Optimal. Leaf size=17 \[ \frac {2 \sinh ^{-1}\left (\frac {\sqrt {b x}}{2}\right )}{b} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 2.00 \[ \frac {2 \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{2}\right )}{\sqrt {b} \sqrt {b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 25, normalized size = 1.47 \[ -\frac {\log \left (-b x + \sqrt {b x + 4} \sqrt {b x} - 2\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 21, normalized size = 1.24 \[ -\frac {2 \, \log \left (\sqrt {b x + 4} - \sqrt {b x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 60, normalized size = 3.53 \[ \frac {\sqrt {\left (b x +4\right ) b x}\, \ln \left (\frac {b^{2} x +2 b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+4 b x}\right )}{\sqrt {b x}\, \sqrt {b x +4}\, \sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 32, normalized size = 1.88 \[ \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 4 \, b x} b + 4 \, b\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 33, normalized size = 1.94 \[ -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {b\,x+4}-2\right )}{\sqrt {b\,x}\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, 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.
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