Integrand size = 16, antiderivative size = 48 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{9} \sqrt {4-12 x+9 x^2}-\frac {2 (2-3 x) \log (2-3 x)}{9 \sqrt {4-12 x+9 x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 622, 31} \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{9} \sqrt {9 x^2-12 x+4}-\frac {2 (2-3 x) \log (2-3 x)}{9 \sqrt {9 x^2-12 x+4}} \]
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Rule 31
Rule 622
Rule 654
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.69 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {(-2+3 x) \left (\frac {x}{3}+\frac {2}{9} \log (2-3 x)\right )}{\sqrt {(-2+3 x)^2}} \]
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Time = 2.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 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 | \(\frac {-\frac {2 x}{3}-\frac {4 \ln \left (1-\frac {3 x}{2}\right )}{9}}{\sqrt {\left (-2+3 x \right )^{2}}}-\frac {2 x \left (-\frac {3 x}{2}-\ln \left (1-\frac {3 x}{2}\right )\right )}{3 \sqrt {\left (-2+3 x \right )^{2}}}\) | \(49\) |
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Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.25 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{3} \, x + \frac {2}{9} \, \log \left (3 \, x - 2\right ) \]
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Time = 0.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {2 \left (x - \frac {2}{3}\right ) \log {\left (x - \frac {2}{3} \right )}}{9 \sqrt {\left (x - \frac {2}{3}\right )^{2}}} + \frac {\sqrt {9 x^{2} - 12 x + 4}}{9} \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.44 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{9} \, \sqrt {9 \, x^{2} - 12 \, x + 4} + \frac {2}{9} \, \log \left (x - \frac {2}{3}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.52 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{3} \, x \mathrm {sgn}\left (3 \, x - 2\right ) + \frac {2}{9} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm {sgn}\left (3 \, x - 2\right ) \]
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Time = 10.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {2\,\ln \left (x+\frac {\sqrt {{\left (3\,x-2\right )}^2}}{3}-\frac {2}{3}\right )}{9}+\frac {\sqrt {9\,x^2-12\,x+4}}{9} \]
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