\(\int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx\) [2851]

Optimal result

Integrand size = 24, antiderivative size = 41 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-\frac {2 \sqrt {2+x} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-2-x}} \]

[Out]

-2/3*EllipticF(3/(6+3*x)^(1/2),1/3*15^(1/2))*(2+x)^(1/2)*3^(1/2)/(-2-x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {122, 119} \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-\frac {2 \sqrt {x+2} \operatorname {EllipticF}\left (\arcsin \left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]

[In]

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

[Out]

(-2*Sqrt[2 + x]*EllipticF[ArcSin[1/Sqrt[2/3 + x/3]], 5/3])/(Sqrt[3]*Sqrt[-2 - x])

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)
]

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {2+x} \int \frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}} \sqrt {-3+x} \sqrt {-1+x}} \, dx}{\sqrt {3} \sqrt {-2-x}} \\ & = -\frac {2 \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{3}+\frac {x}{3}}}\right )|\frac {5}{3}\right )}{\sqrt {3} \sqrt {-2-x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=\frac {2 i \sqrt {\frac {-3+x}{-1+x}} \sqrt {\frac {-1+x}{2+x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {3}}{\sqrt {-2-x}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {\frac {-3+x}{2+x}}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(32)=64\).

Time = 4.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.76

method	result	size
default	\(-\frac {2 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {-1+x}\, \sqrt {2+x}\, \sqrt {3}\, \sqrt {1-x}\, \sqrt {3-x}\, F\left (\frac {\sqrt {10+5 x}}{5}, \frac {\sqrt {15}}{3}\right )}{3 \left (x^{3}-2 x^{2}-5 x +6\right )}\)	\(72\)
elliptic	\(\frac {2 \sqrt {-\left (-1+x \right ) \left (-3+x \right ) \left (2+x \right )}\, \sqrt {10+5 x}\, \sqrt {3-3 x}\, \sqrt {15-5 x}\, F\left (\frac {\sqrt {10+5 x}}{5}, \frac {\sqrt {15}}{3}\right )}{15 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {-1+x}\, \sqrt {-x^{3}+2 x^{2}+5 x -6}}\)	\(86\)

[In]

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

[Out]

-2/3*(-2-x)^(1/2)*(-3+x)^(1/2)*(-1+x)^(1/2)*(2+x)^(1/2)*3^(1/2)*(1-x)^(1/2)*(3-x)^(1/2)*EllipticF(1/5*(10+5*x)
^(1/2),1/3*15^(1/2))/(x^3-2*x^2-5*x+6)

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 = 0.20 \[ \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx=-2 i \, {\rm weierstrassPInverse}\left (\frac {76}{3}, -\frac {224}{27}, x - \frac {2}{3}\right ) \]

[In]

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

[Out]

-2*I*weierstrassPInverse(76/3, -224/27, x - 2/3)

Sympy [F]

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

[In]

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

[Out]

Integral(1/(sqrt(-x - 2)*sqrt(x - 3)*sqrt(x - 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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