Integrand size = 13, antiderivative size = 16 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2 x}{3}-\frac {1}{9} \log (2+3 x) \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2 x}{3}-\frac {1}{9} \log (3 x+2) \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{3}-\frac {1}{3 (2+3 x)}\right ) \, dx \\ & = \frac {2 x}{3}-\frac {1}{9} \log (2+3 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {1}{9} (4+6 x-\log (2+3 x)) \]
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Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {2 x}{3}-\frac {\ln \left (\frac {2}{3}+x \right )}{9}\) | \(11\) |
default | \(\frac {2 x}{3}-\frac {\ln \left (2+3 x \right )}{9}\) | \(13\) |
norman | \(\frac {2 x}{3}-\frac {\ln \left (2+3 x \right )}{9}\) | \(13\) |
meijerg | \(-\frac {\ln \left (1+\frac {3 x}{2}\right )}{9}+\frac {2 x}{3}\) | \(13\) |
risch | \(\frac {2 x}{3}-\frac {\ln \left (2+3 x \right )}{9}\) | \(13\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2}{3} \, x - \frac {1}{9} \, \log \left (3 \, x + 2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2 x}{3} - \frac {\log {\left (3 x + 2 \right )}}{9} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2}{3} \, x - \frac {1}{9} \, \log \left (3 \, x + 2\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2}{3} \, x - \frac {1}{9} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {1+2 x}{2+3 x} \, dx=\frac {2\,x}{3}-\frac {\ln \left (x+\frac {2}{3}\right )}{9} \]
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