$\frac{d}{dx}\left(x^2\cdot 2^x\ln\left(2x\right)\right)=2^{\left(x+1\right)}x\ln\left(2x\right)+x^2\left(\ln\left(2\right)2^x\ln\left(2x\right)+\frac{2^{\left(x+1\right)}}{2x}\right)$
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Integral
$\int x^2\cdot 2^x\ln\left(2x\right)dx=\frac{x^{3}\ln\left(2x\right)}{3}-\frac{1}{9}x^{3}+C_0$
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