3.2855 \(\int \frac {1}{\sqrt {-2-x} \sqrt {-1-x} \sqrt {-3+x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac {2 \sqrt {x+1} \sqrt {x+2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right ),\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \sqrt {-x-1}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {121, 118} \[ -\frac {2 \sqrt {x+1} \sqrt {x+2} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \sqrt {-x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + x]*Sqrt[2 + x]*EllipticF[ArcSin[1/Sqrt[2/5 + x/5]], 1/5])/(Sqrt[5]*Sqrt[-2 - x]*Sqrt[-1 - 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 {-1-x} \sqrt {-3+x}} \, dx &=\frac {\sqrt {2+x} \int \frac {1}{\sqrt {-1-x} \sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {-3+x}} \, dx}{\sqrt {5} \sqrt {-2-x}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {2+x}\right ) \int \frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}} \sqrt {\frac {1}{4}+\frac {x}{4}} \sqrt {-3+x}} \, dx}{2 \sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ &=-\frac {2 \sqrt {1+x} \sqrt {2+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {2}{5}+\frac {x}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-2-x} \sqrt {-1-x}}\\ \end {align*}

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Mathematica [C]  time = 0.08, size = 69, normalized size = 1.21 \[ \frac {i \sqrt {\frac {4}{x-3}+1} \sqrt {\frac {5}{x-3}+1} (x-3) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {x-3}}\right ),\frac {5}{4}\right )}{\sqrt {-x-2} \sqrt {-x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[1 + 4/(-3 + x)]*Sqrt[1 + 5/(-3 + x)]*(-3 + x)*EllipticF[I*ArcSinh[2/Sqrt[-3 + x]], 5/4])/(Sqrt[-2 - x]
*Sqrt[-1 - x])

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fricas [F]  time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x - 3} \sqrt {-x - 1} \sqrt {-x - 2}}{x^{3} - 7 \, x - 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x - 3)*sqrt(-x - 1)*sqrt(-x - 2)/(x^3 - 7*x - 6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.04, size = 46, normalized size = 0.81 \[ \frac {\sqrt {x +2}\, \sqrt {-x +3}\, \sqrt {x -3}\, \sqrt {-x -2}\, \EllipticF \left (\sqrt {-x -1}, \frac {i}{2}\right )}{x^{2}-x -6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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