Optimal. Leaf size=17 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]
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Rubi [A] time = 0.0038164, 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, Rules used = {63, 215} \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b x} \sqrt{4+b x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{4+x^2}} \, dx,x,\sqrt{b x}\right )}{b}\\ &=\frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0100814, 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.
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Maple [B] time = 0.006, size = 60, normalized size = 3.5 \begin{align*}{\sqrt{bx \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}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11496, size = 59, normalized size = 3.47 \begin{align*} -\frac{\log \left (-b x + \sqrt{b x + 4} \sqrt{b x} - 2\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36611, size = 15, normalized size = 0.88 \begin{align*} \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{2} \right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1591, size = 30, normalized size = 1.76 \begin{align*} -\frac{2 \, \log \left ({\left | -\sqrt{b x + 4} + \sqrt{b x} \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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