Optimal. Leaf size=211 \[ -\frac{1}{2 a^5 x^2}+\frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^7}+\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^7}-\frac{\log (a+x)}{5 a^7}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^7}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7} \]
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Rubi [A] time = 0.327494, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {325, 293, 634, 618, 204, 628, 31} \[ -\frac{1}{2 a^5 x^2}+\frac{\left (1+\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2\right )}{20 a^7}+\frac{\left (1-\sqrt{5}\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )}{20 a^7}-\frac{\log (a+x)}{5 a^7}-\frac{\sqrt{\frac{1}{2} \left (5-\sqrt{5}\right )} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{5 a^7}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7} \]
Antiderivative was successfully verified.
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Rule 325
Rule 293
Rule 634
Rule 618
Rule 204
Rule 628
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a^5+x^5\right )} \, dx &=-\frac{1}{2 a^5 x^2}-\frac{\int \frac{x^2}{a^5+x^5} \, dx}{a^5}\\ &=-\frac{1}{2 a^5 x^2}-\frac{\int \frac{1}{a+x} \, dx}{5 a^7}-\frac{2 \int \frac{\frac{1}{4} \left (-1-\sqrt{5}\right ) a-\frac{1}{4} \left (1+\sqrt{5}\right ) x}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{5 a^7}-\frac{2 \int \frac{\frac{1}{4} \left (-1+\sqrt{5}\right ) a-\frac{1}{4} \left (1-\sqrt{5}\right ) x}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{5 a^7}\\ &=-\frac{1}{2 a^5 x^2}-\frac{\log (a+x)}{5 a^7}+\frac{\left (1-\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1+\sqrt{5}\right ) a+2 x}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{20 a^7}+\frac{\left (1+\sqrt{5}\right ) \int \frac{-\frac{1}{2} \left (1-\sqrt{5}\right ) a+2 x}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{20 a^7}+\frac{\int \frac{1}{a^2-\frac{1}{2} \left (1-\sqrt{5}\right ) a x+x^2} \, dx}{2 \sqrt{5} a^6}-\frac{\int \frac{1}{a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2} \, dx}{2 \sqrt{5} a^6}\\ &=-\frac{1}{2 a^5 x^2}-\frac{\log (a+x)}{5 a^7}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^2-a x-\sqrt{5} a x+2 x^2\right )}{20 a^7}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^2-a x+\sqrt{5} a x+2 x^2\right )}{20 a^7}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5-\sqrt{5}\right ) a^2-x^2} \, dx,x,-\frac{1}{2} \left (1+\sqrt{5}\right ) a+2 x\right )}{\sqrt{5} a^6}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{2} \left (5+\sqrt{5}\right ) a^2-x^2} \, dx,x,-\frac{1}{2} \left (1-\sqrt{5}\right ) a+2 x\right )}{\sqrt{5} a^6}\\ &=-\frac{1}{2 a^5 x^2}-\frac{\sqrt{\frac{2}{5 \left (5+\sqrt{5}\right )}} \tan ^{-1}\left (\frac{\left (1-\sqrt{5}\right ) a-4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )}{a^7}+\frac{\sqrt{\frac{1}{2} \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{\sqrt{\frac{1}{10} \left (5+\sqrt{5}\right )} \left (\left (1+\sqrt{5}\right ) a-4 x\right )}{2 a}\right )}{5 a^7}-\frac{\log (a+x)}{5 a^7}+\frac{\left (1-\sqrt{5}\right ) \log \left (2 a^2-a x-\sqrt{5} a x+2 x^2\right )}{20 a^7}+\frac{\left (1+\sqrt{5}\right ) \log \left (2 a^2-a x+\sqrt{5} a x+2 x^2\right )}{20 a^7}\\ \end{align*}
Mathematica [A] time = 0.145139, size = 174, normalized size = 0.82 \[ -\frac{\frac{10 a^2}{x^2}-\left (1+\sqrt{5}\right ) \log \left (a^2+\frac{1}{2} \left (\sqrt{5}-1\right ) a x+x^2\right )+\left (\sqrt{5}-1\right ) \log \left (a^2-\frac{1}{2} \left (1+\sqrt{5}\right ) a x+x^2\right )+4 \log (a+x)-2 \sqrt{10-2 \sqrt{5}} \tan ^{-1}\left (\frac{\left (\sqrt{5}-1\right ) a+4 x}{\sqrt{2 \left (5+\sqrt{5}\right )} a}\right )+2 \sqrt{2 \left (5+\sqrt{5}\right )} \tan ^{-1}\left (\frac{4 x-\left (1+\sqrt{5}\right ) a}{\sqrt{10-2 \sqrt{5}} a}\right )}{20 a^7} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.008, size = 105, normalized size = 0.5 \begin{align*}{\frac{1}{5\,{a}^{7}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}-a{{\it \_Z}}^{3}+{a}^{2}{{\it \_Z}}^{2}-{a}^{3}{\it \_Z}+{a}^{4} \right ) }{\frac{ \left ({{\it \_R}}^{3}-2\,{{\it \_R}}^{2}a-2\,{\it \_R}\,{a}^{2}+{a}^{3} \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}-3\,{{\it \_R}}^{2}a+2\,{\it \_R}\,{a}^{2}-{a}^{3}}}}-{\frac{1}{2\,{a}^{5}{x}^{2}}}-{\frac{\ln \left ( a+x \right ) }{5\,{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42332, size = 373, normalized size = 1.77 \begin{align*} -\frac{\frac{\log \left (x +{\left (a^{5}\right )}^{\frac{1}{5}}\right )}{{\left (a^{5}\right )}^{\frac{2}{5}}} + \frac{\sqrt{5} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} + 1\right )} - 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{2 \, \sqrt{5} - 10}}\right )}{{\left (a^{5}\right )}^{\frac{2}{5}} \sqrt{2 \, \sqrt{5} - 10}} - \frac{\sqrt{5} \log \left (\frac{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x -{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}{{\left (a^{5}\right )}^{\frac{1}{5}}{\left (\sqrt{5} - 1\right )} + 4 \, x +{\left (a^{5}\right )}^{\frac{1}{5}} \sqrt{-2 \, \sqrt{5} - 10}}\right )}{{\left (a^{5}\right )}^{\frac{2}{5}} \sqrt{-2 \, \sqrt{5} - 10}} + \frac{\log \left (-{\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} + 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{{\left (a^{5}\right )}^{\frac{2}{5}}{\left (\sqrt{5} + 1\right )}} - \frac{\log \left ({\left (a^{5}\right )}^{\frac{1}{5}} x{\left (\sqrt{5} - 1\right )} + 2 \, x^{2} + 2 \,{\left (a^{5}\right )}^{\frac{2}{5}}\right )}{{\left (a^{5}\right )}^{\frac{2}{5}}{\left (\sqrt{5} - 1\right )}}}{5 \, a^{5}} - \frac{1}{2 \, a^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.363622, size = 51, normalized size = 0.24 \begin{align*} - \frac{1}{2 a^{5} x^{2}} + \frac{- \frac{\log{\left (a + x \right )}}{5} + \operatorname{RootSum}{\left (625 t^{4} - 125 t^{3} + 25 t^{2} - 5 t + 1, \left ( t \mapsto t \log{\left (25 t^{2} a + x \right )} \right )\right )}}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09023, size = 250, normalized size = 1.18 \begin{align*} \frac{\sqrt{-2 \, \sqrt{5} + 10} \arctan \left (\frac{a{\left (\sqrt{5} - 1\right )} + 4 \, x}{a \sqrt{2 \, \sqrt{5} + 10}}\right )}{10 \, a^{7}} - \frac{\sqrt{2 \, \sqrt{5} + 10} \arctan \left (-\frac{a{\left (\sqrt{5} + 1\right )} - 4 \, x}{a \sqrt{-2 \, \sqrt{5} + 10}}\right )}{10 \, a^{7}} - \frac{\sqrt{5} \log \left (a^{2} - \frac{1}{2} \,{\left (\sqrt{5} a + a\right )} x + x^{2}\right )}{20 \, a^{7}} + \frac{\sqrt{5} \log \left (a^{2} + \frac{1}{2} \,{\left (\sqrt{5} a - a\right )} x + x^{2}\right )}{20 \, a^{7}} + \frac{\log \left ({\left | a^{4} - a^{3} x + a^{2} x^{2} - a x^{3} + x^{4} \right |}\right )}{20 \, a^{7}} - \frac{\log \left ({\left | a + x \right |}\right )}{5 \, a^{7}} - \frac{1}{2 \, a^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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