Answer
$\frac{1}{3}(1+x^4)^{3/4}+C$
Work Step by Step
Step 1. Examine the integrand. The power $-\frac{1}{4}$ is likely from a power of $\frac{3}{4}$ and the term $x^3$ from the differentiation of $x^4$.
Step 2. Test: $f(x)=(1+x^4)^{3/4}$. We have $f'(x)=\frac{3}{4}(1+x^4)^{-1/4}(4x^3)=3x^3(1+x^4)^{-1/4}$
Step 3. Thus, we have $\int x^3(1+x^4)^{-1/4}=\frac{1}{3}(1+x^4)^{3/4}+C$, where $C$ is a constant.
