3.29.55 \(\int \frac {1}{\sqrt {-2-x} \sqrt {-1-x} \sqrt {-3+x}} \, dx\) [2855]

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

[Out]

-2/5*EllipticF(5/(10+5*x)^(1/2),1/5*5^(1/2))*(1+x)^(1/2)*(2+x)^(1/2)*5^(1/2)/(-2-x)^(1/2)/(-1-x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 = {122, 119} \begin {gather*} -\frac {2 \sqrt {x+1} \sqrt {x+2} F\left (\text {ArcSin}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {2}{5}}}\right )|\frac {1}{5}\right )}{\sqrt {5} \sqrt {-x-2} \sqrt {-x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.51, size = 69, normalized size = 1.21 \begin {gather*} \frac {i \sqrt {1+\frac {4}{-3+x}} \sqrt {1+\frac {5}{-3+x}} (-3+x) F\left (i \sinh ^{-1}\left (\frac {2}{\sqrt {-3+x}}\right )|\frac {5}{4}\right )}{\sqrt {-2-x} \sqrt {-1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.10, size = 51, normalized size = 0.89

method	result	size
default	\(-\frac {2 \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right ) \sqrt {5}\, \sqrt {3-x}\, \sqrt {2+x}\, \sqrt {-3+x}\, \sqrt {-2-x}}{5 \left (x^{2}-x -6\right )}\)	\(51\)
elliptic	\(\frac {2 \sqrt {\left (-3+x \right ) \left (1+x \right ) \left (2+x \right )}\, \sqrt {2+x}\, \sqrt {15-5 x}\, \EllipticF \left (\sqrt {2+x}, \frac {\sqrt {5}}{5}\right )}{5 \sqrt {-2-x}\, \sqrt {-3+x}\, \sqrt {x^{3}-7 x -6}}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2-x)^(1/2)/(-1-x)^(1/2)/(-3+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.22, size = 6, normalized size = 0.11 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (28, 24, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*weierstrassPInverse(28, 24, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-x - 2)*sqrt(-x - 1)*sqrt(x - 3)), 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/(-2-x)^(1/2)/(-1-x)^(1/2)/(-3+x)^(1/2),x, algorithm="giac")

[Out]

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