Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]
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Rubi [A] time = 0.0104789, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]
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 (a+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,x^5\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^5\right )}{5 a}-\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,x^5\right )}{5 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0048936, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a+b x^5\right )}{5 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{5}+a \right ) }{5\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04988, size = 31, normalized size = 1.41 \begin{align*} -\frac{\log \left (b x^{5} + a\right )}{5 \, a} + \frac{\log \left (x^{5}\right )}{5 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67588, size = 49, normalized size = 2.23 \begin{align*} -\frac{\log \left (b x^{5} + a\right ) - 5 \, \log \left (x\right )}{5 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.341211, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{5} \right )}}{5 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17286, size = 30, normalized size = 1.36 \begin{align*} -\frac{\log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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