Solve x⁵ multiplied by (x³+1) raised to power 1/2

Let $u = x^3+1$.
Then, differentiate $u$ with respect to $x$: $du = 3x^2 dx$.
From this, we can express $x^2 dx$ as $\frac{1}{3} du$.
Also, from $u = x^3+1$, we can write $x^3 = u-1$.

The integral can be written as $\int x^3 \cdot x^2 (x^3+1)^{1/2} dx$.
Substitute the expressions from Step 1:

Distribute $u^{1/2}$ inside the parenthesis:

Simplify the fractions:

Distribute the $\frac{1}{3}$:

The final answer is $\frac{2}{15}(x^3+1)^{5/2} - \frac{2}{9}(x^3+1)^{3/2} + C$.