Integrand size = 13, antiderivative size = 13 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{b \left (a+\frac {b}{x}\right )} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{b \left (a+\frac {b}{x}\right )} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{b \left (a+\frac {b}{x}\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a (b+a x)} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00
method | result | size |
gosper | \(-\frac {1}{\left (a x +b \right ) a}\) | \(13\) |
default | \(-\frac {1}{\left (a x +b \right ) a}\) | \(13\) |
norman | \(\frac {x}{b \left (a x +b \right )}\) | \(13\) |
risch | \(-\frac {1}{\left (a x +b \right ) a}\) | \(13\) |
parallelrisch | \(-\frac {1}{\left (a x +b \right ) a}\) | \(13\) |
derivativedivides | \(\frac {1}{b \left (a +\frac {b}{x}\right )}\) | \(14\) |
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none
Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a^{2} x + a b} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=- \frac {1}{a^{2} x + a b} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{{\left (a + \frac {b}{x}\right )} b} \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=\frac {1}{{\left (a + \frac {b}{x}\right )} b} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^2} \, dx=-\frac {1}{a\,\left (b+a\,x\right )} \]
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