Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (2+b x)^{3/2}} \]
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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 = {37} \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (b x+2)^{3/2}} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2}}{3 (2+b x)^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (2+b x)^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(\frac {x^{\frac {3}{2}}}{3 \left (b x +2\right )^{\frac {3}{2}}}\) | \(13\) |
meijerg | \(\frac {x^{\frac {3}{2}} \sqrt {2}}{12 \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}\) | \(17\) |
default | \(-\frac {\sqrt {x}}{b \left (b x +2\right )^{\frac {3}{2}}}+\frac {\frac {\sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {b x +2}}}{b}\) | \(46\) |
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {\sqrt {b x + 2} x^{\frac {3}{2}}}{3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \]
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Time = 1.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{\frac {3}{2}}}{3 b x \sqrt {b x + 2} + 6 \sqrt {b x + 2}} \]
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none
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{\frac {3}{2}}}{3 \, {\left (b x + 2\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 4.56 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {4 \, {\left (3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt {b} + 4 \, b^{\frac {5}{2}}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}} \]
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Time = 0.42 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {x}}{(2+b x)^{5/2}} \, dx=\frac {x^{3/2}}{3\,{\left (b\,x+2\right )}^{3/2}} \]
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