3.10.47 \(\int \frac {1}{x \sqrt {6 x-x^2}} \, dx\) [947]

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 = {664} \begin {gather*} -\frac {\sqrt {6 x-x^2}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[6*x - x^2]),x]

[Out]

-1/3*Sqrt[6*x - x^2]/x

Rule 664

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 \begin {gather*} \frac {-6+x}{3 \sqrt {-((-6+x) x)}} \end {gather*}

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 = 0.23, size = 17, normalized size = 0.85

method	result	size
risch	\(\frac {-6+x}{3 \sqrt {-x \left (-6+x \right )}}\)	\(14\)
gosper	\(\frac {-6+x}{3 \sqrt {-x^{2}+6 x}}\)	\(17\)
default	\(-\frac {\sqrt {-x^{2}+6 x}}{3 x}\)	\(17\)
trager	\(-\frac {\sqrt {-x^{2}+6 x}}{3 x}\)	\(17\)
meijerg	\(-\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {1-\frac {x}{6}}}{3 \sqrt {x}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 16, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {-x^{2} + 6 \, x}}{3 \, x} \end {gather*}

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 = 0.36, size = 16, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {-x^{2} + 6 \, x}}{3 \, x} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- x \left (x - 6\right )}}\, dx \end {gather*}

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 = 4.96, size = 25, normalized size = 1.25 \begin {gather*} \frac {2}{3 \, {\left (\frac {\sqrt {-x^{2} + 6 \, x} - 3}{x - 3} - 1\right )}} \end {gather*}

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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Mupad [B]
time = 3.51, size = 16, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {6\,x-x^2}}{3\,x} \end {gather*}

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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