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### Indefinite integrals from a differential binomial

The expression

$$x^{m}(ax^n + b)^p \tag{1}$$

where $m,n,p \in \mathbb{Q}$ and $a,b$ are real non-zero constants is called a differential binomial

The integrals from a differential binomial have the form

$$\int x^{m}(ax^n + b)^p dx \tag{2}$$

They can be solved in elementary functions in three cases (detailed below). In all other cases these integrals cannot be solved in elementary functions. Here are the 3 cases in which the integral $(2)$ is solvable.

1) $p \in \mathbb{Z}$

In this case we take $k = LCM(m,n)$ and we apply the substitution $x=t^k$, this leads us to an integral from a rational function of $t$.

2) $\frac{m+1}{n} \in \mathbb{Z}$

In this case to solve the integral we apply the substitution: $ax^n + b = t$. In this case another (rather obvious) second substitution may be needed $t = g(u)$ to arrive at a rational function of $u$.

3) $\frac{m+1}{n} + p \in \mathbb{Z}$

In this case to solve the integral we apply the substitution: $a + bx^{-n} = t$. In this case another (rather obvious) second substitution may be needed $t = g(u)$ to arrive at a rational function of $u$.