\(\int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx\) [1552]

Optimal result

Integrand size = 15, antiderivative size = 22 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=\sqrt {2} \text {arcsinh}\left (\frac {\sqrt {-3+2 x}}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, 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, 221} \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=\sqrt {2} \text {arcsinh}\left (\frac {\sqrt {2 x-3}}{\sqrt {3}}\right ) \]

[In]

Int[1/(Sqrt[x]*Sqrt[-3 + 2*x]),x]

[Out]

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

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 221

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*} \text {integral}& = \sqrt {2} \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=-\sqrt {2} \log \left (-\sqrt {2} \sqrt {x}+\sqrt {-3+2 x}\right ) \]

[In]

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

[Out]

-(Sqrt[2]*Log[-(Sqrt[2]*Sqrt[x]) + Sqrt[-3 + 2*x]])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41

method	result	size
meijerg	\(\frac {\sqrt {2}\, \sqrt {-\operatorname {signum}\left (x -\frac {3}{2}\right )}\, \arcsin \left (\frac {\sqrt {x}\, \sqrt {3}\, \sqrt {2}}{3}\right )}{\sqrt {\operatorname {signum}\left (x -\frac {3}{2}\right )}}\)	\(31\)
default	\(\frac {\sqrt {x \left (-3+2 x \right )}\, \ln \left (\frac {\left (-\frac {3}{2}+2 x \right ) \sqrt {2}}{2}+\sqrt {2 x^{2}-3 x}\right ) \sqrt {2}}{2 \sqrt {x}\, \sqrt {-3+2 x}}\)	\(48\)

[In]

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

[Out]

2^(1/2)/signum(x-3/2)^(1/2)*(-signum(x-3/2))^(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 = 26, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sqrt {2 \, x - 3} \sqrt {x} - 4 \, x + 3\right ) \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=\begin {cases} \sqrt {2} \operatorname {acosh}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {for}\: \left |{x}\right | > \frac {3}{2} \\- \sqrt {2} i \operatorname {asin}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).

Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {\sqrt {2 \, x - 3}}{\sqrt {x}}}{\sqrt {2} + \frac {\sqrt {2 \, x - 3}}{\sqrt {x}}}\right ) \]

[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)))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=-\sqrt {2} \log \left (\sqrt {2} \sqrt {x} - \sqrt {2 \, x - 3}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx=-2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\sqrt {x}}\right ) \]

[In]

int(1/(x^(1/2)*(2*x - 3)^(1/2)),x)

[Out]

-2*2^(1/2)*atanh((2^(1/2)*(3^(1/2)*1i - (2*x - 3)^(1/2)))/(2*x^(1/2)))