3.2851 \(\int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx\)

Optimal. Leaf size=41 \[ -\frac {2 \sqrt {x+2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]

[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)

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {121, 118} \[ -\frac {2 \sqrt {x+2} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{3}+\frac {2}{3}}}\right )|\frac {5}{3}\right )}{\sqrt {3} \sqrt {-x-2}} \]

Antiderivative was successfully verified.

[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 118

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(-2*Sqrt[d/
f]*EllipticF[ArcSin[Rt[-((b*e - a*f)/f), 2]/Sqrt[a + b*x]], (f*(b*c - a*d))/(d*(b*e - a*f))])/(d*Rt[-((b*e - a
*f)/f), 2]), 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 121

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*} \int \frac {1}{\sqrt {-2-x} \sqrt {-3+x} \sqrt {-1+x}} \, dx &=\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*}

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Mathematica [C]  time = 0.13, size = 72, normalized size = 1.76 \[ \frac {2 i \sqrt {\frac {x-3}{x-1}} \sqrt {\frac {x-1}{x+2}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {-x-2}}\right ),\frac {5}{3}\right )}{\sqrt {3} \sqrt {\frac {x-3}{x+2}}} \]

Warning: Unable to verify antiderivative.

[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)])

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}{x^{3} - 2 \, x^{2} - 5 \, x + 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(x - 1)*sqrt(x - 3)*sqrt(-x - 2)/(x^3 - 2*x^2 - 5*x + 6), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.04, size = 72, normalized size = 1.76 \[ -\frac {2 \sqrt {-x -2}\, \sqrt {x -3}\, \sqrt {x -1}\, \sqrt {x +2}\, \sqrt {3}\, \sqrt {-x +1}\, \sqrt {-x +3}\, \EllipticF \left (\frac {\sqrt {5 x +10}}{5}, \frac {\sqrt {15}}{3}\right )}{3 \left (x^{3}-2 x^{2}-5 x +6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 1} \sqrt {x - 3} \sqrt {-x - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x-1}\,\sqrt {-x-2}\,\sqrt {x-3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- x - 2} \sqrt {x - 3} \sqrt {x - 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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