3.29.54 \(\int \frac {1}{\sqrt {-3-x} \sqrt {-1-x} \sqrt {-2+x}} \, dx\) [2854]

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

[Out]

-2/5*EllipticF(5/(15+5*x)^(1/2),1/5*10^(1/2))*(1+x)^(1/2)*(3+x)^(1/2)*5^(1/2)/(-3-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+3} F\left (\text {ArcSin}\left (\frac {1}{\sqrt {\frac {x}{5}+\frac {3}{5}}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-x-3} \sqrt {-x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 + x]*Sqrt[3 + x]*EllipticF[ArcSin[1/Sqrt[3/5 + x/5]], 2/5])/(Sqrt[5]*Sqrt[-3 - 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 {-3-x} \sqrt {-1-x} \sqrt {-2+x}} \, dx &=\frac {\sqrt {3+x} \int \frac {1}{\sqrt {-1-x} \sqrt {\frac {3}{5}+\frac {x}{5}} \sqrt {-2+x}} \, dx}{\sqrt {5} \sqrt {-3-x}}\\ &=\frac {\left (\sqrt {1+x} \sqrt {3+x}\right ) \int \frac {1}{\sqrt {\frac {3}{5}+\frac {x}{5}} \sqrt {\frac {1}{3}+\frac {x}{3}} \sqrt {-2+x}} \, dx}{\sqrt {15} \sqrt {-3-x} \sqrt {-1-x}}\\ &=-\frac {2 \sqrt {1+x} \sqrt {3+x} F\left (\sin ^{-1}\left (\frac {1}{\sqrt {\frac {3}{5}+\frac {x}{5}}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-3-x} \sqrt {-1-x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.51, size = 75, normalized size = 1.32 \begin {gather*} \frac {2 i \sqrt {1+\frac {3}{-2+x}} \sqrt {1+\frac {5}{-2+x}} (-2+x) F\left (i \sinh ^{-1}\left (\frac {\sqrt {3}}{\sqrt {-2+x}}\right )|\frac {5}{3}\right )}{\sqrt {-15-3 (-2+x)} \sqrt {-1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[1 + 3/(-2 + x)]*Sqrt[1 + 5/(-2 + x)]*(-2 + x)*EllipticF[I*ArcSinh[Sqrt[3]/Sqrt[-2 + x]], 5/3])/(Sq
rt[-15 - 3*(-2 + x)]*Sqrt[-1 - x])

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Maple [A]
time = 0.10, size = 53, normalized size = 0.93

method	result	size
default	\(-\frac {2 \EllipticF \left (\frac {\sqrt {6+2 x}}{2}, \frac {\sqrt {10}}{5}\right ) \sqrt {2-x}\, \sqrt {5}\, \sqrt {3+x}\, \sqrt {-2+x}\, \sqrt {-3-x}}{5 \left (x^{2}+x -6\right )}\)	\(53\)
elliptic	\(\frac {\sqrt {\left (-2+x \right ) \left (1+x \right ) \left (3+x \right )}\, \sqrt {6+2 x}\, \sqrt {10-5 x}\, \sqrt {-2-2 x}\, \EllipticF \left (\frac {\sqrt {6+2 x}}{2}, \frac {\sqrt {10}}{5}\right )}{5 \sqrt {-3-x}\, \sqrt {-1-x}\, \sqrt {-2+x}\, \sqrt {x^{3}+2 x^{2}-5 x -6}}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*EllipticF(1/2*(6+2*x)^(1/2),1/5*10^(1/2))*(2-x)^(1/2)*5^(1/2)*(3+x)^(1/2)*(-2+x)^(1/2)*(-3-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/(-3-x)^(1/2)/(-1-x)^(1/2)/(-2+x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.32, size = 8, normalized size = 0.14 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (\frac {76}{3}, \frac {224}{27}, x + \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*weierstrassPInverse(76/3, 224/27, x + 2/3)

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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 - 1} \sqrt {x - 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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