Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]
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Rubi [A] time = 0.0242778, 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} \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^9 \left (a+b x^8\right )} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,x^8\right )\\ &=\frac{1}{8} \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^8\right )\\ &=-\frac{1}{8 a x^8}-\frac{b \log (x)}{a^2}+\frac{b \log \left (a+b x^8\right )}{8 a^2}\\ \end{align*}
Mathematica [A] time = 0.0064687, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^8\right )}{8 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{8 a x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,a{x}^{8}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{8}+a \right ) }{8\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968044, size = 45, normalized size = 1.29 \begin{align*} \frac{b \log \left (b x^{8} + a\right )}{8 \, a^{2}} - \frac{b \log \left (x^{8}\right )}{8 \, a^{2}} - \frac{1}{8 \, a x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28701, size = 80, normalized size = 2.29 \begin{align*} \frac{b x^{8} \log \left (b x^{8} + a\right ) - 8 \, b x^{8} \log \left (x\right ) - a}{8 \, a^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.82565, size = 31, normalized size = 0.89 \begin{align*} - \frac{1}{8 a x^{8}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{8} \right )}}{8 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16192, size = 58, normalized size = 1.66 \begin{align*} -\frac{b \log \left (x^{8}\right )}{8 \, a^{2}} + \frac{b \log \left ({\left | b x^{8} + a \right |}\right )}{8 \, a^{2}} + \frac{b x^{8} - a}{8 \, a^{2} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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