So, I'm practicing reduction formula and so I searched wikipedia for some integrals. I came across this one and even after trying it for hours,I couldn't find a reduction formula for it.
$$\int\frac{dx}{{[{ax+b}]^{m}}{{[p x+q}]^{n}}}$$
I have no clue how to do it. I tried integration by parts but it was useless.
Answer is in this link https://en.m.wikipedia.org/wiki/Integration_by_reduction_formulae in the tables of integral reduction. I would've pasted the answer but couldn't because my phone is not supporting it. Any hint is appreciated. Thanks in advance.
Best Answer
WLOG, $a=c=1$. By parts,
$$\int\frac{dx}{(x+b)^m(x+d)^n}= \\-\frac1{(m-1)(x+b)^{m-1}(x+d)^n}-\frac{n}{m-1}\int\frac{dx}{(x+b)^{m-1}(x+d)^{n+1}}.$$
Notice that $m$ decreases. When it reaches $1$, you have
$$\int\frac{dx}{(x+b)(x+d)^k}=\frac1{b-d}\int\frac{(x+b-x-d)dx}{(x+b)(x+d)^k}=\\\frac1{b-d}\int\frac{dx}{(x+d)^k}-\frac1{b-d}\int\frac{dx}{(x+b)(x+d)^{k-1}}.$$
The first terms integrates immediately and the second is reduced.