Integrand size = 15, antiderivative size = 18 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2 \left (\frac {b+a x}{x}\right )^{5/2}}{5 b} \]
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Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \left (a +\frac {b}{x}\right )^{\frac {5}{2}}}{5 b}\) | \(15\) |
gosper | \(-\frac {2 \left (a x +b \right ) \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}{5 x b}\) | \(25\) |
risch | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a^{2} x^{2}+2 a b x +b^{2}\right )}{5 x^{2} b}\) | \(36\) |
trager | \(-\frac {2 \left (a^{2} x^{2}+2 a b x +b^{2}\right ) \sqrt {-\frac {-a x -b}{x}}}{5 x^{2} b}\) | \(40\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (a \,x^{2}+b x \right )^{\frac {3}{2}} \left (a x +b \right )}{5 x^{3} b \sqrt {x \left (a x +b \right )}}\) | \(45\) |
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{5 \, b x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (14) = 28\).
Time = 0.57 (sec) , antiderivative size = 65, normalized size of antiderivative = 3.61 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=- \frac {2 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}}{5 b} - \frac {4 a^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}}{5 x} - \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x}}}{5 x^{2}} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (14) = 28\).
Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 8.06 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=\frac {2 \, {\left (5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} \mathrm {sgn}\left (x\right ) + 10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b \mathrm {sgn}\left (x\right ) + 10 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} \mathrm {sgn}\left (x\right ) + 5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} \mathrm {sgn}\left (x\right ) + b^{4} \mathrm {sgn}\left (x\right )\right )}}{5 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{5}} \]
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Time = 6.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{x^2} \, dx=-\frac {2\,\sqrt {a+\frac {b}{x}}\,{\left (b+a\,x\right )}^2}{5\,b\,x^2} \]
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