Optimal. Leaf size=31 \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]
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Rubi [A] time = 0.0148877, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 44} \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (5+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (5+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{5 x^3}-\frac{1}{25 x^2}+\frac{1}{125 x}-\frac{1}{125 (5+x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{20 x^4}+\frac{1}{50 x^2}+\frac{\log (x)}{125}-\frac{1}{250} \log \left (5+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0036833, size = 31, normalized size = 1. \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 24, normalized size = 0.8 \begin{align*} -{\frac{1}{20\,{x}^{4}}}+{\frac{1}{50\,{x}^{2}}}+{\frac{\ln \left ( x \right ) }{125}}-{\frac{\ln \left ({x}^{2}+5 \right ) }{250}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931927, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, x^{2} - 5}{100 \, x^{4}} - \frac{1}{250} \, \log \left (x^{2} + 5\right ) + \frac{1}{250} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29247, size = 84, normalized size = 2.71 \begin{align*} -\frac{2 \, x^{4} \log \left (x^{2} + 5\right ) - 4 \, x^{4} \log \left (x\right ) - 10 \, x^{2} + 25}{500 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112281, size = 24, normalized size = 0.77 \begin{align*} \frac{\log{\left (x \right )}}{125} - \frac{\log{\left (x^{2} + 5 \right )}}{250} + \frac{2 x^{2} - 5}{100 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14959, size = 43, normalized size = 1.39 \begin{align*} -\frac{3 \, x^{4} - 10 \, x^{2} + 25}{500 \, x^{4}} - \frac{1}{250} \, \log \left (x^{2} + 5\right ) + \frac{1}{250} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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