3.2847 \(\int \frac {1}{\sqrt {1-x} \sqrt {2-x} \sqrt {3+x}} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {x+3}}{2}\right ),\frac {4}{5}\right )}{\sqrt {5}} \]

[Out]

2/5*EllipticF(1/2*(3+x)^(1/2),2/5*5^(1/2))*5^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {119} \[ \frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {x+3}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticF[ArcSin[Sqrt[3 + x]/2], 4/5])/Sqrt[5]

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 &=\frac {2 F\left (\sin ^{-1}\left (\frac {\sqrt {3+x}}{2}\right )|\frac {4}{5}\right )}{\sqrt {5}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 65, normalized size = 2.83 \[ -\frac {2 i \sqrt {1-\frac {4}{1-x}} \sqrt {\frac {1}{1-x}+1} (1-x) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {1-x}}\right ),-4\right )}{\sqrt {-((x-2) (x+3))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 3} \sqrt {-x + 2} \sqrt {-x + 1}}{x^{3} - 7 \, x + 6}, x\right ) \]

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 - 7*x + 6), x)

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

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 = 16, normalized size = 0.70 \[ \EllipticF \left (\frac {\sqrt {5 x +15}}{5}, \frac {\sqrt {5}}{2}\right ) \]

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]

EllipticF(1/5*(5*x+15)^(1/2),1/2*5^(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 = -0.04 \[ \int \frac {1}{\sqrt {1-x}\,\sqrt {2-x}\,\sqrt {x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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(2 - x)*sqrt(x + 3)), x)

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