3.2885 \(\int \frac {1}{\sqrt {4-x} \sqrt {5-x} \sqrt {-3+x}} \, dx\)

Optimal. Leaf size=18 \[ \sqrt {2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {x-3}\right ),\frac {1}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, 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} \[ \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {x-3}\right )|\frac {1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2]*EllipticF[ArcSin[Sqrt[-3 + x]], 1/2]

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

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Mathematica [C]  time = 0.02, size = 28, normalized size = 1.56 \[ -2 \sqrt {4-x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(x-4)^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[4 - x]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, (-4 + x)^2]

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fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x - 3} \sqrt {-x + 5} \sqrt {-x + 4}}{x^{3} - 12 \, x^{2} + 47 \, x - 60}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/(5-x)^(1/2)/(-3+x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)/(x^3 - 12*x^2 + 47*x - 60), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 3} \sqrt {-x + 5} \sqrt {-x + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/(5-x)^(1/2)/(-3+x)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)), x)

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maple [C]  time = 0.04, size = 13, normalized size = 0.72 \[ -2 \EllipticF \left (\sqrt {-x +4}, i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x+4)^(1/2)/(5-x)^(1/2)/(x-3)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x - 3} \sqrt {-x + 5} \sqrt {-x + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)^(1/2)/(5-x)^(1/2)/(-3+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {x-3}\,\sqrt {4-x}\,\sqrt {5-x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x - 3)^(1/2)*(4 - x)^(1/2)*(5 - x)^(1/2)),x)

[Out]

int(1/((x - 3)^(1/2)*(4 - x)^(1/2)*(5 - x)^(1/2)), x)

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sympy [C]  time = 8.07, size = 66, normalized size = 3.67 \[ \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{\left (x - 4\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{\left (x - 4\right )^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-x)**(1/2)/(5-x)**(1/2)/(-3+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 - 4)**(-2))/(4*pi**(3/2)) - meijer
g(((-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 - 4)**2)/(4*pi**(3/
2))

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