3.29.48 \(\int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx\) [2848]

Optimal. Leaf size=14 \[ 2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \]

[Out]

2*EllipticF(1/(3-x)^(1/2),2^(1/2))

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Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {119} \begin {gather*} 2 F\left (\left .\text {ArcSin}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[1/Sqrt[3 - x]], 2]

Rule 119

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx &=2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.00, size = 67, normalized size = 4.79 \begin {gather*} \frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-2+x}{-1+x}} (-1+x) F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {1-x}}\right )\right |2\right )}{\sqrt {2-x} \sqrt {3-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(13)=26\).
time = 0.11, size = 55, normalized size = 3.93

method	result	size
default	\(-\frac {\EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right ) \sqrt {-2+x}\, \sqrt {-2+2 x}\, \sqrt {2-x}\, \sqrt {2}\, \sqrt {2-2 x}}{2 \left (x^{2}-3 x +2\right )}\)	\(55\)
elliptic	\(-\frac {\sqrt {-\left (-1+x \right ) \left (-2+x \right ) \left (-3+x \right )}\, \sqrt {-2+2 x}\, \sqrt {-2+x}\, \EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {1-x}\, \sqrt {2-x}\, \sqrt {-x^{3}+6 x^{2}-11 x +6}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*EllipticF((3-x)^(1/2),1/2*2^(1/2))*(-2+x)^(1/2)*(-2+2*x)^(1/2)*(2-x)^(1/2)*2^(1/2)*(2-2*x)^(1/2)/(x^2-3*x
+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/(1-x)^(1/2)/(2-x)^(1/2)/(3-x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x + 3)*sqrt(-x + 2)*sqrt(-x + 1)), 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/(1-x)^(1/2)/(2-x)^(1/2)/(3-x)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [C] Result contains complex when optimal does not.
time = 12.28, size = 66, normalized size = 4.71 \begin {gather*} \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), exp_polar(-2*I*pi)/(x - 2)**2)/(4*pi*
*(3/2)) - meijerg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), (x - 2)**(-2))/(4*pi**(3/
2))

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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/(1-x)^(1/2)/(2-x)^(1/2)/(3-x)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {3-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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