3.2888 \(\int \frac{1}{\sqrt{6-x} \sqrt{-2+x} \sqrt{-1+x}} \, dx\)

Optimal. Leaf size=16 \[ 2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{x-2}}{2}\right ),-4\right ) \]

[Out]

2*EllipticF[ArcSin[Sqrt[-2 + x]/2], -4]

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Rubi [A]  time = 0.0048509, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {119} \[ 2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{x-2}}{2}\right )\right |-4\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[6 - x]*Sqrt[-2 + x]*Sqrt[-1 + x]),x]

[Out]

2*EllipticF[ArcSin[Sqrt[-2 + x]/2], -4]

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{6-x} \sqrt{-2+x} \sqrt{-1+x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-2+x}}{2}\right )\right |-4\right )\\ \end{align*}

Mathematica [C]  time = 0.0409578, size = 74, normalized size = 4.62 \[ \frac{i \sqrt{\frac{4}{x-6}+1} \sqrt{\frac{5}{x-6}+1} (x-6)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{x-6}}\right ),\frac{5}{4}\right )}{\sqrt{-(x-6) (x-2)} \sqrt{x-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[6 - x]*Sqrt[-2 + x]*Sqrt[-1 + x]),x]

[Out]

(I*Sqrt[1 + 4/(-6 + x)]*Sqrt[1 + 5/(-6 + x)]*(-6 + x)^(3/2)*EllipticF[I*ArcSinh[2/Sqrt[-6 + x]], 5/4])/(Sqrt[-
((-6 + x)*(-2 + x))]*Sqrt[-1 + x])

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Maple [A]  time = 0.036, size = 21, normalized size = 1.3 \begin{align*} -{\frac{2\,\sqrt{5}}{5}{\it EllipticF} \left ({\frac{1}{2}\sqrt{6-x}},{\frac{2\,\sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(6-x)^(1/2)/(-2+x)^(1/2)/(-1+x)^(1/2),x)

[Out]

-2/5*EllipticF(1/2*(6-x)^(1/2),2/5*5^(1/2))*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x - 2} \sqrt{-x + 6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(6-x)^(1/2)/(-2+x)^(1/2)/(-1+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x - 1)*sqrt(x - 2)*sqrt(-x + 6)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{x - 1} \sqrt{x - 2} \sqrt{-x + 6}}{x^{3} - 9 \, x^{2} + 20 \, x - 12}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(6-x)^(1/2)/(-2+x)^(1/2)/(-1+x)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(x - 1)*sqrt(x - 2)*sqrt(-x + 6)/(x^3 - 9*x^2 + 20*x - 12), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{6 - x} \sqrt{x - 2} \sqrt{x - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(6-x)**(1/2)/(-2+x)**(1/2)/(-1+x)**(1/2),x)

[Out]

Integral(1/(sqrt(6 - x)*sqrt(x - 2)*sqrt(x - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x - 2} \sqrt{-x + 6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(6-x)^(1/2)/(-2+x)^(1/2)/(-1+x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x - 1)*sqrt(x - 2)*sqrt(-x + 6)), x)