3.2845 \(\int \frac {1}{\sqrt {1-x} \sqrt {2+x} \sqrt {3+x}} \, dx\)

Optimal. Leaf size=18 \[ 2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {x+2}}{\sqrt {3}}\right ),-3\right ) \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {119} \[ 2 F\left (\left .\sin ^{-1}\left (\frac {\sqrt {x+2}}{\sqrt {3}}\right )\right |-3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*EllipticF[ArcSin[Sqrt[2 + x]/Sqrt[3]], -3]

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(b/d)] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-(d/b), 0]) &&  !(SimplerQ[c +
d*x, a + b*x] && GtQ[(-(b*e) + a*f)/f, 0] && GtQ[-(f/b), 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[(-(d*e)
+ c*f)/f, 0] && GtQ[(-(b*e) + a*f)/f, 0] && (PosQ[-(f/d)] || PosQ[-(f/b)]))

Rubi steps

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

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Mathematica [C]  time = 0.14, size = 78, normalized size = 4.33 \[ -\frac {2 i \sqrt {-((x-1) (x+2))} \sqrt {x+3} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {x-1}}\right ),\frac {4}{3}\right )}{\sqrt {\frac {9}{x-1}+3} (x-1)^{3/2} \sqrt {\frac {x+3}{x-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*Sqrt[-((-1 + x)*(2 + x))]*Sqrt[3 + x]*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-1 + x]], 4/3])/(Sqrt[3 + 9/(-1
+ x)]*(-1 + x)^(3/2)*Sqrt[(3 + x)/(-1 + x)])

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fricas [F]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x + 3} \sqrt {x + 2} \sqrt {-x + 1}}{x^{3} + 4 \, x^{2} + x - 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 32, normalized size = 1.78 \[ -\frac {2 \sqrt {-x -2}\, \sqrt {3}\, \EllipticF \left (\sqrt {-x -2}, \frac {i \sqrt {3}}{3}\right )}{3 \sqrt {x +2}} \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {1-x}\,\sqrt {x+2}\,\sqrt {x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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