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

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

[Out]

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

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Rubi [A]  time = 0.0053562, 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, 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*}

Mathematica [C]  time = 0.017986, size = 28, normalized size = 1.56 \[ -2 \sqrt{4-x} \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},(x-4)^2\right ) \]

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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Maple [C]  time = 0.031, size = 13, normalized size = 0.7 \begin{align*} -2\,{\it EllipticF} \left ( \sqrt{-x+4},i \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x - 3} \sqrt{-x + 5} \sqrt{-x + 4}}{x^{3} - 12 \, x^{2} + 47 \, x - 60}, x\right ) \end{align*}

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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Sympy [C]  time = 7.57268, size = 66, normalized size = 3.67 \begin{align*} \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}}} \end{align*}

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

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)