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

Optimal result

Integrand size = 15, antiderivative size = 20 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=\sqrt {2} \arcsin \left (\sqrt {\frac {2}{3}} \sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {56, 222} \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=\sqrt {2} \arcsin \left (\sqrt {\frac {2}{3}} \sqrt {x}\right ) \]

[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*} \text {integral}& = 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*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {3}-\sqrt {3-2 x}}\right ) \]

[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])]

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x + 3}}{2 \, \sqrt {x}}\right ) \]

[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))

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=\begin {cases} - \sqrt {2} i \operatorname {acosh}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {for}\: \left |{x}\right | > \frac {3}{2} \\\sqrt {2} \operatorname {asin}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x + 3}}{2 \, \sqrt {x}}\right ) \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=\sqrt {2} \arcsin \left (\frac {1}{3} \, \sqrt {6} \sqrt {x}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt {3-2 x} \sqrt {x}} \, dx=2\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (\sqrt {3}-\sqrt {3-2\,x}\right )}{2\,\sqrt {x}}\right ) \]

[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)))