Find the indefinite integral.
\[
\int x^{4} e^{x^{5}} d x
\]

Step 1 ：First, we recognize that this integral is in the form of an integral that can be solved using the method of substitution. We can let \(u = x^5\).

Step 2 ：Differentiating both sides with respect to \(x\), we get \(du = 5x^4 dx\).

Step 3 ：We can rearrange this to get \(dx = \frac{1}{5x^4} du\).

Step 4 ：Substituting \(u = x^5\) and \(dx = \frac{1}{5x^4} du\) into the integral, we get \(\int x^{4} e^{x^{5}} dx = \int e^u \frac{1}{5x^4} du\).

Step 5 ：Notice that \(x^4\) in the denominator cancels out with \(x^4\) in the numerator, so we are left with \(\int e^u \frac{1}{5} du\).

Step 6 ：This integral can be easily solved to get \(\frac{1}{5} e^u + C\), where \(C\) is the constant of integration.

Step 7 ：Substituting \(u = x^5\) back into the integral, we get the final answer \(\frac{1}{5} e^{x^5} + C\).

Step 8 ：So, the indefinite integral \(\int x^{4} e^{x^{5}} dx = \boxed{\frac{1}{5} e^{x^5} + C}\).