Integrand size = 16, antiderivative size = 62 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {49, 56, 222} \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=2 b^{3/2} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {2-b x}}{\sqrt {x}} \]
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Rule 49
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}-b \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx \\ & = \frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx \\ & = \frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 b \sqrt {2-b x}}{\sqrt {x}}-\frac {2 (2-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\frac {4 \sqrt {2-b x} (-1+2 b x)}{3 x^{3/2}}-4 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.13
method | result | size |
meijerg | \(-\frac {3 \left (-b \right )^{\frac {5}{2}} \left (-\frac {16 \sqrt {\pi }\, \sqrt {2}\, \left (-2 b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{9 x^{\frac {3}{2}} \left (-b \right )^{\frac {3}{2}}}+\frac {8 \sqrt {\pi }\, b^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }\, b}\) | \(70\) |
risch | \(-\frac {4 \left (2 b^{2} x^{2}-5 b x +2\right ) \sqrt {\left (-b x +2\right ) x}}{3 x^{\frac {3}{2}} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\) | \(98\) |
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Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.79 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\left [\frac {3 \, \sqrt {-b} b x^{2} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + 4 \, {\left (2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, b^{\frac {3}{2}} x^{2} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - 2 \, {\left (2 \, b x - 1\right )} \sqrt {-b x + 2} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.92 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.97 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\begin {cases} \frac {8 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\frac {1}{\sqrt {b} \sqrt {x}} \right )} + 2 b^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {4 \sqrt {b} \sqrt {-1 + \frac {2}{b x}}}{3 x} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\\frac {8 i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {1}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {2}{b x}} + 1 \right )} - \frac {4 i \sqrt {b} \sqrt {1 - \frac {2}{b x}}}{3 x} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.79 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=-2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + \frac {2 \, \sqrt {-b x + 2} b}{\sqrt {x}} - \frac {2 \, {\left (-b x + 2\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \]
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Time = 5.79 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.42 \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, b^{2} \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b}} + \frac {2 \, {\left (2 \, {\left (b x - 2\right )} b^{3} + 3 \, b^{3}\right )} \sqrt {-b x + 2}}{{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac {3}{2}}}\right )} b}{3 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(2-b x)^{3/2}}{x^{5/2}} \, dx=\int \frac {{\left (2-b\,x\right )}^{3/2}}{x^{5/2}} \,d x \]
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