Integrand size = 15, antiderivative size = 39 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=2 \sqrt {-a+b x}-2 \sqrt {a} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 65, 211} \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=2 \sqrt {b x-a}-2 \sqrt {a} \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
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Rule 52
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = 2 \sqrt {-a+b x}-a \int \frac {1}{x \sqrt {-a+b x}} \, dx \\ & = 2 \sqrt {-a+b x}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{b} \\ & = 2 \sqrt {-a+b x}-2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=2 \sqrt {-a+b x}-2 \sqrt {a} \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {b x -a}\) | \(32\) |
default | \(-2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {b x -a}\) | \(32\) |
pseudoelliptic | \(-2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {b x -a}\) | \(32\) |
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Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=\left [\sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a}, -2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a}\right ] \]
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Result contains complex when optimal does not.
Time = 1.40 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.79 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=\begin {cases} - 2 i \sqrt {a} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 i a}{\sqrt {b} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {2 i \sqrt {b} \sqrt {x}}{\sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\2 \sqrt {a} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {2 a}{\sqrt {b} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {2 \sqrt {b} \sqrt {x}}{\sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=-2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=-2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} \]
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Time = 0.12 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {-a+b x}}{x} \, dx=2\,\sqrt {b\,x-a}-2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right ) \]
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