Integrand size = 15, antiderivative size = 67 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=-\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221} \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\frac {3 \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {3 \sqrt {x} \sqrt {b x+2}}{2 b^2}+\frac {x^{3/2} \sqrt {b x+2}}{2 b} \]
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Rule 52
Rule 56
Rule 221
Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2} \sqrt {2+b x}}{2 b}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2+b x}} \, dx}{2 b} \\ & = -\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2+b x}} \, dx}{2 b^2} \\ & = -\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2+b x^2}} \, dx,x,\sqrt {x}\right )}{b^2} \\ & = -\frac {3 \sqrt {x} \sqrt {2+b x}}{2 b^2}+\frac {x^{3/2} \sqrt {2+b x}}{2 b}+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\frac {\sqrt {x} (-3+b x) \sqrt {2+b x}}{2 b^2}-\frac {6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2+b x}}\right )}{b^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82
method | result | size |
meijerg | \(\frac {-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \sqrt {b}\, \left (-5 b x +15\right ) \sqrt {\frac {b x}{2}+1}}{10}+3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{b^{\frac {5}{2}} \sqrt {\pi }}\) | \(55\) |
risch | \(\frac {\left (b x -3\right ) \sqrt {x}\, \sqrt {b x +2}}{2 b^{2}}+\frac {3 \ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{2 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +2}}\) | \(68\) |
default | \(\frac {x^{\frac {3}{2}} \sqrt {b x +2}}{2 b}-\frac {3 \left (\frac {\sqrt {x}\, \sqrt {b x +2}}{b}-\frac {\ln \left (\frac {b x +1}{\sqrt {b}}+\sqrt {b \,x^{2}+2 x}\right ) \sqrt {x \left (b x +2\right )}}{b^{\frac {3}{2}} \sqrt {x}\, \sqrt {b x +2}}\right )}{2 b}\) | \(83\) |
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Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\left [\frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} + 3 \, \sqrt {b} \log \left (b x + \sqrt {b x + 2} \sqrt {b} \sqrt {x} + 1\right )}{2 \, b^{3}}, \frac {{\left (b^{2} x - 3 \, b\right )} \sqrt {b x + 2} \sqrt {x} - 6 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x + 2} \sqrt {-b}}{b \sqrt {x}}\right )}{2 \, b^{3}}\right ] \]
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Time = 3.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.12 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\frac {x^{\frac {5}{2}}}{2 \sqrt {b x + 2}} - \frac {x^{\frac {3}{2}}}{2 b \sqrt {b x + 2}} - \frac {3 \sqrt {x}}{b^{2} \sqrt {b x + 2}} + \frac {3 \operatorname {asinh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (48) = 96\).
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.52 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\frac {\frac {5 \, \sqrt {b x + 2} b}{\sqrt {x}} - \frac {3 \, {\left (b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{4} - \frac {2 \, {\left (b x + 2\right )} b^{3}}{x} + \frac {{\left (b x + 2\right )}^{2} b^{2}}{x^{2}}} - \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + 2}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + 2}}{\sqrt {x}}}\right )}{2 \, b^{\frac {5}{2}}} \]
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Time = 5.85 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\frac {{\left (\sqrt {{\left (b x + 2\right )} b - 2 \, b} \sqrt {b x + 2} {\left (b x - 3\right )} - 6 \, \sqrt {b} \log \left ({\left | -\sqrt {b x + 2} \sqrt {b} + \sqrt {{\left (b x + 2\right )} b - 2 \, b} \right |}\right )\right )} {\left | b \right |}}{2 \, b^{4}} \]
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Timed out. \[ \int \frac {x^{3/2}}{\sqrt {2+b x}} \, dx=\int \frac {x^{3/2}}{\sqrt {b\,x+2}} \,d x \]
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