Integrand size = 17, antiderivative size = 44 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {4 b}{a^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \]
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Time = 0.01 (sec) , antiderivative size = 44, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {4 b}{a^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}}-\frac {(2 b) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{a} \\ & = \frac {4 b}{a^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}+\frac {2 \sqrt {x}}{a \sqrt {a+\frac {b}{x}}} \\ \end{align*}
Time = 5.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (2 b+a x)}{a^2 (b+a x)} \]
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Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(\frac {2 \left (a x +b \right ) \left (a x +2 b \right )}{a^{2} x^{\frac {3}{2}} \left (\frac {a x +b}{x}\right )^{\frac {3}{2}}}\) | \(32\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +2 b \right )}{\left (a x +b \right ) a^{2}}\) | \(34\) |
risch | \(\frac {2 a x +2 b}{a^{2} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}+\frac {2 b}{a^{2} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\) | \(46\) |
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none
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {2 \, {\left (a x + 2 \, b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a^{3} x + a^{2} b} \]
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Time = 1.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {2 x}{a \sqrt {b} \sqrt {\frac {a x}{b} + 1}} + \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a x}{b} + 1}} \]
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none
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {2 \, \sqrt {a + \frac {b}{x}} \sqrt {x}}{a^{2}} + \frac {2 \, b}{\sqrt {a + \frac {b}{x}} a^{2} \sqrt {x}} \]
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Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {2 \, {\left (\frac {\sqrt {a x + b}}{a} + \frac {b}{\sqrt {a x + b} a}\right )}}{a} \]
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Time = 6.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx=\frac {\sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{3/2}}{a^2}+\frac {4\,b\,\sqrt {x}}{a^3}\right )}{x+\frac {b}{a}} \]
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