Integrand size = 18, antiderivative size = 12 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (9+x \left (\frac {2}{5}+2 x\right )\right ) \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {642} \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (10 x^2+2 x+45\right ) \]
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Rule 642
Rubi steps \begin{align*} \text {integral}& = \log \left (45+2 x+10 x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (45+2 x+10 x^2\right ) \]
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Time = 1.66 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\ln \left (x^{2}+\frac {1}{5} x +\frac {9}{2}\right )\) | \(10\) |
derivativedivides | \(\ln \left (10 x^{2}+2 x +45\right )\) | \(12\) |
default | \(\ln \left (10 x^{2}+2 x +45\right )\) | \(12\) |
norman | \(\ln \left (10 x^{2}+2 x +45\right )\) | \(12\) |
risch | \(\ln \left (10 x^{2}+2 x +45\right )\) | \(12\) |
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none
Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (10 \, x^{2} + 2 \, x + 45\right ) \]
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Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log {\left (10 x^{2} + 2 x + 45 \right )} \]
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none
Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (10 \, x^{2} + 2 \, x + 45\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\log \left (10 \, x^{2} + 2 \, x + 45\right ) \]
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Time = 0.04 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \frac {2+20 x}{45+2 x+10 x^2} \, dx=\ln \left (x^2+\frac {x}{5}+\frac {9}{2}\right ) \]
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