Integrand size = 16, antiderivative size = 56 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=\frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{x^{3/2} \sqrt {2-b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2-b x}} \, dx \\ & = \frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}+\frac {1}{3} (2 b) \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx \\ & = \frac {1}{x^{3/2} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 x^{3/2}}-\frac {2 b \sqrt {2-b x}}{3 \sqrt {x}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=\frac {-1-2 b x+2 b^2 x^2}{3 x^{3/2} \sqrt {2-b x}} \]
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Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(\frac {2 b^{2} x^{2}-2 b x -1}{3 x^{\frac {3}{2}} \sqrt {-b x +2}}\) | \(28\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-2 b^{2} x^{2}+2 b x +1\right )}{6 x^{\frac {3}{2}} \sqrt {-\frac {b x}{2}+1}}\) | \(31\) |
default | \(-\frac {1}{3 x^{\frac {3}{2}} \sqrt {-b x +2}}+\frac {2 b \left (-\frac {1}{\sqrt {x}\, \sqrt {-b x +2}}+\frac {\sqrt {x}\, b}{\sqrt {-b x +2}}\right )}{3}\) | \(45\) |
risch | \(\frac {\left (5 b^{2} x^{2}-8 b x -4\right ) \sqrt {\left (-b x +2\right ) x}}{12 x^{\frac {3}{2}} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {b^{2} \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{4 \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}\) | \(85\) |
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none
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=-\frac {{\left (2 \, b^{2} x^{2} - 2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b x^{3} - 2 \, x^{2}\right )}} \]
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Result contains complex when optimal does not.
Time = 2.87 (sec) , antiderivative size = 355, normalized size of antiderivative = 6.34 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=\begin {cases} - \frac {2 b^{\frac {15}{2}} x^{3} \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac {6 b^{\frac {13}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac {3 b^{\frac {11}{2}} x \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac {2 b^{\frac {9}{2}} \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {2 i b^{\frac {15}{2}} x^{3} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac {6 i b^{\frac {13}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac {3 i b^{\frac {11}{2}} x \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac {2 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=\frac {b^{2} \sqrt {x}}{4 \, \sqrt {-b x + 2}} - \frac {\sqrt {-b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (-b x + 2\right )}^{\frac {3}{2}}}{12 \, x^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (40) = 80\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.71 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=-\frac {\sqrt {-b} b^{3}}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} {\left | b \right |}} - \frac {{\left (5 \, {\left (b x - 2\right )} b^{2} {\left | b \right |} + 12 \, b^{2} {\left | b \right |}\right )} \sqrt {-b x + 2}}{12 \, {\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {3}{2}}} \]
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Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{5/2} (2-b x)^{3/2}} \, dx=\frac {\sqrt {2-b\,x}\,\left (\frac {2\,x}{3}-\frac {2\,b\,x^2}{3}+\frac {1}{3\,b}\right )}{x^{5/2}-\frac {2\,x^{3/2}}{b}} \]
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