Integrate [((log(5) x)^2 + sqrt 2*(log(5) x) + 9)/x] with respect to x.

We have to integrate: [((log(5) x)^2 + sqrt 2*(log(5) x) +
9)/x] with respect to x.

use the relation log(a)x = ln x/ln
a

[((log(5) x)^2 + sqrt 2*(log(5) x) +
9)/x]

=> [((ln x/ln 5)^2 + sqrt 2*(ln x/ln 5) +
9)/x]

=> (1/ln 5)(ln x)^2/x + (sqrt 2/ln 5)(ln x)/x
+ 9/x

Int[(1/ln 5)(ln x)^2/x + (sqrt 2/ln 5)(ln x)/x +
9/x]dx

To find Int[(ln x)^2/x
dx]

let y = ln x

dy/dx = 1/x
or dy = dx/x

=> Int[y^2
dy]

=> y^3/3

substitute
y = ln x

=> (ln
x)^3/3

Similarly lnt[ln x/x dx] = (ln
x)^2/2

=> (1/ln 5)(ln x)^3/3 + (sqrt 2/ln 5)(ln
x)^2/2 + 9 ln x + C

The required integral is
(1/ln 5)(ln x)^3/3 + (sqrt 2/ln 5)(ln x)^2/2 + 9 ln x +
C