Integrand size = 16, antiderivative size = 46 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {a-b x}+\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\frac {a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}}+\sqrt {x} \sqrt {a-b x} \]
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = \sqrt {x} \sqrt {a-b x}+\frac {1}{2} a \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx \\ & = \sqrt {x} \sqrt {a-b x}+a \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {a-b x}+a \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right ) \\ & = \sqrt {x} \sqrt {a-b x}+\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{\sqrt {b}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {a-b x}+\frac {2 a \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{\sqrt {b}} \]
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Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43
method | result | size |
default | \(\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(66\) |
risch | \(\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\) | \(66\) |
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none
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\left [-\frac {a \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, \sqrt {-b x + a} b \sqrt {x}}{2 \, b}, -\frac {a \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} b \sqrt {x}}{b}\right ] \]
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Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.59 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\begin {cases} - \frac {i \sqrt {a} \sqrt {x}}{\sqrt {-1 + \frac {b x}{a}}} - \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} + \frac {i b x^{\frac {3}{2}}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\sqrt {a} \sqrt {x} \sqrt {1 - \frac {b x}{a}} + \frac {a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=-\frac {a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}} + \frac {\sqrt {-b x + a} a}{{\left (b - \frac {b x - a}{x}\right )} \sqrt {x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (34) = 68\).
Time = 76.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\frac {{\left (\frac {a \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} + \frac {\sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b}\right )} b}{{\left | b \right |}} \]
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Time = 0.65 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a-b x}}{\sqrt {x}} \, dx=\sqrt {x}\,\sqrt {a-b\,x}+\frac {2\,a\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a-b\,x}-\sqrt {a}}\right )}{\sqrt {b}} \]
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