Optimal. Leaf size=23 \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]
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Rubi [A] time = 0.0087873, antiderivative size = 23, 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} \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x \left (2 b+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x (2 b+b x)} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{2 b+b x} \, dx,x,x^5\right )\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )}{10 b}\\ &=\frac{\log (x)}{2 b}-\frac{\log \left (2+x^5\right )}{10 b}\\ \end{align*}
Mathematica [A] time = 0.005415, size = 23, normalized size = 1. \[ \frac{\log (x)}{2 b}-\frac{\log \left (x^5+2\right )}{10 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{2\,b}}-{\frac{\ln \left ({x}^{5}+2 \right ) }{10\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994236, size = 28, normalized size = 1.22 \begin{align*} -\frac{\log \left (x^{5} + 2\right )}{10 \, b} + \frac{\log \left (x^{5}\right )}{10 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70775, size = 47, normalized size = 2.04 \begin{align*} -\frac{\log \left (x^{5} + 2\right ) - 5 \, \log \left (x\right )}{10 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.345973, size = 15, normalized size = 0.65 \begin{align*} \frac{\log{\left (x \right )}}{2 b} - \frac{\log{\left (x^{5} + 2 \right )}}{10 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19682, size = 28, normalized size = 1.22 \begin{align*} -\frac{\log \left ({\left | x^{5} + 2 \right |}\right )}{10 \, b} + \frac{\log \left ({\left | x \right |}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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