 0.15, size = 31, normalized size = 1.55

method	result	size
meijerg	\(\sqrt {2}\, \arcsin \left (\frac {\sqrt {x}\, \sqrt {3}\, \sqrt {2}}{3}\right )\)	\(17\)
default	\(\frac {\sqrt {\left (3-2 x \right ) x}\, \sqrt {2}\, \arcsin \left (\frac {4 x}{3}-1\right )}{2 \sqrt {3-2 x}\, \sqrt {x}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.34, size = 21, normalized size = 1.05 \begin {gather*} -\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x + 3}}{2 \, \sqrt {x}}\right ) \end {gather*}

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 = 0.29, size = 21, normalized size = 1.05 \begin {gather*} -\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x + 3}}{2 \, \sqrt {x}}\right ) \end {gather*}

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 = 0.55, size = 42, normalized size = 2.10 \begin {gather*} \begin {cases} - \sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {for}\: \left |{x}\right | > \frac {3}{2} \\\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {otherwise} \end {cases} \end {gather*}

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), Abs(x) > 3/2), (sqrt(2)*asin(sqrt(6)*sqrt(x)/3), True))

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Giac [A]
time = 0.00, size = 27, normalized size = 1.35 \begin {gather*} -\frac {2 \arcsin \left (\frac {\sqrt {-2 x+3}}{\sqrt {3}}\right )}{\sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.30, size = 27, normalized size = 1.35 \begin {gather*} 2\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (\sqrt {3}-\sqrt {3-2\,x}\right )}{2\,\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(3 - 2*x)^(1/2)),x)

[Out]

2*2^(1/2)*atan((2^(1/2)*(3^(1/2) - (3 - 2*x)^(1/2)))/(2*x^(1/2)))

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