We are asked to evaluate the integral:

\[
\int \frac{-6\sqrt{x}}{1 + x^3} \, dx
\]

First, rewrite the square root as a fractional exponent:

\[
\int \frac{-6x^{1/2}}{1 + x^3} \, dx
\]

This integral does not have an elementary solution. However, there are ways to approach it through special functions or numerical methods. Here's a step-by-step guide for setting it up:

### Substitution attempt:
Let’s try a substitution to simplify the expression. One substitution to attempt is based on the term \( x^3 \).

Let:
\[
u = x^{3/2} \quad \text{which implies} \quad x = u^{2/3}, \quad dx = \frac{2}{3} u^{-1/3} \, du
\]

This substitution might not immediately work well for this problem. Instead, the most efficient approach is to recognize that the integral is best evaluated using more advanced techniques (e.g., series expansion, numerical integration, or special functions such as the hypergeometric function).

Would you like me to continue with a more specific method, like a series expansion or perhaps using a numerical approach?

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**Related questions:**
1. How do we compute integrals involving square roots of polynomials?
2. What is the method of series expansion for integrals?
3. How do substitutions help simplify complex integrals?
4. What are special functions used for in integrals?
5. When should numerical integration be used?

**Tip**: If an integral seems unsolvable with basic techniques, consider substitutions or special functions!

Explore the integration of (-6√x) / (1 + x^3) using advanced calculus techniques. Learn how substitution, series expansion, and special functions like the hypergeometric function can be applied. This step-by-step guide is ideal for university-level students and those studying advanced calculus.

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