Integrand size = 15, antiderivative size = 65 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=-\frac {2 x^{3/2}}{3 b (2+b x)^{3/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {2+b x}}+\frac {2 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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Time = 0.01 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {49, 56, 221} \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {2 \sqrt {x}}{b^2 \sqrt {b x+2}}-\frac {2 x^{3/2}}{3 b (b x+2)^{3/2}} \]
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Rule 49
Rule 56
Rule 221
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 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=-\frac {4 \sqrt {x} (3+2 b x)}{3 b^2 (2+b x)^{3/2}}-\frac {2 \log \left (-\sqrt {b} \sqrt {x}+\sqrt {2+b x}\right )}{b^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (10 b x +15\right )}{15 \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}+2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(55\) |
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none
Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.63 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right ) - 4 \, {\left (2 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right ) + 2 \, {\left (2 \, b^{2} x + 3 \, b\right )} \sqrt {b x + 2} \sqrt {x}\right )}}{3 \, {\left (b^{5} x^{2} + 4 \, b^{4} x + 4 \, b^{3}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (61) = 122\).
Time = 2.17 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.95 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=- \frac {8 b^{\frac {11}{2}} x^{8}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} - \frac {12 b^{\frac {9}{2}} x^{7}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} + \frac {6 b^{5} x^{\frac {15}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} + \frac {12 b^{4} x^{\frac {13}{2}} \sqrt {b x + 2} \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{3 b^{\frac {15}{2}} x^{\frac {15}{2}} \sqrt {b x + 2} + 6 b^{\frac {13}{2}} x^{\frac {13}{2}} \sqrt {b x + 2}} \]
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none
Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.06 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=-\frac {2 \, {\left (b + \frac {3 \, {\left (b x + 2\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + 2\right )}^{\frac {3}{2}} b^{2}} - \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{b^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (48) = 96\).
Time = 1.63 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.37 \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=-\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{\sqrt {b}} + \frac {16 \, {\left (3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt {b} + 6 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac {3}{2}} + 8 \, b^{\frac {5}{2}}\right )}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}}\right )} {\left | b \right |}}{3 \, b^{3}} \]
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Timed out. \[ \int \frac {x^{3/2}}{(2+b x)^{5/2}} \, dx=\int \frac {x^{3/2}}{{\left (b\,x+2\right )}^{5/2}} \,d x \]
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