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we have $$\int_0^1 x^2(1-x)^2(a+bx)e^xdx=2(7a-32b)e+2(-19a+87b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$. (Values here)

we have $$\int_0^1 x^2(1-x)^2(a+bx)e^xdx=2(7a-32b)e+2(-19a+87b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$. (Values here)

Edit: $$\int_0^1 \frac{x^3(1-x)^3(a+bx)}{1+x}dx=a\left(\frac{111}{20}-8\ln 2\right)+b\left(8\ln 2-\frac{194}{35}\right)=\ln 2-\frac{551}{800}>0$$ Values here

we have $$\int_0^1 x^2(1-x)(a+bx)e^xdx=2(7a-32b)e+2(-19a+87b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$. (Values here)

we have $$\int_0^1 x^2(1-x)^2(a+bx)e^xdx=2(7a-32b)e+2(-19a+87b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$. (Values here)

Set $a=0.03+\ln 2$ and $b=-0.97+\ln 2$. we have $$\int_0^1 x(1-x)(a+bx)e^xdx=(-a+3b)e+(3a-8b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$.

we have $$\int_0^1 x^2(1-x)(a+bx)e^xdx=2(7a-32b)e+2(-19a+87b)=0.03-e+\ln 2+2>0$$ since $a> 0$ and $a+b>0$. (Values here)