Optimal. Leaf size=17 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]
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Rubi [A] time = 0.0173335, antiderivative size = 17, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[b*x]*Sqrt[4 + b*x]),x]
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Rubi in Sympy [A] time = 3.58326, size = 12, normalized size = 0.71 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b x}}{2} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x)**(1/2)/(b*x+4)**(1/2),x)
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Mathematica [A] time = 0.0202466, size = 34, normalized size = 2. \[ \frac{2 \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{2}\right )}{\sqrt{b} \sqrt{b x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[b*x]*Sqrt[4 + b*x]),x]
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Maple [B] time = 0.01, size = 60, normalized size = 3.5 \[{1\sqrt{xb \left ( bx+4 \right ) }\ln \left ({({b}^{2}x+2\,b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+4\,bx} \right ){\frac{1}{\sqrt{bx}}}{\frac{1}{\sqrt{bx+4}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x)^(1/2)/(b*x+4)^(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 + 4)*sqrt(b*x)),x, algorithm="maxima")
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Fricas [A] time = 0.215262, size = 34, normalized size = 2. \[ -\frac{\log \left (-b x + \sqrt{b x + 4} \sqrt{b x} - 2\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 4)*sqrt(b*x)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.17145, size = 15, normalized size = 0.88 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{2} \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x)**(1/2)/(b*x+4)**(1/2),x)
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GIAC/XCAS [A] time = 0.254723, size = 30, normalized size = 1.76 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 4} + \sqrt{b x} \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + 4)*sqrt(b*x)),x, algorithm="giac")
[Out]