Optimal. Leaf size=19 \[ -\log \left (2+b \sqrt {x}\right )+\frac {\log (x)}{2} \]
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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 = {272, 36, 29, 31}
\begin {gather*} \frac {\log (x)}{2}-\log \left (b \sqrt {x}+2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (2+b \sqrt {x}\right ) x} \, dx &=2 \text {Subst}\left (\int \frac {1}{x (2+b x)} \, dx,x,\sqrt {x}\right )\\ &=-\left (b \text {Subst}\left (\int \frac {1}{2+b x} \, dx,x,\sqrt {x}\right )\right )+\text {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 \begin {gather*} -\log \left (2+b \sqrt {x}\right )+\log \left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 16, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\ln \left (x \right )}{2}-\ln \left (2+b \sqrt {x}\right )\) | \(16\) |
default | \(\frac {\ln \left (x \right )}{2}-\ln \left (2+b \sqrt {x}\right )\) | \(16\) |
meijerg | \(-\ln \left (1+\frac {b \sqrt {x}}{2}\right )+\frac {\ln \left (x \right )}{2}-\ln \left (2\right )+\ln \left (b \right )\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 0.79 \begin {gather*} -\log \left (b \sqrt {x} + 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 15, normalized size = 0.79 \begin {gather*} -\log \left (b \sqrt {x} + 2\right ) + \log \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 19, normalized size = 1.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 17, normalized size = 0.89 \begin {gather*} -\log \left ({\left | b \sqrt {x} + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
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 \begin {gather*} -2\,\mathrm {atanh}\left (b\,\sqrt {x}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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