Answer
$$ \displaystyle\int {x}^{3}{\cdot}{\left(3+4{x}^{3}\right)}^{8}\, \mathrm d x = {{x^4\,\left(4451532800\,x^{24}+29914300416\,x^{21}+89232998400\,x ^{18}+154983628800\,x^{15}+172540368000\,x^{12}+127414425600\,x^9+ 62114532480\,x^6+19014652800\,x^3+3119591475\right)}\over{1901900}} $$
Explanation
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