To solve the integral

$I = \int \frac{x^3 e^{x^2}}{(x^2 + 1)^2} \, dx,$

let's try substitution to simplify the expression.

Step 1: Substitution

Let $u = x^2 + 1$, which implies that:

$du = 2x \, dx \quad \text{or} \quad \frac{du}{2} = x \, dx.$

Now, observe that $x^2 = u - 1$, so:

$x^3 = x \cdot x^2 = x(u - 1).$

Thus, the integral becomes:

$I = \int \frac{x(u - 1) e^{u - 1}}{u^2} \cdot \frac{du}{2}.$

Step 2: Simplifying the Integral

We can now factor constants and simplify:

$I = \frac{1}{2} \int \frac{x(u - 1) e^{u - 1}}{u^2} \, du.$

Let’s now evaluate or proceed with any further simplification.

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Tip: Substitutions are often helpful in integrals when the inner function in a composite function can simplify both the expression and its derivative.