3.29.88 \(\int \frac {1}{\sqrt {6-x} \sqrt {-2+x} \sqrt {-1+x}} \, dx\) [2888]

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

[Out]

2*EllipticF(1/2*(-2+x)^(1/2),2*I)

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {120} \begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\frac {\sqrt {x-2}}{2}\right )\right |-4\right ) \end {gather*}

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 120

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.61, size = 74, normalized size = 4.62 \begin {gather*} \frac {i \sqrt {1+\frac {4}{-6+x}} \sqrt {1+\frac {5}{-6+x}} (-6+x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-6+x}}\right )|\frac {5}{4}\right )}{\sqrt {-((-6+x) (-2+x))} \sqrt {-1+x}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(12)=24\).
time = 0.13, size = 29, normalized size = 1.81

method	result	size
default	\(\frac {2 \sqrt {5}\, \sqrt {2-x}\, \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {-2+x}}\)	\(29\)
elliptic	\(\frac {2 \sqrt {-\left (-6+x \right ) \left (-2+x \right ) \left (-1+x \right )}\, \sqrt {30-5 x}\, \sqrt {2-x}\, \EllipticF \left (\sqrt {-1+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {6-x}\, \sqrt {-2+x}\, \sqrt {-x^{3}+9 x^{2}-20 x +12}}\)	\(70\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {6 - x} \sqrt {x - 2} \sqrt {x - 1}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{\sqrt {x-1}\,\sqrt {x-2}\,\sqrt {6-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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