Evaluate the indefinite integral.
\[
\int x^{3} \sqrt{11+x^{4}} d x
\]

Step 1 ：First, we recognize that this integral is in the form of a standard formula for integration by substitution. The standard formula is \(\int x^n f(x^{n+1}) dx = \frac{1}{n+1} \int f(u) du\), where \(u = x^{n+1}\).

Step 2 ：In our case, \(n = 3\) and \(f(x) = \sqrt{11+x}\). So, we let \(u = x^4\), then \(du = 4x^3 dx\).

Step 3 ：Substitute \(u\) and \(du\) into the integral, we get \(\frac{1}{4} \int \sqrt{11+u} du\).

Step 4 ：Now, we can use the power rule for integration to solve this integral. The power rule states that \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.

Step 5 ：Applying the power rule, we get \(\frac{1}{4} \cdot \frac{2}{3} (11+u)^{3/2} + C\).

Step 6 ：Substitute \(u = x^4\) back into the equation, we get \(\frac{1}{6} (11+x^4)^{3/2} + C\).

Step 7 ：So, the indefinite integral of \(x^3 \sqrt{11+x^4}\) with respect to \(x\) is \(\boxed{\frac{1}{6} (11+x^4)^{3/2} + C}\).