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

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 = 0.0049185, 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, Rules used = {650} \[ -\frac{\sqrt{6 x-x^2}}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[6*x - x^2]),x]

[Out]

-Sqrt[6*x - x^2]/(3*x)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{6 x-x^2}} \, dx &=-\frac{\sqrt{6 x-x^2}}{3 x}\\ \end{align*}

Mathematica [A]  time = 0.0057691, 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 = 0.002, size = 17, normalized size = 0.9 \begin{align*}{\frac{-6+x}{3}{\frac{1}{\sqrt{-{x}^{2}+6\,x}}}} \end{align*}

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 = 1.4938, size = 22, normalized size = 1.1 \begin{align*} -\frac{\sqrt{-x^{2} + 6 \, x}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(-x^2 + 6*x)/x

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Fricas [A]  time = 1.45785, size = 34, normalized size = 1.7 \begin{align*} -\frac{\sqrt{-x^{2} + 6 \, x}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-x^2+6*x)^(1/2),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. \begin{align*} \int \frac{1}{x \sqrt{- x \left (x - 6\right )}}\, dx \end{align*}

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 [A]  time = 1.11879, size = 34, normalized size = 1.7 \begin{align*} \frac{2}{3 \,{\left (\frac{\sqrt{-x^{2} + 6 \, x} - 3}{x - 3} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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