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

Optimal. Leaf size=22 \[ \sqrt {2} \sinh ^{-1}\left (\frac {\sqrt {-3+2 x}}{\sqrt {3}}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, 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, 221} \begin {gather*} \sqrt {2} \sinh ^{-1}\left (\frac {\sqrt {2 x-3}}{\sqrt {3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\sqrt {x} \sqrt {-3+2 x}} \, dx &=\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*}

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Mathematica [A]
time = 0.03, size = 30, normalized size = 1.36 \begin {gather*} -\sqrt {2} \log \left (-\sqrt {2} \sqrt {x}+\sqrt {-3+2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2]*Log[-(Sqrt[2]*Sqrt[x]) + 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 = 2.16, size = 34, normalized size = 1.55 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\sqrt {2} \text {ArcCosh}\left [\frac {\sqrt {6} \sqrt {x}}{3}\right ],\text {Abs}\left [x\right ]>\frac {3}{2}\right \}\right \},-I \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[{{Sqrt[2] ArcCosh[Sqrt[6] Sqrt[x] / 3], Abs[x] > 3 / 2}}, -I 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. \(47\) vs. \(2(17)=34\).
time = 0.16, size = 48, normalized size = 2.18

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*(2*x-3))^(1/2)/x^(1/2)/(2*x-3)^(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 [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).
time = 0.38, size = 41, normalized size = 1.86 \begin {gather*} -\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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Fricas [A]
time = 0.29, size = 26, normalized size = 1.18 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-2 \, \sqrt {2} \sqrt {2 \, x - 3} \sqrt {x} - 4 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.55, size = 42, normalized size = 1.91 \begin {gather*} \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} \end {gather*}

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

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Giac [A]
time = 0.00, size = 33, normalized size = 1.50 \begin {gather*} -\frac {2 \ln \left |\sqrt {2 x-3}-\sqrt {2} \sqrt {x}\right |}{\sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*log(abs(-sqrt(2)*sqrt(x) + sqrt(2*x - 3)))

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Mupad [B]
time = 0.44, size = 30, normalized size = 1.36 \begin {gather*} -2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\left (-\sqrt {2\,x-3}+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,\sqrt {x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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