Integration through substitution help please

[cheetahday]

\(\displaystyle \displaystyle{ \int \,}\)\(\displaystyle \dfrac{dx}{x\, \left(x^p\, +\, a\right)},\) where \(\displaystyle a,\, p\, \neq\, 0.\) Use the substitution \(\displaystyle u\, =\, 1\, +\, ax^{-p}.\)

So far I've only gotten
x^p=(u-1)/a
x=((u-1)/a)^(1/p)

I'm stuck everything I plug in just becomes really messy

 

[Ishuda]

\(\displaystyle \displaystyle{ \int \,}\)\(\displaystyle \dfrac{dx}{x\, \left(x^p\, +\, a\right)},\) where \(\displaystyle a,\, p\, \neq\, 0.\) Use the substitution \(\displaystyle u\, =\, 1\, +\, ax^{-p}.\)

So far I've only gotten
x^p=(u-1)/a
x=((u-1)/a)^(1/p)

I'm stuck everything I plug in just becomes really messy

\(\displaystyle u\, =\, 1\, +\, a\, x^{-p}\)

\(\displaystyle du\, =\, -pa\, x^{-(p\, +\, 1)}\, dx\)

or

\(\displaystyle dx\, =\, -(pa)^{-1}\, x^{p\, +\,1}\)

and

\(\displaystyle \dfrac{dx}{x\left(x^p \,+\, a\right)} \,=\, \dfrac{-(pa)^{-1} \,x^{p+1} \,du}{x^{p+1}\left(1\, +\, a x^{-p}\right)}\,=\, \dfrac{-(pa)^{-1}\, du}{u}\)