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

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 = 0.0379717, 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 = 5.08702, size = 44, normalized size = 0.92 \[ - \frac{2 \left (- 9 x + 6\right ) \log{\left (- 3 x + 2 \right )}}{27 \sqrt{- 9 x^{2} + 12 x - 4}} - \frac{\sqrt{- 9 x^{2} + 12 x - 4}}{9} \]

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 = 0.0191862, size = 35, normalized size = 0.73 \[ \frac{(3 x-2) (3 x+2 \log (2-3 x)-2)}{9 \sqrt{-(2-3 x)^2}} \]

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 = 0.004, size = 31, normalized size = 0.7 \[{\frac{ \left ( -2+3\,x \right ) \left ( 3\,x+2\,\ln \left ( -2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{- \left ( -2+3\,x \right ) ^{2}}}}} \]

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 = 0.821781, size = 28, normalized size = 0.58 \[ -\frac{1}{9} \, \sqrt{-9 \, x^{2} + 12 \, x - 4} + \frac{2}{9} i \, \log \left (x - \frac{2}{3}\right ) \]

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*I*log(x - 2/3)

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Fricas [A]  time = 0.221141, size = 14, normalized size = 0.29 \[ -\frac{1}{3} i \, x - \frac{2}{9} i \, \log \left (x - \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*I*x - 2/9*I*log(x - 2/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- \left (3 x - 2\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(-(3*x - 2)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef