Optimal. Leaf size=19 \[ \frac {\log (x)}{2}-\log \left (b \sqrt {x}+2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 266
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\log (x)}{2}-\log \left (b \sqrt {x}+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 15, normalized size = 0.79 \[ -\log \left (b \sqrt {x} + 2\right ) + \log \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 17, normalized size = 0.89 \[ -\log \left ({\left | b \sqrt {x} + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.84 \[ \frac {\ln \relax (x )}{2}-\ln \left (b \sqrt {x}+2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 15, normalized size = 0.79 \[ -\log \left (b \sqrt {x} + 2\right ) + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 10, normalized size = 0.53 \[ -2\,\mathrm {atanh}\left (b\,\sqrt {x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 19, normalized size = 1.00 \[ \begin {cases} \frac {\log {\relax (x )}}{2} - \log {\left (\sqrt {x} + \frac {2}{b} \right )} & \text {for}\: b \neq 0 \\\frac {\log {\relax (x )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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