Optimal. Leaf size=19 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{\sqrt{2}}\right )}{b} \]
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Rubi [A] time = 0.0195618, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{\sqrt{2}}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[b*x]*Sqrt[2 + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 3.60663, size = 17, normalized size = 0.89 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b x}}{2} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x)**(1/2)/(b*x+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0188604, size = 36, normalized size = 1.89 \[ \frac{2 \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{b x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[b*x]*Sqrt[2 + b*x]),x]
[Out]
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Maple [B] time = 0.007, size = 58, normalized size = 3.1 \[{1\sqrt{xb \left ( bx+2 \right ) }\ln \left ({({b}^{2}x+b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,bx} \right ){\frac{1}{\sqrt{bx}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x)^(1/2)/(b*x+2)^(1/2),x)
[Out]
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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")
[Out]
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Fricas [A] time = 0.204132, size = 34, normalized size = 1.79 \[ -\frac{\log \left (-b x + \sqrt{b x + 2} \sqrt{b x} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 2)*sqrt(b*x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.21744, size = 20, normalized size = 1.05 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x)**(1/2)/(b*x+2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.25721, size = 30, normalized size = 1.58 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 2} + \sqrt{b x} \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 2)*sqrt(b*x)),x, algorithm="giac")
[Out]