U-substitution $\int \left (1+e^{at} \right )^{\frac{3}{2}}e^{-at}dt$

Question

How can I solve this integral only with u-substitution? Is this possible?

$$\int \left (1+e^{at} \right )^{\frac{3}{2}}e^{-at}dt$$

I know how to evaluate this integral, my question is how can I obtain this primitive without partial fractions, just substitution. Thank you.

Answer 1

$$\int \left(e^{a t}+1\right)^{3/2} e^{-a t} \, dt$$ substitute $ e^{-a t} =u$ so that $-ae^{-a t} dt=du\rightarrow dt=\dfrac{du}{-au}$

The integral becomes $$-\frac{1}{a}\int \left(\frac{1}{u}+1\right)^{3/2} \, du$$

Answer 2

Let $$ u = e^{at}, \qquad du = a e^{at} dt $$ Therefore $$ dt = \frac{1}{a u} du $$

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$$ \int \left( 1 + e^{at} \right)^{\frac{3}{2}} e^{-at} dt \quad \Rightarrow \quad \int \frac{\left( 1 + e^{at} \right)^{\frac{3}{2}}}{a u^{2}} du $$