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Answer
$$ \displaystyle\int 2{x}^{5}{\cdot}{\left({x}^{2}+1\right)}^{20}\, \mathrm d x = {{x^6\,\left(231\,x^{40}+4830\,x^{38}+48070\,x^{36}+302841\,x^{34}+ 1354815\,x^{32}+4576264\,x^{30}+12113640\,x^{28}+25741485\,x^{26}+ 44618574\,x^{24}+63740820\,x^{22}+75508356\,x^{20}+74364290\,x^{18}+ 60843510\,x^{16}+41186376\,x^{14}+22881320\,x^{12}+10296594\,x^{10}+ 3677355\,x^8+1009470\,x^6+201894\,x^4+26565\,x^2+1771\right)}\over{ 5313}} $$
Explanation
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