Optimal. Leaf size=20 \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]
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Rubi [A] time = 0.0209205, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-(b*x)]*Sqrt[2 - b*x]),x]
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Rubi in Sympy [A] time = 5.34128, size = 20, normalized size = 1. \[ - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{- b x}}{2} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x)**(1/2)/(-b*x+2)**(1/2),x)
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Mathematica [A] time = 0.0169434, size = 37, normalized size = 1.85 \[ \frac{2 \sqrt{x} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{-b x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-(b*x)]*Sqrt[2 - b*x]),x]
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Maple [B] time = 0.009, size = 64, normalized size = 3.2 \[{1\sqrt{-xb \left ( -bx+2 \right ) }\ln \left ({({b}^{2}x-b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-2\,bx} \right ){\frac{1}{\sqrt{-bx}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x)^(1/2)/(-b*x+2)^(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(1/(sqrt(-b*x + 2)*sqrt(-b*x)),x, algorithm="maxima")
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Fricas [A] time = 0.202253, size = 36, normalized size = 1.8 \[ -\frac{\log \left (-b x + \sqrt{-b x + 2} \sqrt{-b x} + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x + 2)*sqrt(-b*x)),x, algorithm="fricas")
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Sympy [A] time = 2.32438, size = 53, normalized size = 2.65 \[ \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x)**(1/2)/(-b*x+2)**(1/2),x)
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GIAC/XCAS [A] time = 0.262722, size = 32, normalized size = 1.6 \[ \frac{2 \,{\rm ln}\left ({\left | -\sqrt{-b x + 2} + \sqrt{-b x} \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x + 2)*sqrt(-b*x)),x, algorithm="giac")
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