3.29.49 \(\int \frac {1}{\sqrt {-3+x} \sqrt {-2+x} \sqrt {-1+x}} \, dx\) [2849]

Optimal. Leaf size=12 \[ -2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {-1+x}}\right )\right |2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {119} \begin {gather*} -2 F\left (\left .\text {ArcSin}\left (\frac {1}{\sqrt {x-1}}\right )\right |2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*EllipticF[ArcSin[1/Sqrt[-1 + x]], 2]

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)
]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.58, size = 59, normalized size = 4.92 \begin {gather*} \frac {2 i \sqrt {1+\frac {1}{-3+x}} \sqrt {1+\frac {2}{-3+x}} (-3+x) F\left (\left .i \sinh ^{-1}\left (\frac {1}{\sqrt {-3+x}}\right )\right |2\right )}{\sqrt {-2+x} \sqrt {-1+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(11)=22\).
time = 0.12, size = 44, normalized size = 3.67

method	result	size
default	\(-\frac {\EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \left (-1+x \right ) \sqrt {3-x}\, \sqrt {-3+x}}{x^{2}-4 x +3}\)	\(44\)
elliptic	\(-\frac {\sqrt {\left (-1+x \right ) \left (-2+x \right ) \left (-3+x \right )}\, \sqrt {3-x}\, \sqrt {-2+2 x}\, \EllipticF \left (\sqrt {3-x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {-1+x}\, \sqrt {-3+x}\, \sqrt {x^{3}-6 x^{2}+11 x -6}}\)	\(67\)

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]

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

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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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.21, size = 8, normalized size = 0.67 \begin {gather*} 2 \, {\rm weierstrassPInverse}\left (4, 0, x - 2\right ) \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]

2*weierstrassPInverse(4, 0, x - 2)

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Sympy [C] Result contains complex when optimal does not.
time = 12.26, size = 65, normalized size = 5.42 \begin {gather*} - \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{2 i \pi }}{\left (x - 2\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \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]

-meijerg(((1/2, 1, 1), (3/4, 3/4, 5/4)), ((1/4, 1/2, 3/4, 1, 5/4), (0,)), (x - 2)**(-2))/(4*pi**(3/2)) + meije
rg(((-1/4, 0, 1/4, 1/2, 3/4), (1,)), ((0, 1/2, 0), (-1/4, 1/4, 1/4)), exp_polar(2*I*pi)/(x - 2)**2)/(4*pi**(3/
2))

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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 = -0.08 \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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