Optimal. Leaf size=42 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]
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Rubi [A] time = 0.0103159, antiderivative size = 42, 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 = {47, 63, 205} \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{-a+b x}}{x^2} \, dx &=-\frac{\sqrt{-a+b x}}{x}+\frac{1}{2} b \int \frac{1}{x \sqrt{-a+b x}} \, dx\\ &=-\frac{\sqrt{-a+b x}}{x}+\operatorname{Subst}\left (\int \frac{1}{\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{-a+b x}\right )\\ &=-\frac{\sqrt{-a+b x}}{x}+\frac{b \tan ^{-1}\left (\frac{\sqrt{-a+b x}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.0477617, size = 52, normalized size = 1.24 \[ \frac{-b x \sqrt{1-\frac{b x}{a}} \tanh ^{-1}\left (\sqrt{1-\frac{b x}{a}}\right )+a-b x}{x \sqrt{b x-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 35, normalized size = 0.8 \begin{align*}{b\arctan \left ({\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{x}\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 = 1.53313, size = 224, normalized size = 5.33 \begin{align*} \left [-\frac{\sqrt{-a} b x \log \left (\frac{b x - 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \, \sqrt{b x - a} a}{2 \, a x}, \frac{\sqrt{a} b x \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) - \sqrt{b x - a} a}{a x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.73324, size = 124, normalized size = 2.95 \begin{align*} \begin{cases} - \frac{i a}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\- \frac{\sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1966, size = 55, normalized size = 1.31 \begin{align*} \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{\sqrt{b x - a} b}{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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