Evaluate $\int \frac{2-x^3}{(1+x^3)^{3/2}} dx$

$$\int \frac{2-x^3}{(1+x^3)^{3/2}} dx=\int \frac{2x^{-3}-1}{(x^{-2}+x)^{3/2}} dx$$ Now substitute $t=x^{-2}+x$.

@ClaudeLeibovici has a point, because what you did is a well-worn technique, that of using an Ansatz. The basic idea is to make an educated guess as to the form of the solution, then make it more specific with your calculations, as you did. So it's worth understanding what makes a specific Ansatz a sensible starting point:

• It makes sense to assume a $(1+x^2)^{-3/2}$ factor results from differentiating $(1+x^2)^{-1/2}$: if nothing else, integration by parts makes sense of that.
• You could have started with a more general Ansatz, $I(x)=f(x)(1+x^3)^{-3/2}$, so the problem is equivalent to $(x^3+1)f^\prime-\tfrac32x^2f=2-x^3$. Since constant $f$ doesn't solve this, it's natural to try a linear $f$ next, which worked for you.

Maybe this answer isn't what you were looking for, but it's important to understand how to use an Ansatz as more than an accident.

Split the integral and integrate by parts:

$$\begin{align}I = \int\dfrac{2 - x^3}{\left(1 + x^3\right)^{3/2}}\,\mathrm dx &\equiv\int\dfrac2{\left(1 + x^3\right)^{3/2}}\,\mathrm dx - \int\dfrac{x^2}{\left(1 + x^3\right)^{3/2}}x\,\mathrm dx \\ &= \int\dfrac2{\left(1 + x^3\right)^{3/2}}\,\mathrm dx - \left(-\dfrac{2x}{3\left(1 + x^3\right)^{1/2}} + \dfrac23\int\dfrac{1 + x^3}{\left(1 + x^3\right)^{3/2}}\,\mathrm dx\right) \\ &= \int\dfrac{6 - 2 - 2x^3}{3\left(1 + x^3\right)^{3/2}}\,\mathrm dx + \dfrac{2x}{3\sqrt{1 + x^3}} \\ &= \dfrac23\int\dfrac{2 - x^3}{\left(1 + x^3\right)^{3/2}}\,\mathrm dx + \dfrac{2x}{3\sqrt{1 + x^3}} \\ &= \dfrac23I + \dfrac{2x}{3\sqrt{1 + x^3}} \\ \implies I &= \dfrac{2x}{\sqrt{1 + x^3}}\end{align}$$