We can solve the integral:

$I = \int -\frac{6x^3}{\left(1-5x^4\right)^3} \, dx$

To proceed, we'll perform a substitution to simplify the expression.

Step 1: Substitution

Let: $u = 1 - 5x^4$

Then, differentiate $u$ with respect to $x$: $du = -20x^3 \, dx \quad \text{or} \quad dx = \frac{du}{-20x^3}$

Step 2: Substituting in the integral

Now substitute in the integral:

$I = \int -\frac{6x^3}{u^3} \cdot \frac{du}{-20x^3}$

The $x^3$ terms cancel out, and the constants simplify:

$I = \int \frac{6}{20} \cdot \frac{1}{u^3} \, du$

Simplify the constants:

$I = \frac{3}{10} \int u^{-3} \, du$

Step 3: Integrating

The integral of $u^{-3}$ is:

$\int u^{-3} \, du = \frac{u^{-2}}{-2} = -\frac{1}{2u^2}$

Thus, the integral becomes:

$I = \frac{3}{10} \cdot \left(-\frac{1}{2u^2}\right)$

Simplifying:

$I = -\frac{3}{20u^2}$

Step 4: Substitute $u = 1 - 5x^4$ back in:

Now, substitute $u = 1 - 5x^4$ back into the result:

$I = -\frac{3}{20(1 - 5x^4)^2} + C$

Thus, the final answer is:

$I = -\frac{3}{20(1 - 5x^4)^2} + C$

Would you like any further clarification or more details?

Here are 5 related questions for further exploration:

1. How do you perform integration by parts?
2. How does substitution help simplify integrals?
3. What are some common techniques for solving rational functions?
4. How can you recognize when to use trigonometric substitution?
5. What are some strategies for solving integrals involving powers of trigonometric functions?

Tip: When integrating complex fractions, substitution can often turn a difficult integral into something more manageable by simplifying the expression.