Optimal. Leaf size=21 \[ \frac {\log (x)}{2}-\frac {3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 21, 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} \[ \frac {\log (x)}{2}-\frac {3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]
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 [3]{x}\right ) x} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x (2+b x)} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\sqrt [3]{x}\right )-\frac {1}{2} (3 b) \operatorname {Subst}\left (\int \frac {1}{2+b x} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3}{2} \log \left (2+b \sqrt [3]{x}\right )+\frac {\log (x)}{2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ \frac {\log (x)}{2}-\frac {3}{2} \log \left (b \sqrt [3]{x}+2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 17, normalized size = 0.81 \[ -\frac {3}{2} \, \log \left (b x^{\frac {1}{3}} + 2\right ) + \frac {3}{2} \, \log \left (x^{\frac {1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 17, normalized size = 0.81 \[ -\frac {3}{2} \, \log \left ({\left | b x^{\frac {1}{3}} + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.76 \[ \frac {\ln \relax (x )}{2}-\frac {3 \ln \left (b \,x^{\frac {1}{3}}+2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 15, normalized size = 0.71 \[ -\frac {3}{2} \, \log \left (b x^{\frac {1}{3}} + 2\right ) + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 10, normalized size = 0.48 \[ -3\,\mathrm {atanh}\left (b\,x^{1/3}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 22, normalized size = 1.05 \[ \begin {cases} \frac {\log {\relax (x )}}{2} - \frac {3 \log {\left (\sqrt [3]{x} + \frac {2}{b} \right )}}{2} & \text {for}\: b \neq 0 \\\frac {\log {\relax (x )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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