Integrand size = 16, antiderivative size = 40 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=-\frac {(2-b x)^{3/2}}{5 x^{5/2}}-\frac {b (2-b x)^{3/2}}{15 x^{3/2}} \]
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Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=-\frac {b (2-b x)^{3/2}}{15 x^{3/2}}-\frac {(2-b x)^{3/2}}{5 x^{5/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {(2-b x)^{3/2}}{5 x^{5/2}}+\frac {1}{5} b \int \frac {\sqrt {2-b x}}{x^{5/2}} \, dx \\ & = -\frac {(2-b x)^{3/2}}{5 x^{5/2}}-\frac {b (2-b x)^{3/2}}{15 x^{3/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=\frac {\sqrt {2-b x} \left (-6+b x+b^2 x^2\right )}{15 x^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48
method | result | size |
gosper | \(-\frac {\left (b x +3\right ) \left (-b x +2\right )^{\frac {3}{2}}}{15 x^{\frac {5}{2}}}\) | \(19\) |
meijerg | \(-\frac {2 \sqrt {2}\, \left (-\frac {1}{6} b^{2} x^{2}-\frac {1}{6} b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{5 x^{\frac {5}{2}}}\) | \(31\) |
default | \(-\frac {2 \sqrt {-b x +2}}{5 x^{\frac {5}{2}}}-\frac {b \left (-\frac {\sqrt {-b x +2}}{3 x^{\frac {3}{2}}}-\frac {b \sqrt {-b x +2}}{3 \sqrt {x}}\right )}{5}\) | \(46\) |
risch | \(-\frac {\sqrt {\left (-b x +2\right ) x}\, \left (b^{3} x^{3}-b^{2} x^{2}-8 b x +12\right )}{15 x^{\frac {5}{2}} \sqrt {-b x +2}\, \sqrt {-x \left (b x -2\right )}}\) | \(55\) |
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none
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=\frac {{\left (b^{2} x^{2} + b x - 6\right )} \sqrt {-b x + 2}}{15 \, x^{\frac {5}{2}}} \]
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Result contains complex when optimal does not.
Time = 2.93 (sec) , antiderivative size = 194, normalized size of antiderivative = 4.85 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=\begin {cases} \frac {b^{\frac {9}{2}} x^{2} \sqrt {-1 + \frac {2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac {b^{\frac {7}{2}} x \sqrt {-1 + \frac {2}{b x}}}{15 b^{2} x^{2} - 30 b x} - \frac {8 b^{\frac {5}{2}} \sqrt {-1 + \frac {2}{b x}}}{15 b^{2} x^{2} - 30 b x} + \frac {12 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{x \left (15 b^{2} x^{2} - 30 b x\right )} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\\frac {i b^{\frac {5}{2}} \sqrt {1 - \frac {2}{b x}}}{15} + \frac {i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{15 x} - \frac {2 i \sqrt {b} \sqrt {1 - \frac {2}{b x}}}{5 x^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=-\frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b}{6 \, x^{\frac {3}{2}}} - \frac {{\left (-b x + 2\right )}^{\frac {5}{2}}}{10 \, x^{\frac {5}{2}}} \]
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none
Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=\frac {{\left ({\left (b x - 2\right )} b^{5} + 5 \, b^{5}\right )} {\left (b x - 2\right )} \sqrt {-b x + 2} b}{15 \, {\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {5}{2}} {\left | b \right |}} \]
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt {2-b x}}{x^{7/2}} \, dx=\frac {\sqrt {2-b\,x}\,\left (\frac {b^2\,x^2}{15}+\frac {b\,x}{15}-\frac {2}{5}\right )}{x^{5/2}} \]
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