Optimal. Leaf size=41 \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0295683, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2 - b*x]/Sqrt[x],x]
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Rubi in Sympy [A] time = 5.00571, size = 37, normalized size = 0.9 \[ \sqrt{x} \sqrt{- b x + 2} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x+2)**(1/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0213109, size = 41, normalized size = 1. \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2 - b*x]/Sqrt[x],x]
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Maple [B] time = 0.007, size = 63, normalized size = 1.5 \[ \sqrt{x}\sqrt{-bx+2}+{1\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x+2)^(1/2)/x^(1/2),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 + 2)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.218371, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{\sqrt{-b}}, \frac{\sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x + 2)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 6.05513, size = 121, normalized size = 2.95 \[ \begin{cases} \frac{i b x^{\frac{3}{2}}}{\sqrt{b x - 2}} - \frac{2 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b x^{\frac{3}{2}}}{\sqrt{- b x + 2}} + \frac{2 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x+2)**(1/2)/x**(1/2),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x + 2)/sqrt(x),x, algorithm="giac")
[Out]