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 = 0.0371955, 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 \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-a + b*x]/x^2,x]
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Rubi in Sympy [A] time = 5.27163, size = 31, normalized size = 0.74 \[ - \frac{\sqrt{- a + b x}}{x} + \frac{b \operatorname{atan}{\left (\frac{\sqrt{- a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x-a)**(1/2)/x**2,x)
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Mathematica [A] time = 0.0258018, size = 42, normalized size = 1. \[ \frac{b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b x-a}}{x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-a + b*x]/x^2,x]
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Maple [A] time = 0.012, size = 35, normalized size = 0.8 \[{b\arctan \left ({1\sqrt{bx-a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}-{\frac{1}{x}\sqrt{bx-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x-a)^(1/2)/x^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x - a)/x^2,x, algorithm="maxima")
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Fricas [A] time = 0.227303, size = 1, normalized size = 0.02 \[ \left [\frac{b x \log \left (\frac{{\left (b x - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b x - a} a}{x}\right ) - 2 \, \sqrt{b x - a} \sqrt{-a}}{2 \, \sqrt{-a} x}, -\frac{b x \arctan \left (\frac{\sqrt{a}}{\sqrt{b x - a}}\right ) + \sqrt{b x - a} \sqrt{a}}{\sqrt{a} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x - a)/x^2,x, algorithm="fricas")
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Sympy [A] time = 6.71264, size = 121, normalized size = 2.88 \[ \begin{cases} - \frac{i a}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{i \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} - 1}} + \frac{i b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{\sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{\sqrt{x}} - \frac{b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x-a)**(1/2)/x**2,x)
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GIAC/XCAS [A] time = 0.211147, size = 55, normalized size = 1.31 \[ \frac{\frac{b^{2} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{\sqrt{a}} - \frac{\sqrt{b x - a} b}{x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x - a)/x^2,x, algorithm="giac")
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