Answer
$$ \displaystyle\int x{\cdot}{\left(2x+1\right)}^{15}\, \mathrm d x = {{x^2\,\left(65536\,x^{15}+522240\,x^{14}+1949696\,x^{13}+4526080\, x^{12}+7311360\,x^{11}+8712704\,x^{10}+7920640\,x^9+5601024\,x^8+ 3111680\,x^7+1361360\,x^6+466752\,x^5+123760\,x^4+24752\,x^3+3570\,x ^2+340\,x+17\right)}\over{34}} $$
Explanation
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