Integrand size = 16, antiderivative size = 67 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\frac {2 x^{3/2}}{3 b (2-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2-b x}}+\frac {2 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 67, 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} \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\frac {2 \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2-b x}}+\frac {2 x^{3/2}}{3 b (2-b x)^{3/2}} \]
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Rule 49
Rule 56
Rule 222
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{3/2}}{3 b (2-b x)^{3/2}}-\frac {\int \frac {\sqrt {x}}{(2-b x)^{3/2}} \, dx}{b} \\ & = \frac {2 x^{3/2}}{3 b (2-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2-b x}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b^2} \\ & = \frac {2 x^{3/2}}{3 b (2-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2-b x}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = \frac {2 x^{3/2}}{3 b (2-b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2-b x}}+\frac {2 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\frac {4 \sqrt {x} (-3+2 b x)}{3 b^2 (2-b x)^{3/2}}-\frac {4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right )}{b^{5/2}} \]
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Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09
method | result | size |
meijerg | \(-\frac {4 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (-10 b x +15\right )}{20 b^{2} \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{2 b^{\frac {5}{2}}}\right )}{3 \left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\) | \(73\) |
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Time = 0.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.58 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )} \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - 4 \, {\left (2 \, b^{2} x - 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{5} x^{2} - 4 \, b^{4} x + 4 \, b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - 2 \, {\left (2 \, b^{2} x - 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} - 4 \, b^{4} x + 4 \, b^{3}\right )}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.95 (sec) , antiderivative size = 648, normalized size of antiderivative = 9.67 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\begin {cases} \frac {8 i b^{\frac {11}{2}} x^{8}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} - \frac {12 i b^{\frac {9}{2}} x^{7}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} - \frac {6 i b^{5} x^{\frac {15}{2}} \sqrt {b x - 2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} + \frac {3 \pi b^{5} x^{\frac {15}{2}} \sqrt {b x - 2}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} + \frac {12 i b^{4} x^{\frac {13}{2}} \sqrt {b x - 2} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} - \frac {6 \pi b^{4} x^{\frac {13}{2}} \sqrt {b x - 2}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x - 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {8 b^{\frac {11}{2}} x^{8}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {- b x + 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {- b x + 2}} + \frac {12 b^{\frac {9}{2}} x^{7}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {- b x + 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {- b x + 2}} + \frac {6 b^{5} x^{\frac {15}{2}} \sqrt {- b x + 2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {- b x + 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {- b x + 2}} - \frac {12 b^{4} x^{\frac {13}{2}} \sqrt {- b x + 2} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {- b x + 2} - 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.75 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\frac {2 \, {\left (b + \frac {3 \, {\left (b x - 2\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (50) = 100\).
Time = 1.53 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.66 \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b}} + \frac {16 \, {\left (3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt {-b} - 6 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt {-b} b + 8 \, \sqrt {-b} b^{2}\right )}}{{\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3}}\right )} {\left | b \right |}}{3 \, b^{3}} \]
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Timed out. \[ \int \frac {x^{3/2}}{(2-b x)^{5/2}} \, dx=\int \frac {x^{3/2}}{{\left (2-b\,x\right )}^{5/2}} \,d x \]
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