3.216 \(\int \frac{x}{\sqrt{4-12 x+9 x^2}} \, dx\)

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

[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])

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Rubi [A]  time = 0.0378473, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{1}{9} \sqrt{9 x^2-12 x+4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{9 x^2-12 x+4}} \]

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])

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Rubi in Sympy [A]  time = 4.87924, size = 42, normalized size = 0.88 \[ - \frac{2 \left (- 9 x + 6\right ) \log{\left (- 3 x + 2 \right )}}{27 \sqrt{9 x^{2} - 12 x + 4}} + \frac{\sqrt{9 x^{2} - 12 x + 4}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/((-2+3*x)**2)**(1/2),x)

[Out]

-2*(-9*x + 6)*log(-3*x + 2)/(27*sqrt(9*x**2 - 12*x + 4)) + sqrt(9*x**2 - 12*x +
4)/9

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Mathematica [A]  time = 0.0276005, size = 33, normalized size = 0.69 \[ \frac{(3 x-2) (3 x+2 \log (3 x-2)-2)}{9 \sqrt{(2-3 x)^2}} \]

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 = 0.006, size = 29, normalized size = 0.6 \[{\frac{ \left ( -2+3\,x \right ) \left ( 3\,x+2\,\ln \left ( -2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{ \left ( -2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/((-2+3*x)^2)^(1/2),x)

[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 = 0.834375, size = 28, normalized size = 0.58 \[ \frac{1}{9} \, \sqrt{9 \, x^{2} - 12 \, x + 4} + \frac{2}{9} \, \log \left (x - \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt((3*x - 2)^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 = 0.225078, size = 16, normalized size = 0.33 \[ \frac{1}{3} \, x + \frac{2}{9} \, \log \left (3 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt((3*x - 2)^2),x, algorithm="fricas")

[Out]

1/3*x + 2/9*log(3*x - 2)

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Sympy [A]  time = 0.166254, size = 12, normalized size = 0.25 \[ \frac{x}{3} + \frac{2 \log{\left (3 x - 2 \right )}}{9} \]

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/XCAS [A]  time = 0.20659, size = 34, normalized size = 0.71 \[ \frac{1}{3} \, x{\rm sign}\left (3 \, x - 2\right ) + \frac{2}{9} \,{\rm ln}\left ({\left | 3 \, x - 2 \right |}\right ){\rm sign}\left (3 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt((3*x - 2)^2),x, algorithm="giac")

[Out]

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