Integrand size = 17, antiderivative size = 46 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {4 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {4 b}{3 a^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {2}{a \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}+\frac {(2 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}} \, dx}{a} \\ & = -\frac {4 b}{3 a^2 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2}{a \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \\ \end{align*}
Time = 6.34 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (2 b+3 a x)}{3 a^2 (b+a x)^2} \]
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Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72
method | result | size |
gosper | \(-\frac {2 \left (3 a x +2 b \right ) \left (a x +b \right )}{3 a^{2} x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\) | \(33\) |
default | \(-\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (3 a x +2 b \right )}{3 \left (a x +b \right )^{2} a^{2}}\) | \(35\) |
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none
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{4} x^{2} + 2 \, a^{3} b x + a^{2} b^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (39) = 78\).
Time = 5.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=- \frac {6 a x}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} - \frac {4 b}{3 a^{3} \sqrt {b} x \sqrt {\frac {a x}{b} + 1} + 3 a^{2} b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}} \]
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Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x - b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2} x^{\frac {3}{2}}} \]
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Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {2 \, {\left (3 \, a x + 2 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2}} \]
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Time = 6.36 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^3}+\frac {4\,b\,\sqrt {x}}{3\,a^4}\right )}{x^2+\frac {b^2}{a^2}+\frac {2\,b\,x}{a}} \]
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