Integrand size = 15, antiderivative size = 18 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=-\frac {6-x}{2 \sqrt {4+x^2}} \]
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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 = {651} \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=-\frac {6-x}{2 \sqrt {x^2+4}} \]
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Rule 651
Rubi steps \begin{align*} \text {integral}& = -\frac {6-x}{2 \sqrt {4+x^2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {-6+x}{2 \sqrt {4+x^2}} \]
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Time = 2.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {-6+x}{2 \sqrt {x^{2}+4}}\) | \(13\) |
trager | \(\frac {-6+x}{2 \sqrt {x^{2}+4}}\) | \(13\) |
risch | \(\frac {-6+x}{2 \sqrt {x^{2}+4}}\) | \(13\) |
default | \(\frac {x}{2 \sqrt {x^{2}+4}}-\frac {3}{\sqrt {x^{2}+4}}\) | \(21\) |
meijerg | \(\frac {x}{4 \sqrt {1+\frac {x^{2}}{4}}}+\frac {\frac {3 \sqrt {\pi }}{2}-\frac {3 \sqrt {\pi }}{2 \sqrt {1+\frac {x^{2}}{4}}}}{\sqrt {\pi }}\) | \(37\) |
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x^{2} + \sqrt {x^{2} + 4} {\left (x - 6\right )} + 4}{2 \, {\left (x^{2} + 4\right )}} \]
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Time = 1.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x}{2 \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]
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none
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x}{2 \, \sqrt {x^{2} + 4}} - \frac {3}{\sqrt {x^{2} + 4}} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x - 6}{2 \, \sqrt {x^{2} + 4}} \]
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Time = 9.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {2+3 x}{\left (4+x^2\right )^{3/2}} \, dx=\frac {x-6}{2\,\sqrt {x^2+4}} \]
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