Integrand size = 16, antiderivative size = 82 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=-\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-15 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 52, 56, 222} \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=-15 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {15}{2} b \sqrt {x} \sqrt {2-b x} \]
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(5 b) \int \frac {(2-b x)^{3/2}}{\sqrt {x}} \, dx \\ & = -\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {\sqrt {2-b x}}{\sqrt {x}} \, dx \\ & = -\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-\frac {1}{2} (15 b) \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx \\ & = -\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-(15 b) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {15}{2} b \sqrt {x} \sqrt {2-b x}-\frac {5}{2} b \sqrt {x} (2-b x)^{3/2}-\frac {2 (2-b x)^{5/2}}{\sqrt {x}}-15 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=\frac {\sqrt {2-b x} \left (-16-9 b x+b^2 x^2\right )}{2 \sqrt {x}}+30 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.95
method | result | size |
meijerg | \(\frac {15 \left (-b \right )^{\frac {3}{2}} \left (\frac {16 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {1}{16} b^{2} x^{2}+\frac {9}{16} b x +1\right ) \sqrt {-\frac {b x}{2}+1}}{15 \sqrt {x}\, \sqrt {-b}}+\frac {2 \sqrt {\pi }\, \sqrt {b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {-b}}\right )}{2 \sqrt {\pi }\, b}\) | \(78\) |
risch | \(-\frac {\left (b^{3} x^{3}-11 b^{2} x^{2}+2 b x +32\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}-\frac {15 \sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{2 \sqrt {x}\, \sqrt {-b x +2}}\) | \(106\) |
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Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=\left [\frac {15 \, \sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}, \frac {30 \, \sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x^{2} - 9 \, b x - 16\right )} \sqrt {-b x + 2} \sqrt {x}}{2 \, x}\right ] \]
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Result contains complex when optimal does not.
Time = 4.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.45 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=\begin {cases} 15 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {i b^{3} x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {11 i b^{2} x^{\frac {3}{2}}}{2 \sqrt {b x - 2}} + \frac {i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {16 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- 15 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {b^{3} x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {11 b^{2} x^{\frac {3}{2}}}{2 \sqrt {- b x + 2}} - \frac {b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {16}{\sqrt {x} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.17 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=15 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {\frac {7 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {9 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}}}{b^{2} - \frac {2 \, {\left (b x - 2\right )} b}{x} + \frac {{\left (b x - 2\right )}^{2}}{x^{2}}} - \frac {8 \, \sqrt {-b x + 2}}{\sqrt {x}} \]
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Time = 5.72 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01 \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=\frac {{\left (\frac {{\left ({\left (b x - 2\right )} {\left (b x - 7\right )} - 30\right )} \sqrt {-b x + 2}}{\sqrt {{\left (b x - 2\right )} b + 2 \, b}} - \frac {30 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b}}\right )} b^{2}}{2 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(2-b x)^{5/2}}{x^{3/2}} \, dx=\int \frac {{\left (2-b\,x\right )}^{5/2}}{x^{3/2}} \,d x \]
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