\(I=\int\left[\left(x^{3} d x\right) /\left\\{\sqrt{\left. \left.\left(1+x^{8}\right)\right\\}\right]}\right.\right.\) (a) \(\log \mid x^{4}+\sqrt{\left(1+x^{8}\right) \mid+c}\) (b) \(\log \mid \sqrt{\left(x^{8}+1\right) \mid+c}\) (c) \((1 / 4) \log \mid \mathrm{x}^{4}+\sqrt{\left(1+\mathrm{x}^{8}\right) \mid+\mathrm{c}}\) (d) none of these

Let's make the substitution $$u = 1 + x^8.$$ With this substitution, we'll be able to simplify the denominator of the integrand. Now, we need to find the differential of the new variable, \(du\).

Differentiate \(u\) with respect to \(x\) to get the relationship between \(du\) and \(dx\): $$\frac{du}{dx} = \frac{d}{dx}(1+x^8) = 8x^7.$$ Now, solve for \(dx\) in terms of \(du\): $$dx = \frac{du}{8x^7}.$$

Now that we have a substitution and a differential expression for \(dx\), let's substitute them into our integral. Remember that we need to transform both the integrand and the differential: $$I=\int\frac{x^{3} dx}{\sqrt{1+x^{8}}} = \int\frac{x^3}{\sqrt{u}}\cdot \frac{du}{8x^7} = \frac{1}{8}\int\frac{du}{u^{\frac{1}{2}}x^4}.$$

Notice that in our new integral, we have \(x^4\) in the denominator. To simplify further, we need to express \(x^4\) in terms of \(u\). Since \(u = 1 + x^8\), we can rewrite it as: $$x^8 = u - 1,$$ then take the square root on both sides: $$x^4 = \sqrt{u-1}.$$ Now, substitute this expression for \(x^4\) in the integrand: $$I = \frac{1}{8}\int\frac{du}{u^{\frac{1}{2}}\sqrt{u-1}} = \frac{1}{8}\int\frac{du}{u\sqrt{u-1}}.$$ Now we have a simpler integral. To solve this, notice that the expression inside the integral is a derivative of a logarithmic function. Specifically, the derivative of \(\log\mid\sqrt{u}+\sqrt{u-1}\mid\) is \(\frac{1}{u\sqrt{u-1}}\). So, $$I = \frac{1}{8}\int\frac{du}{u\sqrt{u-1}} = \frac{1}{8}\log\mid\sqrt{u}+\sqrt{u-1}\mid+c.$$

Finally, we need to convert our result back to the original variable, \(x\). Recall that \(u = 1 + x^8\). Substitute this expression back into our result: $$I = \frac{1}{8}\log\mid\sqrt{1+x^8}+\sqrt{1+x^8-1}\mid+c = \frac{1}{8}\log\mid\sqrt{1+x^8}+\sqrt{x^8}\mid+c.$$ Now, compare this result to the given options (a), (b), and (c). Our result exactly matches option (c). Therefore, the correct answer is (c) $$\frac{1}{4}\log\mid x^{4}+\sqrt{\left(1+x^{8}\right)}\mid+c.$$