Integrand size = 13, antiderivative size = 16 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {\left (a+\frac {b}{x}\right )^4}{4 b} \]
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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} \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {\left (a+\frac {b}{x}\right )^4}{4 b} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+\frac {b}{x}\right )^4}{4 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(39\) vs. \(2(16)=32\).
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {b^3}{4 x^4}-\frac {a b^2}{x^3}-\frac {3 a^2 b}{2 x^2}-\frac {a^3}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(14)=28\).
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12
method | result | size |
gosper | \(-\frac {4 a^{3} x^{3}+6 a^{2} b \,x^{2}+4 a \,b^{2} x +b^{3}}{4 x^{4}}\) | \(34\) |
norman | \(\frac {-a^{3} x^{3}-\frac {3}{2} a^{2} b \,x^{2}-a \,b^{2} x -\frac {1}{4} b^{3}}{x^{4}}\) | \(35\) |
risch | \(\frac {-a^{3} x^{3}-\frac {3}{2} a^{2} b \,x^{2}-a \,b^{2} x -\frac {1}{4} b^{3}}{x^{4}}\) | \(35\) |
default | \(-\frac {a \,b^{2}}{x^{3}}-\frac {a^{3}}{x}-\frac {3 a^{2} b}{2 x^{2}}-\frac {b^{3}}{4 x^{4}}\) | \(36\) |
parallelrisch | \(\frac {-4 a^{3} x^{3}-6 a^{2} b \,x^{2}-4 a \,b^{2} x -b^{3}}{4 x^{4}}\) | \(36\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (14) = 28\).
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {4 \, a^{3} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a b^{2} x + b^{3}}{4 \, x^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (10) = 20\).
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=\frac {- 4 a^{3} x^{3} - 6 a^{2} b x^{2} - 4 a b^{2} x - b^{3}}{4 x^{4}} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {{\left (a + \frac {b}{x}\right )}^{4}}{4 \, b} \]
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none
Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {{\left (a + \frac {b}{x}\right )}^{4}}{4 \, b} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^2} \, dx=-\frac {a^3\,x^3+\frac {3\,a^2\,b\,x^2}{2}+a\,b^2\,x+\frac {b^3}{4}}{x^4} \]
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