Integrand size = 16, antiderivative size = 131 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222} \[ \int x^{5/2} (2-b x)^{3/2} \, dx=\frac {3 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{20} x^{7/2} \sqrt {2-b x}-\frac {x^{5/2} \sqrt {2-b x}}{20 b} \]
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Rule 52
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{5} \int x^{5/2} \sqrt {2-b x} \, dx \\ & = \frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3}{20} \int \frac {x^{5/2}}{\sqrt {2-b x}} \, dx \\ & = -\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {\int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx}{4 b} \\ & = -\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{8 b^2} \\ & = -\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{8 b^3} \\ & = -\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^3} \\ & = -\frac {3 \sqrt {x} \sqrt {2-b x}}{8 b^3}-\frac {x^{3/2} \sqrt {2-b x}}{8 b^2}-\frac {x^{5/2} \sqrt {2-b x}}{20 b}+\frac {3}{20} x^{7/2} \sqrt {2-b x}+\frac {1}{5} x^{7/2} (2-b x)^{3/2}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{4 b^{7/2}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.72 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=-\frac {\sqrt {x} \sqrt {2-b x} \left (15+5 b x+2 b^2 x^2-22 b^3 x^3+8 b^4 x^4\right )}{40 b^3}-\frac {3 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right )}{2 b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.74
method | result | size |
meijerg | \(-\frac {24 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {7}{2}} \left (56 b^{4} x^{4}-154 b^{3} x^{3}+14 b^{2} x^{2}+35 b x +105\right ) \sqrt {-\frac {b x}{2}+1}}{6720 b^{3}}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {7}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{32 b^{\frac {7}{2}}}\right )}{\left (-b \right )^{\frac {5}{2}} \sqrt {\pi }\, b}\) | \(97\) |
risch | \(\frac {\left (8 b^{4} x^{4}-22 b^{3} x^{3}+2 b^{2} x^{2}+5 b x +15\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{40 b^{3} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{8 b^{\frac {7}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(123\) |
default | \(-\frac {x^{\frac {5}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{5 b}+\frac {-\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {-\frac {\sqrt {x}\, \left (-b x +2\right )^{\frac {5}{2}}}{4 b}+\frac {\frac {\left (-b x +2\right )^{\frac {3}{2}} \sqrt {x}}{2}+\frac {3 \sqrt {x}\, \sqrt {-b x +2}}{2}+\frac {3 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right )}{2 \sqrt {-b x +2}\, \sqrt {x}\, \sqrt {b}}}{4 b}}{b}}{b}\) | \(143\) |
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Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.20 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=\left [-\frac {{\left (8 \, b^{5} x^{4} - 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} + 5 \, b^{2} x + 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 15 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{40 \, b^{4}}, -\frac {{\left (8 \, b^{5} x^{4} - 22 \, b^{4} x^{3} + 2 \, b^{3} x^{2} + 5 \, b^{2} x + 15 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 30 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{40 \, b^{4}}\right ] \]
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Result contains complex when optimal does not.
Time = 80.75 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.21 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=\begin {cases} - \frac {i b^{2} x^{\frac {11}{2}}}{5 \sqrt {b x - 2}} + \frac {19 i b x^{\frac {9}{2}}}{20 \sqrt {b x - 2}} - \frac {23 i x^{\frac {7}{2}}}{20 \sqrt {b x - 2}} - \frac {i x^{\frac {5}{2}}}{40 b \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{8 b^{2} \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{4 b^{3} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {7}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {b^{2} x^{\frac {11}{2}}}{5 \sqrt {- b x + 2}} - \frac {19 b x^{\frac {9}{2}}}{20 \sqrt {- b x + 2}} + \frac {23 x^{\frac {7}{2}}}{20 \sqrt {- b x + 2}} + \frac {x^{\frac {5}{2}}}{40 b \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{8 b^{2} \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{4 b^{3} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{4 b^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.37 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=\frac {\frac {15 \, \sqrt {-b x + 2} b^{4}}{\sqrt {x}} + \frac {70 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{3}}{x^{\frac {3}{2}}} - \frac {128 \, {\left (-b x + 2\right )}^{\frac {5}{2}} b^{2}}{x^{\frac {5}{2}}} - \frac {70 \, {\left (-b x + 2\right )}^{\frac {7}{2}} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (-b x + 2\right )}^{\frac {9}{2}}}{x^{\frac {9}{2}}}}{20 \, {\left (b^{8} - \frac {5 \, {\left (b x - 2\right )} b^{7}}{x} + \frac {10 \, {\left (b x - 2\right )}^{2} b^{6}}{x^{2}} - \frac {10 \, {\left (b x - 2\right )}^{3} b^{5}}{x^{3}} + \frac {5 \, {\left (b x - 2\right )}^{4} b^{4}}{x^{4}} - \frac {{\left (b x - 2\right )}^{5} b^{3}}{x^{5}}\right )}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {7}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (92) = 184\).
Time = 17.07 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.58 \[ \int x^{5/2} (2-b x)^{3/2} \, dx=-\frac {3 \, {\left ({\left ({\left (2 \, {\left (b x - 2\right )} {\left ({\left (b x - 2\right )} {\left (\frac {4 \, {\left (b x - 2\right )}}{b^{4}} + \frac {41}{b^{4}}\right )} + \frac {171}{b^{4}}\right )} + \frac {745}{b^{4}}\right )} {\left (b x - 2\right )} + \frac {965}{b^{4}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {630 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{3}}\right )} {\left | b \right |} - \frac {20 \, {\left ({\left ({\left (b x - 2\right )} {\left (2 \, {\left (b x - 2\right )} {\left (\frac {3 \, {\left (b x - 2\right )}}{b^{3}} + \frac {25}{b^{3}}\right )} + \frac {163}{b^{3}}\right )} + \frac {279}{b^{3}}\right )} \sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} - \frac {210 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b^{2}}\right )} {\left | b \right |}}{b} + \frac {80 \, {\left (\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2} {\left ({\left (b x - 2\right )} {\left (\frac {2 \, {\left (b x - 2\right )}}{b^{2}} + \frac {13}{b^{2}}\right )} + \frac {33}{b^{2}}\right )} - \frac {30 \, \log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b} b}\right )} {\left | b \right |}}{b^{2}}}{120 \, b} \]
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Timed out. \[ \int x^{5/2} (2-b x)^{3/2} \, dx=\int x^{5/2}\,{\left (2-b\,x\right )}^{3/2} \,d x \]
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