Integrand size = 17, antiderivative size = 50 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{\sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {344, 201, 223, 212} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{\sqrt {b}}-\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}} \]
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Rule 201
Rule 212
Rule 223
Rule 344
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-a \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-a \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {x}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 5.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}} \left (\sqrt {b+a x}+\frac {a x \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {b+a x}} \]
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Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a x +\sqrt {a x +b}\, \sqrt {b}\right )}{\sqrt {x}\, \sqrt {a x +b}\, \sqrt {b}}\) | \(54\) |
risch | \(-\frac {\sqrt {\frac {a x +b}{x}}}{\sqrt {x}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{\sqrt {b}\, \sqrt {a x +b}}\) | \(57\) |
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none
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\left [\frac {a \sqrt {b} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, b x}, \frac {a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) - b \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b x}\right ] \]
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Time = 1.43 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=- \frac {\sqrt {a} \sqrt {1 + \frac {b}{a x}}}{\sqrt {x}} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{\sqrt {b}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\frac {a \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{2 \, \sqrt {b}} - \frac {\sqrt {a + \frac {b}{x}} a \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]
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Time = 0.31 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\frac {{\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {\sqrt {a x + b} a}{x}\right )} \mathrm {sgn}\left (x\right )}{a} \]
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Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{x^{3/2}} \,d x \]
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