3.2869 \(\int \frac{1}{\sqrt{4-x} \sqrt{(5-x) (-3+x)}} \, dx\)

Optimal. Leaf size=14 \[ -2 F\left (\left .\sin ^{-1}\left (\sqrt{4-x}\right )\right |-1\right ) \]

[Out]

-2*EllipticF[ArcSin[Sqrt[4 - x]], -1]

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Rubi [A]  time = 0.0649002, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ -2 F\left (\left .\sin ^{-1}\left (\sqrt{4-x}\right )\right |-1\right ) \]

Antiderivative was successfully verified.

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

[Out]

-2*EllipticF[ArcSin[Sqrt[4 - x]], -1]

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Rubi in Sympy [A]  time = 8.49995, size = 14, normalized size = 1. \[ - 2 F\left (\operatorname{asin}{\left (\sqrt{- x + 4} \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*elliptic_f(asin(sqrt(-x + 4)), -1)

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Mathematica [B]  time = 0.0248544, size = 46, normalized size = 3.29 \[ \frac{2 \sqrt{-x^2+8 x-15} F\left (\left .\sin ^{-1}\left (\frac{1}{\sqrt{4-x}}\right )\right |-1\right )}{\sqrt{1-\frac{1}{(x-4)^2}} (x-4)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[4 - x]*Sqrt[(5 - x)*(-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 [B]  time = 0.023, size = 35, normalized size = 2.5 \[ -2\,{\frac{{\it EllipticF} \left ( \sqrt{4-x},i \right ) \sqrt{5-x}\sqrt{-3+x}}{\sqrt{- \left ( -5+x \right ) \left ( -3+x \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*EllipticF((4-x)^(1/2),I)*(5-x)^(1/2)*(-3+x)^(1/2)/(-(-5+x)*(-3+x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{2} + 8 \, x - 15} \sqrt{-x + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^2 + 8*x - 15)*sqrt(-x + 4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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