3.2835 \(\int \frac{1}{\sqrt{-3-x} \sqrt{-2-x} \sqrt{-1+x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.114573, 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 \[ -\frac{\sqrt{x+2} \sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{4}+\frac{3}{4}}}\right )|\frac{1}{4}\right )}{\sqrt{-x-3} \sqrt{-x-2}} \]

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]))

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Rubi in Sympy [A]  time = 9.38881, size = 56, normalized size = 0.61 \[ - \frac{4 i \sqrt{\frac{x}{4} + \frac{3}{4}} \sqrt{\frac{x}{3} + \frac{2}{3}} F\left (i \operatorname{asinh}{\left (\frac{\sqrt{x - 1}}{2} \right )}\middle | \frac{4}{3}\right )}{\sqrt{- x - 3} \sqrt{- x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3-x)**(1/2)/(-2-x)**(1/2)/(-1+x)**(1/2),x)

[Out]

-4*I*sqrt(x/4 + 3/4)*sqrt(x/3 + 2/3)*elliptic_f(I*asinh(sqrt(x - 1)/2), 4/3)/(sq
rt(-x - 3)*sqrt(-x - 2))

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Mathematica [A]  time = 0.121416, size = 75, normalized size = 0.82 \[ \frac{2 i \sqrt{\frac{3}{x-1}+1} \sqrt{\frac{4}{x-1}+1} (x-1) F\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{x-1}}\right )|\frac{4}{3}\right )}{\sqrt{-3 (x-1)-12} \sqrt{-x-2}} \]

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

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Maple [A]  time = 0.053, size = 54, normalized size = 0.6 \[{\frac{2\,\sqrt{3}}{3\,{x}^{2}+6\,x-9}{\it EllipticF} \left ( \sqrt{-2-x},{\frac{i}{3}}\sqrt{3} \right ) \sqrt{3+x}\sqrt{1-x}\sqrt{-1+x}\sqrt{-3-x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3-x)^(1/2)/(-2-x)^(1/2)/(-1+x)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*sqrt(-x - 2)*sqrt(-x - 3)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*sqrt(-x - 2)*sqrt(-x - 3)),x, algorithm="fricas")

[Out]

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

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

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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x - 1} \sqrt{-x - 2} \sqrt{-x - 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 1)*sqrt(-x - 2)*sqrt(-x - 3)),x, algorithm="giac")

[Out]

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