Integrand size = 15, antiderivative size = 16 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+\frac {b}{x^2}}}{b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {\sqrt {a +\frac {b}{x^{2}}}}{b}\) | \(15\) |
trager | \(-\frac {\sqrt {-\frac {-a \,x^{2}-b}{x^{2}}}}{b}\) | \(23\) |
gosper | \(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(29\) |
default | \(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(29\) |
risch | \(-\frac {a \,x^{2}+b}{x^{2} b \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}\) | \(29\) |
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none
Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {\frac {a x^{2} + b}{x^{2}}}}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=\begin {cases} - \frac {\sqrt {a + \frac {b}{x^{2}}}}{b} & \text {for}\: b \neq 0 \\- \frac {1}{2 \sqrt {a} x^{2}} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a + \frac {b}{x^{2}}}}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (14) = 28\).
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=\frac {2 \, \sqrt {a}}{{\left ({\left (\sqrt {a} x - \sqrt {a x^{2} + b}\right )}^{2} - b\right )} \mathrm {sgn}\left (x\right )} \]
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Time = 6.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^3} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}}}{b} \]
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