Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]
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Rubi [A] time = 0.02158, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{5 a x^5}-\frac{b \log (x)}{a^2}+\frac{b \log \left (a+b x^5\right )}{5 a^2}\\ \end{align*}
Mathematica [A] time = 0.0061167, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^5\right )}{5 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,a{x}^{5}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05955, size = 45, normalized size = 1.29 \begin{align*} \frac{b \log \left (b x^{5} + a\right )}{5 \, a^{2}} - \frac{b \log \left (x^{5}\right )}{5 \, a^{2}} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7038, size = 80, normalized size = 2.29 \begin{align*} \frac{b x^{5} \log \left (b x^{5} + a\right ) - 5 \, b x^{5} \log \left (x\right ) - a}{5 \, a^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.997335, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{5 a x^{5}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21634, size = 57, normalized size = 1.63 \begin{align*} \frac{b \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{2}} - \frac{b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{b x^{5} - a}{5 \, a^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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