Analytical expression for $ \int_0^b \frac{(a+x)^k - a^k}{x}dx $ Can an analytical expression be found for the following definite integral:
$$ \int_0^b \frac{(a+x)^k - a^k}{x}dx $$
I have tried in Wolfram Alpha. Unfortunately, it ran out of free calculation time...
In my particular application, I am faced with the following constraints on the various terms:
$$ a,b \in [0,1] \,\, \text{ and } \,\, a = 1-b \,\, \text{ and } \,\, k \in \mathbb{Z}^+$$
It is possible to get the solution using a discrete sum (using the binomial theorem) or using a recursion. However, we are looking for closed-form expressions without sums or integrals. Not sure it is possible...any help would be much appreciated!

$$\begin{align}f(x)
=&(a+x)^k-a^k\\
=&\sum_{n=0}^k{k\choose n}a^{k-n}x^n-a^k\\
=&\sum_{n=1}^k{k\choose n}a^{k-n}x^n
\end{align}$$

$$\frac{f(x)}{x}=\sum_{n=1}^k{k\choose n}a^{k-n}x^{n-1}$$

$$\begin{align}\int_0^b\frac{f(x)}{x}\,\mathrm dx
=&\int_0^b\sum_{n=1}^k{k\choose n}a^{k-n}x^{n-1}\,\mathrm dx\\
=&\sum_{n=1}^k{k\choose n}a^{k-n}\int_0^bx^{n-1}\,\mathrm dx\\
=&\sum_{n=1}^k{k\choose n}a^{k-n}\frac{b^n}{n}
\end{align}$$
