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

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

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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 = {54, 215} \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 54

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 215

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} \operatorname {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 = 0.01, size = 31, normalized size = 1.41 \begin {gather*} \frac {\sqrt {4 x-6} \sin ^{-1}\left (\sqrt {1-\frac {2 x}{3}}\right )}{\sqrt {3-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-6 + 4*x]*ArcSin[Sqrt[1 - (2*x)/3]])/Sqrt[3 - 2*x]

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IntegrateAlgebraic [A]  time = 0.04, size = 30, normalized size = 1.36 \begin {gather*} -\sqrt {2} \log \left (\sqrt {2 x-3}-\sqrt {2} \sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.98, 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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giac [A]  time = 0.92, size = 23, normalized size = 1.05 \begin {gather*} -\sqrt {2} \log \left (\sqrt {2} \sqrt {x} - \sqrt {2 \, 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="giac")

[Out]

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

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maple [B]  time = 0.01, size = 48, normalized size = 2.18 \begin {gather*} \frac {\sqrt {\left (2 x -3\right ) x}\, \sqrt {2}\, \ln \left (\frac {\left (2 x -\frac {3}{2}\right ) \sqrt {2}}{2}+\sqrt {2 x^{2}-3 x}\right )}{2 \sqrt {2 x -3}\, \sqrt {x}} \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]

1/2*(x*(-3+2*x))^(1/2)/x^(1/2)/(-3+2*x)^(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]  time = 2.87, 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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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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sympy [A]  time = 1.03, size = 44, normalized size = 2.00 \begin {gather*} \begin {cases} \sqrt {2} \operatorname {acosh}{\left (\frac {\sqrt {6} \sqrt {x}}{3} \right )} & \text {for}\: \frac {2 \left |{x}\right |}{3} > 1 \\- \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), 2*Abs(x)/3 > 1), (-sqrt(2)*I*asin(sqrt(6)*sqrt(x)/3), True))

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