Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {(2+b x)^{3/2}}{3 x^{3/2}} \]
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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} \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {(b x+2)^{3/2}}{3 x^{3/2}} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {(2+b x)^{3/2}}{3 x^{3/2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {(2+b x)^{3/2}}{3 x^{3/2}} \]
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Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(-\frac {\left (b x +2\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\) | \(13\) |
meijerg | \(-\frac {2 \sqrt {2}\, \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}{3 x^{\frac {3}{2}}}\) | \(17\) |
risch | \(-\frac {b^{2} x^{2}+4 b x +4}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\) | \(26\) |
default | \(-\frac {2 \sqrt {b x +2}}{3 x^{\frac {3}{2}}}-\frac {b \sqrt {b x +2}}{3 \sqrt {x}}\) | \(27\) |
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none
Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
Time = 0.87 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=- \frac {b^{\frac {3}{2}} \sqrt {1 + \frac {2}{b x}}}{3} - \frac {2 \sqrt {b} \sqrt {1 + \frac {2}{b x}}}{3 x} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {{\left (b x + 2\right )}^{\frac {3}{2}} b^{4}}{3 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}} {\left | b \right |}} \]
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {2+b x}}{x^{5/2}} \, dx=-\frac {\sqrt {b\,x+2}\,\left (\frac {b\,x}{3}+\frac {2}{3}\right )}{x^{3/2}} \]
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