Integrand size = 16, antiderivative size = 58 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=\frac {1}{3 \sqrt {x} (2-b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 \sqrt {x}} \]
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Time = 0.00 (sec) , antiderivative size = 58, 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} \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=-\frac {2 \sqrt {2-b x}}{3 \sqrt {x}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}+\frac {1}{3 \sqrt {x} (2-b x)^{3/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3 \sqrt {x} (2-b x)^{3/2}}+\frac {2}{3} \int \frac {1}{x^{3/2} (2-b x)^{3/2}} \, dx \\ & = \frac {1}{3 \sqrt {x} (2-b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}+\frac {2}{3} \int \frac {1}{x^{3/2} \sqrt {2-b x}} \, dx \\ & = \frac {1}{3 \sqrt {x} (2-b x)^{3/2}}+\frac {2}{3 \sqrt {x} \sqrt {2-b x}}-\frac {2 \sqrt {2-b x}}{3 \sqrt {x}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=-\frac {3-6 b x+2 b^2 x^2}{3 \sqrt {x} (2-b x)^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.48
method | result | size |
gosper | \(-\frac {2 b^{2} x^{2}-6 b x +3}{3 \sqrt {x}\, \left (-b x +2\right )^{\frac {3}{2}}}\) | \(28\) |
meijerg | \(-\frac {\sqrt {2}\, \left (2 b^{2} x^{2}-6 b x +3\right )}{12 \sqrt {x}\, \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\) | \(31\) |
default | \(-\frac {1}{\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}+2 b \left (\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}\right )\) | \(45\) |
risch | \(\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{4 \sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}+\frac {b \left (5 b x -12\right ) \sqrt {x}\, \sqrt {\left (-b x +2\right ) x}}{12 \sqrt {-x \left (b x -2\right )}\, \left (b x -2\right ) \sqrt {-b x +2}}\) | \(87\) |
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Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=-\frac {{\left (2 \, b^{2} x^{2} - 6 \, b x + 3\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{3} - 4 \, b x^{2} + 4 \, x\right )}} \]
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Result contains complex when optimal does not.
Time = 2.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.22 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=\begin {cases} - \frac {2 b^{\frac {13}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} + \frac {6 b^{\frac {11}{2}} x \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} - \frac {3 b^{\frac {9}{2}} \sqrt {-1 + \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {2 i b^{\frac {13}{2}} x^{2} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} + \frac {6 i b^{\frac {11}{2}} x \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} - \frac {3 i b^{\frac {9}{2}} \sqrt {1 - \frac {2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=\frac {{\left (b^{2} - \frac {6 \, {\left (b x - 2\right )} b}{x}\right )} x^{\frac {3}{2}}}{12 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} - \frac {\sqrt {-b x + 2}}{4 \, \sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (40) = 80\).
Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.93 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=-\frac {\sqrt {-b x + 2} b^{2}}{4 \, \sqrt {{\left (b x - 2\right )} b + 2 \, b} {\left | b \right |}} - \frac {3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt {-b} b^{2} - 24 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt {-b} b^{3} + 20 \, \sqrt {-b} b^{4}}{3 \, {\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} {\left | b \right |}} \]
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Time = 0.39 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{3/2} (2-b x)^{5/2}} \, dx=\frac {3\,\sqrt {2-b\,x}-6\,b\,x\,\sqrt {2-b\,x}+2\,b^2\,x^2\,\sqrt {2-b\,x}}{\sqrt {x}\,\left (x\,\left (12\,b-3\,b^2\,x\right )-12\right )} \]
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