3.2852 \(\int \frac {1}{\sqrt {-3-x} \sqrt {-2-x} \sqrt {-1+x}} \, dx\)

Optimal. Leaf size=92 \[ -\frac {\sqrt {x+2} \sqrt {x+3} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {2}{\sqrt {x+3}}\right ),\frac {1}{4}\right )}{\sqrt {-x-3} \sqrt {-x-2}}-\frac {2 i K(4) \sqrt {x+2}}{\sqrt {-x-2}}+\frac {K\left (\frac {3}{4}\right ) \sqrt {x+3}}{\sqrt {-x-3}} \]

[Out]

-2*I*EllipticK(4)*(2+x)^(1/2)/(-2-x)^(1/2)+EllipticK(3/4)*(3+x)^(1/2)/(-3-x)^(1/2)-EllipticF(2/(3+x)^(1/2),1/2
)*(2+x)^(1/2)*(3+x)^(1/2)/(-3-x)^(1/2)/(-2-x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 52, normalized size of antiderivative = 0.57, 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} \[ -\frac {\sqrt {x+2} \sqrt {x+3} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {x}{4}+\frac {3}{4}}}\right )|\frac {1}{4}\right )}{\sqrt {-x-3} \sqrt {-x-2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.17, size = 75, normalized size = 0.82 \[ \frac {2 i \sqrt {\frac {3}{x-1}+1} \sqrt {\frac {4}{x-1}+1} (x-1) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {x-1}}\right ),\frac {4}{3}\right )}{\sqrt {-3 (x-1)-12} \sqrt {-x-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[1 + 3/(-1 + x)]*Sqrt[1 + 4/(-1 + x)]*(-1 + x)*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-1 + x]], 4/3])/(Sq
rt[-12 - 3*(-1 + x)]*Sqrt[-2 - x])

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x - 1} \sqrt {-x - 2} \sqrt {-x - 3}}{x^{3} + 4 \, x^{2} + x - 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x - 1)*sqrt(-x - 2)*sqrt(-x - 3)/(x^3 + 4*x^2 + x - 6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 54, normalized size = 0.59 \[ \frac {2 \sqrt {x +3}\, \sqrt {3}\, \sqrt {-x +1}\, \sqrt {x -1}\, \sqrt {-x -3}\, \EllipticF \left (\sqrt {-x -2}, \frac {i \sqrt {3}}{3}\right )}{3 \left (x^{2}+2 x -3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {x-1}\,\sqrt {-x-2}\,\sqrt {-x-3}} \,d x \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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