Are there any closed form solution to this integral? If not are there any good numerical methods to evaluate it?
$$ \int_{0}^{\infty} (x+3)^{2.5} \times (1+x)^{3.5} \times exp(-x) dx$$
I have tried integration by parts and some online integral calculators, but all attempts are failed. Are there any good suggestions?
Povided that $\Re(a)>0\land \Re(c)>-1$ $$\int_0^\infty (x+a)^b \, x^c\,e^{-x}\,dx=\Gamma (c+1)\, a^{b+c+1} \,\,U(c+1,b+c+2,a)$$
Assuming that $(a,b,c)$ are positive numbers, $$\int_0^\infty (x+a)^b \, x^c\,e^{-x}\,dx=\Gamma (c+1)\,\, U(-b,-b-c,a)$$
In both cases appears Kummer's confluent hypergeometric function. So,a series solution is feasible using the binomial expansion.
$$\int_{0}^{\infty} (x+3)^{2.5} \, (x+\color{red}{0})^{3.5} \, e^{-x}\, dx=\frac{105 \sqrt{\pi }}{16}\,\, U\left(-\frac{5}{2},-6,3\right)=2070.92$$