It seems like you're trying to work with an integral expression that may be missing some details, specifically the limits of integration or additional context. From what you've provided, it looks like you want to evaluate an integral of the following form:

$\int \left( \sqrt{x^{\frac{1}{2}}} - 2^{\frac{3}{2}} + x^{\frac{5}{2}} \right) \, dx$

We can break this into three parts:

1. $\sqrt{x^{\frac{1}{2}}} = x^{\frac{1}{4}}$, so the first term is $\int x^{\frac{1}{4}} \, dx$
2. $-2^{\frac{3}{2}}$ is just a constant, so the second term is $-2^{\frac{3}{2}} \int 1 \, dx$
3. $x^{\frac{5}{2}}$, so the third term is $\int x^{\frac{5}{2}} \, dx$

Now let's calculate each term:

1. $\int x^{\frac{1}{4}} \, dx = \frac{x^{\frac{5}{4}}}{\frac{5}{4}} = \frac{4}{5} x^{\frac{5}{4}}$
2. $-2^{\frac{3}{2}} \int 1 \, dx = -2^{\frac{3}{2}} x$
3. $\int x^{\frac{5}{2}} \, dx = \frac{x^{\frac{7}{2}}}{\frac{7}{2}} = \frac{2}{7} x^{\frac{7}{2}}$

Thus, the integral of the expression is:

$\frac{4}{5} x^{\frac{5}{4}} - 2^{\frac{3}{2}} x + \frac{2}{7} x^{\frac{7}{2}} + C$

Where $C$ is the constant of integration.

Would you like to clarify any part or ask for further details? Here are some related questions to help explore the topic further:

1. What is the power rule for integrals and how is it applied?
2. How do constants affect the integration of terms?
3. Can you explain the process of integrating polynomials and their fractions?
4. What is the general formula for integrating expressions with powers of $x$?
5. How can definite integrals be calculated using this same technique?

Tip: Always simplify the expression first before integrating, especially when dealing with powers of $x$, as it helps avoid confusion and errors.