Integration with respect to functions

You can write is as

\begin{align*} \int xd\left( f\left( x\right) \right) & =\int x\frac{df}{dx}dx\\ & =\int xf^{\prime}dx \end{align*}

And Mathematica gives now

 Integrate[x*f'[x], x]

I do not understand how Mathematica operates when Integrating with respect to functions

It looks like you need to reformulate the command yourself like the above so that the integration is with respect to a variable and not a function.

I do not know why Mathematica gave the result it did with your input.

update

fyi, I tried this in Rubi, and Maple 20204 and Fricas and Maxima and XCas/GIAC. Out of these, maxima and giac returned back same result as Mathematica. This is the result

<< Rubi`
Int[x, f[x]]

Maple:

int(x,f(x))

Fricas:

>sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 10.3, Release Date: 2024-03-19                    │
│ Using Python 3.11.1. Type "help()" for help.                       │
└────────────────────────────────────────────────────────────────────┘
sage: var('x')
x
sage: f=function("f",nargs=1)

sage: integrate(f(x),x, algorithm="fricas")
integral(f(x), x)

sage: integrate(x,f(x), algorithm="fricas")

Gives

TypeError: An error occurred when FriCAS evaluated 'integrate(sage4,sage3)':

Maxima:

sage: integrate(x,f(x), algorithm="maxima")
x*f(x)

Giac:

sage: integrate(x,f(x), algorithm="giac")
x*f(x)

The Mathematica result seems to be correct if we assume Integrate[x, f[x]]==Integrate[x f'[x]],x]

Integration by parts

Integrate[u f'[u],{u,x0,x}]==x f[x]-x0 f[x0]-Integrate[f[u],{u,x0,x}]

differentiate result

D[x f[x]-x0 f[x0]-Integrate[f[u],{u,x0,x}],x] (* x Derivative[1][f][x] *)