Optimal. Leaf size=49 \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]
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Rubi [A] time = 0.0267747, antiderivative size = 49, 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} \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^{11} \left (a+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{10 a x^{10}}+\frac{b}{5 a^2 x^5}+\frac{b^2 \log (x)}{a^3}-\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}\\ \end{align*}
Mathematica [A] time = 0.0063012, size = 49, normalized size = 1. \[ -\frac{b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{5 a^2 x^5}-\frac{1}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{10\,a{x}^{10}}}+{\frac{b}{5\,{x}^{5}{a}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04092, size = 63, normalized size = 1.29 \begin{align*} -\frac{b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{3}} + \frac{b^{2} \log \left (x^{5}\right )}{5 \, a^{3}} + \frac{2 \, b x^{5} - a}{10 \, a^{2} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72871, size = 115, normalized size = 2.35 \begin{align*} -\frac{2 \, b^{2} x^{10} \log \left (b x^{5} + a\right ) - 10 \, b^{2} x^{10} \log \left (x\right ) - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38351, size = 42, normalized size = 0.86 \begin{align*} \frac{- a + 2 b x^{5}}{10 a^{2} x^{10}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2008, size = 74, normalized size = 1.51 \begin{align*} -\frac{b^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{3}} + \frac{b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{3 \, b^{2} x^{10} - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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