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 = 0.0105244, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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*}
Mathematica [A] time = 0.0204661, size = 39, normalized size = 1. \[ 2 \sqrt{b x-a}-2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.8 \begin{align*} -2\,\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{bx-a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52038, size = 186, normalized size = 4.77 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.16425, size = 151, normalized size = 3.87 \begin{align*} \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}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18125, size = 42, normalized size = 1.08 \begin{align*} -2 \, \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) + 2 \, \sqrt{b x - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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