3.2868 \(\int \frac{1}{\sqrt{4-x} \sqrt{5-x} \sqrt{-3+x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0423021, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{x-3}\right )|\frac{1}{2}\right ) \]

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]

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Rubi in Sympy [A]  time = 4.55227, size = 15, normalized size = 0.83 \[ \sqrt{2} F\left (\operatorname{asin}{\left (\sqrt{x - 3} \right )}\middle | \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4-x)**(1/2)/(5-x)**(1/2)/(-3+x)**(1/2),x)

[Out]

sqrt(2)*elliptic_f(asin(sqrt(x - 3)), 1/2)

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Mathematica [B]  time = 0.137452, size = 46, normalized size = 2.56 \[ \frac{2 \sqrt{-x^2+8 x-15} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{4-x}}\right )\right |-1\right )}{\sqrt{1-\frac{1}{(x-4)^2}} (x-4)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[-15 + 8*x - x^2]*EllipticF[ArcSin[1/Sqrt[4 - x]], -1])/(Sqrt[1 - (-4 + x
)^(-2)]*(-4 + x))

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Maple [C]  time = 0.085, size = 13, normalized size = 0.7 \[ -2\,{\it EllipticF} \left ( \sqrt{4-x},i \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x - 3} \sqrt{-x + 5} \sqrt{-x + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.0654, 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)) - meijerg(((-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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x - 3} \sqrt{-x + 5} \sqrt{-x + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x - 3)*sqrt(-x + 5)*sqrt(-x + 4)),x, algorithm="giac")

[Out]

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