Integrand size = 15, antiderivative size = 53 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}} \]
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Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {47, 37} \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=-\frac {2 \sqrt {b x+2}}{3 x^{3/2}}+\frac {1}{x^{3/2} \sqrt {b x+2}}+\frac {2 b \sqrt {b x+2}}{3 \sqrt {x}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {1}{x^{3/2} \sqrt {2+b x}}+2 \int \frac {1}{x^{5/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}-\frac {1}{3} (2 b) \int \frac {1}{x^{3/2} \sqrt {2+b x}} \, dx \\ & = \frac {1}{x^{3/2} \sqrt {2+b x}}-\frac {2 \sqrt {2+b x}}{3 x^{3/2}}+\frac {2 b \sqrt {2+b x}}{3 \sqrt {x}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.60 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {-1+2 b x+2 b^2 x^2}{3 x^{3/2} \sqrt {2+b x}} \]
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Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.51
method | result | size |
gosper | \(\frac {2 b^{2} x^{2}+2 b x -1}{3 x^{\frac {3}{2}} \sqrt {b x +2}}\) | \(27\) |
meijerg | \(-\frac {\sqrt {2}\, \left (-2 b^{2} x^{2}-2 b x +1\right )}{6 x^{\frac {3}{2}} \sqrt {\frac {b x}{2}+1}}\) | \(31\) |
default | \(-\frac {1}{3 x^{\frac {3}{2}} \sqrt {b x +2}}-\frac {2 b \left (-\frac {1}{\sqrt {x}\, \sqrt {b x +2}}-\frac {b \sqrt {x}}{\sqrt {b x +2}}\right )}{3}\) | \(43\) |
risch | \(\frac {5 b^{2} x^{2}+8 b x -4}{12 x^{\frac {3}{2}} \sqrt {b x +2}}+\frac {b^{2} \sqrt {x}}{4 \sqrt {b x +2}}\) | \(43\) |
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Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {{\left (2 \, b^{2} x^{2} + 2 \, b x - 1\right )} \sqrt {b x + 2} \sqrt {x}}{3 \, {\left (b x^{3} + 2 \, x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (49) = 98\).
Time = 2.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 3.21 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {2 b^{\frac {15}{2}} x^{3} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {6 b^{\frac {13}{2}} x^{2} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} + \frac {3 b^{\frac {11}{2}} x \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} - \frac {2 b^{\frac {9}{2}} \sqrt {1 + \frac {2}{b x}}}{3 b^{6} x^{3} + 12 b^{5} x^{2} + 12 b^{4} x} \]
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Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {b^{2} \sqrt {x}}{4 \, \sqrt {b x + 2}} + \frac {\sqrt {b x + 2} b}{2 \, \sqrt {x}} - \frac {{\left (b x + 2\right )}^{\frac {3}{2}}}{12 \, x^{\frac {3}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (37) = 74\).
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {b^{\frac {7}{2}}}{{\left ({\left (\sqrt {b x + 2} \sqrt {b} - \sqrt {{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} {\left | b \right |}} + \frac {{\left (5 \, {\left (b x + 2\right )} b^{2} {\left | b \right |} - 12 \, b^{2} {\left | b \right |}\right )} \sqrt {b x + 2}}{12 \, {\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac {3}{2}}} \]
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Time = 0.38 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{5/2} (2+b x)^{3/2}} \, dx=\frac {\sqrt {b\,x+2}\,\left (\frac {2\,x}{3}+\frac {2\,b\,x^2}{3}-\frac {1}{3\,b}\right )}{x^{5/2}+\frac {2\,x^{3/2}}{b}} \]
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