Optimal. Leaf size=63 \[ -\frac{b^2}{5 a^3 x^5}+\frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]
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Rubi [A] time = 0.0325913, antiderivative size = 63, 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} \[ -\frac{b^2}{5 a^3 x^5}+\frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^{16} \left (a+b x^5\right )} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{1}{15 a x^{15}}+\frac{b}{10 a^2 x^{10}}-\frac{b^2}{5 a^3 x^5}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log \left (a+b x^5\right )}{5 a^4}\\ \end{align*}
Mathematica [A] time = 0.006567, size = 63, normalized size = 1. \[ -\frac{b^2}{5 a^3 x^5}+\frac{b^3 \log \left (a+b x^5\right )}{5 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{10 a^2 x^{10}}-\frac{1}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 56, normalized size = 0.9 \begin{align*} -{\frac{1}{15\,a{x}^{15}}}+{\frac{b}{10\,{a}^{2}{x}^{10}}}-{\frac{{b}^{2}}{5\,{a}^{3}{x}^{5}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991964, size = 78, normalized size = 1.24 \begin{align*} \frac{b^{3} \log \left (b x^{5} + a\right )}{5 \, a^{4}} - \frac{b^{3} \log \left (x^{5}\right )}{5 \, a^{4}} - \frac{6 \, b^{2} x^{10} - 3 \, a b x^{5} + 2 \, a^{2}}{30 \, a^{3} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8381, size = 139, normalized size = 2.21 \begin{align*} \frac{6 \, b^{3} x^{15} \log \left (b x^{5} + a\right ) - 30 \, b^{3} x^{15} \log \left (x\right ) - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 38.5353, size = 56, normalized size = 0.89 \begin{align*} - \frac{2 a^{2} - 3 a b x^{5} + 6 b^{2} x^{10}}{30 a^{3} x^{15}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15772, size = 93, normalized size = 1.48 \begin{align*} \frac{b^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} - \frac{b^{3} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{11 \, b^{3} x^{15} - 6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - 2 \, a^{3}}{30 \, a^{4} x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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