Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=\frac {1}{2 b \left (a+\frac {b}{x}\right )^2} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=\frac {1}{2 b \left (a+\frac {b}{x}\right )^2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 b \left (a+\frac {b}{x}\right )^2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=-\frac {b+2 a x}{2 a^2 (b+a x)^2} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {1}{2 b \left (a +\frac {b}{x}\right )^{2}}\) | \(15\) |
gosper | \(-\frac {2 a x +b}{2 \left (a x +b \right )^{2} a^{2}}\) | \(19\) |
parallelrisch | \(\frac {-2 a x -b}{2 a^{2} \left (a x +b \right )^{2}}\) | \(21\) |
risch | \(\frac {-\frac {x}{a}-\frac {b}{2 a^{2}}}{\left (a x +b \right )^{2}}\) | \(22\) |
default | \(-\frac {1}{\left (a x +b \right ) a^{2}}+\frac {b}{2 a^{2} \left (a x +b \right )^{2}}\) | \(27\) |
norman | \(\frac {-\frac {x^{2}}{a}-\frac {b x}{2 a^{2}}}{x \left (a x +b \right )^{2}}\) | \(28\) |
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (14) = 28\).
Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=-\frac {2 \, a x + b}{2 \, {\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (10) = 20\).
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=\frac {- 2 a x - b}{2 a^{4} x^{2} + 4 a^{3} b x + 2 a^{2} b^{2}} \]
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none
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=\frac {1}{2 \, {\left (a + \frac {b}{x}\right )}^{2} b} \]
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none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=\frac {1}{2 \, {\left (a + \frac {b}{x}\right )}^{2} b} \]
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Time = 5.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^2} \, dx=-\frac {\frac {b}{2\,a^2}+\frac {x}{a}}{a^2\,x^2+2\,a\,b\,x+b^2} \]
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