3.29.50 \(\int \frac {1}{\sqrt {-3-x} \sqrt {-2+x} \sqrt {-1+x}} \, dx\) [2850]

Optimal. Leaf size=57 \[ -\frac {\sqrt {3+x} F\left (\sin ^{-1}\left (\frac {2}{\sqrt {3+x}}\right )|\frac {5}{4}\right )}{\sqrt {-3-x}}-\frac {i \sqrt {3+x} K\left (-\frac {1}{4}\right )}{\sqrt {-3-x}} \]

[Out]

-EllipticF(2/(3+x)^(1/2),1/2*5^(1/2))*(3+x)^(1/2)/(-3-x)^(1/2)-I*EllipticK(1/4)*(3+x)^(1/2)/(-3-x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 0.63, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {122, 119} \begin {gather*} -\frac {\sqrt {x+3} F\left (\text {ArcSin}\left (\frac {1}{\sqrt {\frac {x}{4}+\frac {3}{4}}}\right )|\frac {5}{4}\right )}{\sqrt {-x-3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[-2*(Sqrt[d/
f]/(d*Rt[-(b*e - a*f)/f, 2]))*EllipticF[ArcSin[Rt[-(b*e - a*f)/f, 2]/Sqrt[a + b*x]], f*((b*c - a*d)/(d*(b*e -
a*f)))], 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 122

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

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Mathematica [A]
time = 1.07, size = 63, normalized size = 1.11 \begin {gather*} \frac {i \sqrt {\frac {-2+x}{-1+x}} \sqrt {\frac {-1+x}{3+x}} F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-3-x}}\right )|\frac {5}{4}\right )}{\sqrt {\frac {-2+x}{3+x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 65, normalized size = 1.14

method	result	size
default	\(\frac {\sqrt {-3-x}\, \sqrt {-2+x}\, \sqrt {-1+x}\, \sqrt {3+x}\, \sqrt {1-x}\, \sqrt {2-x}\, \EllipticF \left (\frac {\sqrt {15+5 x}}{5}, \frac {\sqrt {5}}{2}\right )}{-x^{3}+7 x -6}\)	\(65\)
elliptic	\(\frac {\sqrt {-\left (-1+x \right ) \left (-2+x \right ) \left (3+x \right )}\, \sqrt {15+5 x}\, \sqrt {1-x}\, \sqrt {10-5 x}\, \EllipticF \left (\frac {\sqrt {15+5 x}}{5}, \frac {\sqrt {5}}{2}\right )}{5 \sqrt {-3-x}\, \sqrt {-2+x}\, \sqrt {-1+x}\, \sqrt {-x^{3}+7 x -6}}\)	\(81\)

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,method=_RETURNVERBOSE)

[Out]

1/(-x^3+7*x-6)*(-3-x)^(1/2)*(-2+x)^(1/2)*(-1+x)^(1/2)*(3+x)^(1/2)*(1-x)^(1/2)*(2-x)^(1/2)*EllipticF(1/5*(15+5*
x)^(1/2),1/2*5^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- x - 3} \sqrt {x - 2} \sqrt {x - 1}}\, dx \end {gather*}

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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x-1}\,\sqrt {x-2}\,\sqrt {-x-3}} \,d x \end {gather*}

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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