Optimal. Leaf size=39 \[ 2 \sqrt {b x-a}-2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {50, 63, 205} \[ 2 \sqrt {b x-a}-2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {\sqrt {-a+b x}}{x} \, dx &=2 \sqrt {-a+b x}-a \int \frac {1}{x \sqrt {-a+b x}} \, dx\\ &=2 \sqrt {-a+b x}-\frac {(2 a) \operatorname {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*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ 2 \sqrt {b x-a}-2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 78, normalized size = 2.00 \[ \left [\sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a}, -2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 31, normalized size = 0.79 \[ -2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.82 \[ -2 \sqrt {a}\, \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )+2 \sqrt {b x -a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 31, normalized size = 0.79 \[ -2 \, \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + 2 \, \sqrt {b x - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 31, normalized size = 0.79 \[ 2\,\sqrt {b\,x-a}-2\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.74, size = 148, normalized size = 3.79 \[ \begin {cases} - 2 i \sqrt {a} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 i a}{\sqrt {b} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {2 i \sqrt {b} \sqrt {x}}{\sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\2 \sqrt {a} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {2 a}{\sqrt {b} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {2 \sqrt {b} \sqrt {x}}{\sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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