Optimal. Leaf size=48 \[ \frac{1}{9} \sqrt{9 x^2-12 x+4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{9 x^2-12 x+4}} \]
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Rubi [A] time = 0.0109042, 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, Rules used = {640, 608, 31} \[ \frac{1}{9} \sqrt{9 x^2-12 x+4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{9 x^2-12 x+4}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 608
Rule 31
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*}
Mathematica [A] time = 0.0162329, 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.
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Maple [A] time = 0.095, size = 29, normalized size = 0.6 \begin{align*}{\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}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75691, size = 28, normalized size = 0.58 \begin{align*} \frac{1}{9} \, \sqrt{9 \, x^{2} - 12 \, x + 4} + \frac{2}{9} \, \log \left (x - \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58107, size = 35, normalized size = 0.73 \begin{align*} \frac{1}{3} \, x + \frac{2}{9} \, \log \left (3 \, x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.250986, size = 12, normalized size = 0.25 \begin{align*} \frac{x}{3} + \frac{2 \log{\left (3 x - 2 \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23288, size = 34, normalized size = 0.71 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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