3.805 \(\int \frac{1}{x \sqrt{6 x-x^2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{\sqrt{6 x-x^2}}{3 x} \]

[Out]

-Sqrt[6*x - x^2]/(3*x)

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Rubi [A]  time = 0.0191091, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{\sqrt{6 x-x^2}}{3 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[6*x - x^2]),x]

[Out]

-Sqrt[6*x - x^2]/(3*x)

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Rubi in Sympy [A]  time = 1.7664, size = 14, normalized size = 0.7 \[ - \frac{\sqrt{- x^{2} + 6 x}}{3 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(-x**2+6*x)**(1/2),x)

[Out]

-sqrt(-x**2 + 6*x)/(3*x)

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Mathematica [A]  time = 0.0135887, size = 17, normalized size = 0.85 \[ \frac{x-6}{3 \sqrt{-(x-6) x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[6*x - x^2]),x]

[Out]

(-6 + x)/(3*Sqrt[-((-6 + x)*x)])

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Maple [A]  time = 0.004, size = 17, normalized size = 0.9 \[{\frac{-6+x}{3}{\frac{1}{\sqrt{-{x}^{2}+6\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^2+6*x)^(1/2),x)

[Out]

1/3*(-6+x)/(-x^2+6*x)^(1/2)

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Maxima [A]  time = 0.783018, size = 22, normalized size = 1.1 \[ -\frac{\sqrt{-x^{2} + 6 \, x}}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 6*x)*x),x, algorithm="maxima")

[Out]

-1/3*sqrt(-x^2 + 6*x)/x

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Fricas [A]  time = 0.261602, size = 22, normalized size = 1.1 \[ -\frac{\sqrt{-x^{2} + 6 \, x}}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 6*x)*x),x, algorithm="fricas")

[Out]

-1/3*sqrt(-x^2 + 6*x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{- x \left (x - 6\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(-x**2+6*x)**(1/2),x)

[Out]

Integral(1/(x*sqrt(-x*(x - 6))), x)

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GIAC/XCAS [A]  time = 0.26747, size = 34, normalized size = 1.7 \[ \frac{2}{3 \,{\left (\frac{\sqrt{-x^{2} + 6 \, x} - 3}{x - 3} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^2 + 6*x)*x),x, algorithm="giac")

[Out]

2/3/((sqrt(-x^2 + 6*x) - 3)/(x - 3) - 1)