3.3.16 \(\int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx\) [216]

Optimal. Leaf size=48 \[ \frac {1}{9} \sqrt {4-12 x+9 x^2}-\frac {2 (2-3 x) \log (2-3 x)}{9 \sqrt {4-12 x+9 x^2}} \]

[Out]

-2/9*(2-3*x)*ln(2-3*x)/((-2+3*x)^2)^(1/2)+1/9*((-2+3*x)^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 622, 31} \begin {gather*} \frac {1}{9} \sqrt {9 x^2-12 x+4}-\frac {2 (2-3 x) \log (2-3 x)}{9 \sqrt {9 x^2-12 x+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

Sqrt[4 - 12*x + 9*x^2]/9 - (2*(2 - 3*x)*Log[2 - 3*x])/(9*Sqrt[4 - 12*x + 9*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 622

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 654

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

Rubi steps

\begin {align*} \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx &=\frac {1}{9} \sqrt {4-12 x+9 x^2}+\frac {2}{3} \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx\\ &=\frac {1}{9} \sqrt {4-12 x+9 x^2}+\frac {(2 (-6+9 x)) \int \frac {1}{-6+9 x} \, dx}{3 \sqrt {4-12 x+9 x^2}}\\ &=\frac {1}{9} \sqrt {4-12 x+9 x^2}-\frac {2 (2-3 x) \log (2-3 x)}{9 \sqrt {4-12 x+9 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.69 \begin {gather*} \frac {(-2+3 x) (-2+3 x+2 \log (-2+3 x))}{9 \sqrt {(2-3 x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

((-2 + 3*x)*(-2 + 3*x + 2*Log[-2 + 3*x]))/(9*Sqrt[(2 - 3*x)^2])

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Maple [A]
time = 0.40, size = 29, normalized size = 0.60

method	result	size
default	\(\frac {\left (-2+3 x \right ) \left (3 x +2 \ln \left (-2+3 x \right )\right )}{9 \sqrt {\left (-2+3 x \right )^{2}}}\)	\(29\)
risch	\(\frac {\sqrt {\left (-2+3 x \right )^{2}}\, x}{-6+9 x}+\frac {2 \sqrt {\left (-2+3 x \right )^{2}}\, \ln \left (-2+3 x \right )}{9 \left (-2+3 x \right )}\)	\(45\)
meijerg	\(\frac {-\frac {2 x}{3}-\frac {4 \ln \left (1-\frac {3 x}{2}\right )}{9}}{\sqrt {\left (-2+3 x \right )^{2}}}-\frac {2 x \left (-\frac {3 x}{2}-\ln \left (1-\frac {3 x}{2}\right )\right )}{3 \sqrt {\left (-2+3 x \right )^{2}}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((-2+3*x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(-2+3*x)*(3*x+2*ln(-2+3*x))/((-2+3*x)^2)^(1/2)

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Maxima [A]
time = 0.49, size = 21, normalized size = 0.44 \begin {gather*} \frac {1}{9} \, \sqrt {9 \, x^{2} - 12 \, x + 4} + \frac {2}{9} \, \log \left (x - \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((-2+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/9*sqrt(9*x^2 - 12*x + 4) + 2/9*log(x - 2/3)

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Fricas [A]
time = 2.10, size = 12, normalized size = 0.25 \begin {gather*} \frac {1}{3} \, x + \frac {2}{9} \, \log \left (3 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((-2+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*x + 2/9*log(3*x - 2)

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Sympy [A]
time = 0.02, size = 12, normalized size = 0.25 \begin {gather*} \frac {x}{3} + \frac {2 \log {\left (3 x - 2 \right )}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((-2+3*x)**2)**(1/2),x)

[Out]

x/3 + 2*log(3*x - 2)/9

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Giac [A]
time = 1.43, size = 25, normalized size = 0.52 \begin {gather*} \frac {1}{3} \, x \mathrm {sgn}\left (3 \, x - 2\right ) + \frac {2}{9} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm {sgn}\left (3 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/((-2+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*x*sgn(3*x - 2) + 2/9*log(abs(3*x - 2))*sgn(3*x - 2)

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Mupad [B]
time = 0.26, size = 32, normalized size = 0.67 \begin {gather*} \frac {2\,\ln \left (x+\frac {\sqrt {{\left (3\,x-2\right )}^2}}{3}-\frac {2}{3}\right )}{9}+\frac {\sqrt {9\,x^2-12\,x+4}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((3*x - 2)^2)^(1/2),x)

[Out]

(2*log(x + ((3*x - 2)^2)^(1/2)/3 - 2/3))/9 + (9*x^2 - 12*x + 4)^(1/2)/9

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