Integrand size = 15, antiderivative size = 37 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2+b x}} \]
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Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {\sqrt {x}}{3 \sqrt {b x+2}}+\frac {\sqrt {x}}{3 (b x+2)^{3/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {x} (2+b x)^{3/2}} \, dx \\ & = \frac {\sqrt {x}}{3 (2+b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2+b x}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {\sqrt {x} (3+b x)}{3 (2+b x)^{3/2}} \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.49
method | result | size |
gosper | \(\frac {\sqrt {x}\, \left (b x +3\right )}{3 \left (b x +2\right )^{\frac {3}{2}}}\) | \(18\) |
meijerg | \(\frac {\sqrt {x}\, \sqrt {2}\, \left (b x +3\right )}{12 \left (\frac {b x}{2}+1\right )^{\frac {3}{2}}}\) | \(22\) |
default | \(\frac {\sqrt {x}}{3 \left (b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {b x +2}}\) | \(26\) |
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none
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {{\left (b x + 3\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (29) = 58\).
Time = 1.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {b x}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} + \frac {3}{3 b^{\frac {3}{2}} x \sqrt {1 + \frac {2}{b x}} + 6 \sqrt {b} \sqrt {1 + \frac {2}{b x}}} \]
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Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=-\frac {{\left (b - \frac {3 \, {\left (b x + 2\right )}}{x}\right )} x^{\frac {3}{2}}}{6 \, {\left (b x + 2\right )}^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {8 \, {\left (3 \, {\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac {5}{2}}}{3 \, {\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} {\left | b \right |}} \]
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Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {x} (2+b x)^{5/2}} \, dx=\frac {3\,\sqrt {x}\,\sqrt {b\,x+2}+b\,x^{3/2}\,\sqrt {b\,x+2}}{3\,b^2\,x^2+12\,b\,x+12} \]
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