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]

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.68, size = 16, normalized size = 0.80 \[ -\frac {\sqrt {-x^{2} + 6 \, x}}{3 \, x} \]

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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giac [A]  time = 0.41, size = 25, normalized size = 1.25 \[ \frac {2}{3 \, {\left (\frac {\sqrt {-x^{2} + 6 \, x} - 3}{x - 3} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 17, normalized size = 0.85 \[ \frac {x -6}{3 \sqrt {-x^{2}+6 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-x^2+6*x)^(1/2),x)

[Out]

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

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maxima [A]  time = 1.43, size = 16, normalized size = 0.80 \[ -\frac {\sqrt {-x^{2} + 6 \, x}}{3 \, x} \]

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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mupad [B]  time = 3.51, size = 16, normalized size = 0.80 \[ -\frac {\sqrt {6\,x-x^2}}{3\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(6*x - x^2)^(1/2)),x)

[Out]

-(6*x - x^2)^(1/2)/(3*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- x \left (x - 6\right )}}\, dx \]

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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