Optimal. Leaf size=19 \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]
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Rubi [A] time = 0.0082974, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (2+b \sqrt{x}\right ) x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (2+b x)} \, dx,x,\sqrt{x}\right )\\ &=-\left (b \operatorname{Subst}\left (\int \frac{1}{2+b x} \, dx,x,\sqrt{x}\right )\right )+\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sqrt{x}\right )\\ &=-\log \left (2+b \sqrt{x}\right )+\frac{\log (x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0045757, size = 19, normalized size = 1. \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-\ln \left ( 2+b\sqrt{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974452, size = 20, normalized size = 1.05 \begin{align*} -\log \left (b \sqrt{x} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30217, size = 49, normalized size = 2.58 \begin{align*} -\log \left (b \sqrt{x} + 2\right ) + \log \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.57503, size = 19, normalized size = 1. \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} - \log{\left (\sqrt{x} + \frac{2}{b} \right )} & \text{for}\: b \neq 0 \\\frac{\log{\left (x \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11872, size = 23, normalized size = 1.21 \begin{align*} -\log \left ({\left | b \sqrt{x} + 2 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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