To find the antiderivative of the expression \( \frac{5x^8 + 3x^3}{4x^2} \), let's first simplify the expression by dividing each term in the numerator by the denominator:

\[
\frac{5x^8}{4x^2} + \frac{3x^3}{4x^2}
\]

Simplifying each term:

1. **Simplify \( \frac{5x^8}{4x^2} \)**:
   \[
   \frac{5x^8}{4x^2} = \frac{5}{4}x^{8-2} = \frac{5}{4}x^6
   \]

2. **Simplify \( \frac{3x^3}{4x^2} \)**:
   \[
   \frac{3x^3}{4x^2} = \frac{3}{4}x^{3-2} = \frac{3}{4}x
   \]

So, the simplified expression is:

\[
\frac{5}{4}x^6 + \frac{3}{4}x
\]

Now, we find the antiderivative of each term:

1. **Antiderivative of \( \frac{5}{4}x^6 \)**:
   \[
   \int \frac{5}{4}x^6 \, dx = \frac{5}{4} \cdot \frac{x^{7}}{7} = \frac{5}{28}x^7
   \]

2. **Antiderivative of \( \frac{3}{4}x \)**:
   \[
   \int \frac{3}{4}x \, dx = \frac{3}{4} \cdot \frac{x^{2}}{2} = \frac{3}{8}x^2
   \]

Finally, combine the results and add the constant of integration \( C \):

\[
\frac{5}{28}x^7 + \frac{3}{8}x^2 + C
\]

### Final Answer:
\[
\int \left(\frac{5x^8 + 3x^3}{4x^2}\right) \, dx = \frac{5}{28}x^7 + \frac{3}{8}x^2 + C
\]

Would you like to see more details or have any other questions?

### Related Questions:
1. How do you simplify an expression before integrating?
2. What is the antiderivative of \( \frac{7x^6 + 4x^2}{2x} \)?
3. Can you explain how to handle integrals of rational functions?
4. What are some strategies for integrating more complex polynomials?
5. How do you find the definite integral of a polynomial function?

### Tip:
Always simplify the expression as much as possible before integrating. This can make the process much easier and help avoid mistakes.

Explore how to find the antiderivative of the expression (5x^8 + 3x^3) / (4x^2) step-by-step. Learn about integration, polynomial division, and antiderivatives with detailed explanations and examples. Suitable for advanced high school students and anyone studying calculus.

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