Evaluate the indefinite integral: integral (6x2)/(sqrt(1 - (2x3 + 5)2)) dx

The integral to be evaluated is:

$$\int \frac{6x^2}{\sqrt{1 - (2x^3 + 5)^2}} \, dx$$

This integral resembles the form $\int \frac{du}{\sqrt{1 - u^2}} = \arcsin(u) + C$.
Let $u = 2x^3 + 5$.

Differentiate $u$ with respect to $x$:

$$\frac{du}{dx} = \frac{d}{dx}(2x^3 + 5) = 6x^2$$

From this, we can write $du = 6x^2 \, dx$.

Substitute $u = 2x^3 + 5$ and $du = 6x^2 \, dx$ into the original integral:

$$\int \frac{du}{\sqrt{1 - u^2}}$$

This is a standard integral:

$$\int \frac{du}{\sqrt{1 - u^2}} = \arcsin(u) + C$$

$$\arcsin(2x^3 + 5) + C$$

The phrase "to its 25th term" is not applicable to this type of integration problem, which yields a direct function. We have found the indefinite integral.