Integrand size = 16, antiderivative size = 17 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {2-b x}}{\sqrt {x}} \]
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Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {2-b x}}{\sqrt {x}} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2-b x}}{\sqrt {x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {2-b x}}{\sqrt {x}} \]
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Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {\sqrt {-b x +2}}{\sqrt {x}}\) | \(14\) |
default | \(-\frac {\sqrt {-b x +2}}{\sqrt {x}}\) | \(14\) |
meijerg | \(-\frac {\sqrt {2}\, \sqrt {-\frac {b x}{2}+1}}{\sqrt {x}}\) | \(17\) |
risch | \(\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}\) | \(38\) |
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none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {-b x + 2}}{\sqrt {x}} \]
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Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=\begin {cases} - \sqrt {b} \sqrt {-1 + \frac {2}{b x}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- i \sqrt {b} \sqrt {1 - \frac {2}{b x}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {-b x + 2}}{\sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (13) = 26\).
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {-b x + 2} b^{2}}{\sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} \]
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Time = 0.35 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx=-\frac {\sqrt {2-b\,x}}{\sqrt {x}} \]
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