I was reading a book on Integral Calculus, and in one chapter, the author dealt with methods of solving Integrals of the form $$\int x^m(a+bx^n)^Pdx$$
The author broke it down into 4 cases:$$$$
$Case 1: \text{If } P\in \Bbb N\text{ , expand the integrand using the Binomial Theorem.} $
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$Case 2: \text{If } P\in \Bbb Z^- \text{, substitute $x=t^k$ where $k$ is the LCM of $m$ and $n$.} $
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$Case 3:\text{If }\dfrac{m+1}{n}$ is an integer and P is a fraction, substitute $(a+bx^n)=t^k$ where $k$ is the denominator of the fraction $P$.
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$Case 4: \text{If }\left(\frac{m+1}{n}+P\right)$ is an integer and P is a fraction substitute $(a+bx^n)=t^kx^n$ where $k$ is the denominator of the fraction $P$.
$$$$ I was able to understand the logic behind the first case: after expanding $(a+bx^n)^P$ Binomially, we would then multiply each term of the expansion by $x^m$, and finally integrate term by term. $$$$However I couldn't understand the motivation and intuition behind the substitutions in cases 2,3 and 4. How and why would the author think of the particular substitutions he made? What suggested these substitutions to the author? $$$$I would be grateful if somebody would please explain the logic and motivation behind making these substitutions. $$$$Many thanks!