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

Optimal result

Integrand size = 28, antiderivative size = 14 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {3-x}}\right ),2\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {119} \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=2 \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {3-x}}\right ),2\right ) \]

[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*} \text {integral}& = 2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {3-x}}\right )\right |2\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 4.79 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-2+x}{-1+x}} (-1+x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {1}{\sqrt {1-x}}\right ),2\right )}{\sqrt {2-x} \sqrt {3-x}} \]

[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(13)=26\).

Time = 1.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.93

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

[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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (4, 0, x - 2\right ) \]

[In]

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

[Out]

-2*I*weierstrassPInverse(4, 0, x - 2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.82 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]

[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))

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3-x}} \, dx=\int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {3-x}} \,d x \]

[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)