Integrand size = 11, antiderivative size = 16 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {-25+4 x^2}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {197} \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {4 x^2-25}} \]
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Rule 197
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{25 \sqrt {-25+4 x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \sqrt {-25+4 x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {x}{25 \sqrt {4 x^{2}-25}}\) | \(13\) |
trager | \(-\frac {x}{25 \sqrt {4 x^{2}-25}}\) | \(13\) |
risch | \(-\frac {x}{25 \sqrt {4 x^{2}-25}}\) | \(13\) |
pseudoelliptic | \(-\frac {x}{25 \sqrt {4 x^{2}-25}}\) | \(13\) |
gosper | \(-\frac {\left (2 x -5\right ) \left (5+2 x \right ) x}{25 \left (4 x^{2}-25\right )^{\frac {3}{2}}}\) | \(23\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (-1+\frac {4 x^{2}}{25}\right )\right )}^{\frac {3}{2}} x}{125 \operatorname {signum}\left (-1+\frac {4 x^{2}}{25}\right )^{\frac {3}{2}} \sqrt {1-\frac {4 x^{2}}{25}}}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {4 \, x^{2} + 2 \, \sqrt {4 \, x^{2} - 25} x - 25}{50 \, {\left (4 \, x^{2} - 25\right )}} \]
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Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.12 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=\begin {cases} - \frac {x}{25 \sqrt {4 x^{2} - 25}} & \text {for}\: \left |{x^{2}}\right | > \frac {25}{4} \\\frac {i x}{25 \sqrt {25 - 4 x^{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.44 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]
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none
Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25 \, \sqrt {4 \, x^{2} - 25}} \]
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Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (-25+4 x^2\right )^{3/2}} \, dx=-\frac {x}{25\,\sqrt {4\,x^2-25}} \]
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