Rational indefinite integration $\int\frac{x^3+5}{(x-5)^3(x-1)}dx$

Question

Finding $\displaystyle \int\frac{x^3+5}{(x-5)^3(x-1)}dx$

what i try

putting $\displaystyle \frac{x-5}=\frac{1}{t}$ and $\displaystyle dx=-\frac{1}{t^2}dt$

$\displaystyle -\frac{(5t+1)^3+5}{5t+1}dt=-\int \frac{1}{5}(5t+1)^3+\ln|5t+1|+c$

but answer is differ from what i am find

help me to rectify it please

Answer 1

No substitution is need here. Just do partial fraction decomposition:$$\frac{x^3+5}{(x-5)^3 (x-1)}=-\frac3{32(x-1)}+\frac{35}{32(x-5)}+\frac{85}{8(x-5)^2}+\frac{65}{2 (x-5)^3}.$$That said, your approach is not bad. There was perhaps some computational error. Your substitution leads to$$-\frac{65}2t-\frac3{8(4 t+1)}-\frac{85}{8}-\frac1t.$$

Answer 2

Hint

Set $x-5=t$

$$\dfrac{(t+5)^3+5}{t^3(t+4)}=\dfrac a{t+4}+\dfrac bt+\dfrac c{t^2}+\dfrac d{t^3}$$

Now compare the coefficients of the different powers of $t$ to find $a,b,c,d$