Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

Question

I was browsing the book Irresistible Integrals and found this gem, at page 97, $$ \int_{0}^{1}x^{n}\ln^{k}(x)dx=\frac{(-1)^{k}k!}{(n+1)^{k+1}} $$ that resembles a previous question of mine here. So, I'm wondering if, somehow, one could find such a similarly simple expression for $$ \int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx $$ So, the question is, can someone find such expression?

Thanks.

EDIT: This is just to clarify the reason I think the present question is different from this one. I'm looking for a simpler expression, someting like a fraction in the line of $\frac{k!}{(n+1)^{k+1}}$, mayby a litle bit mone complicated. This contrasts with the double sum that results from the solution given by Robert Israel.

So what I'm looking for is a simpler expression. I already have an answer, but I'd like to have a simpler one.

EDIT 2 Here are the constraints of the problem: $p, q \in \mathbb{N}$, $a \in \mathbb{N_{>0}}$ and $b \in \mathbb{Q_{>0}}$.

I hope my reference still could be of help here. See. http://math.stackexchange.com/questions/911283/generalized-logarithmic-integral-reference-request/911339#911339 — andre, Jul 28 '15 at 21:09
wasn't your question alreeady answered? Or if you know the result but would like to know if there is a more explicit formula than @RobertIsrael gave, you should say so! :) — Ant, Jul 29 '15 at 06:42
@Ant, yes my question was already answered, but what I'm looking for is something more simple, something like the result of the first integral in this post. — Neves, Jul 29 '15 at 06:51
possible duplicate of Finding the general form for $\int\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ — Mark Viola, Jul 30 '15 at 23:43
@Neves Are there any constraints on the parameters? E.g., may we assume integer exponents? — David H, Jul 31 '15 at 09:28
@David H, yes there are constraints, $p, q \in \mathbb{N}$, $a \in \mathbb{N_{>0}}$ and $b \in \mathbb{Q_{>0}}$. — Neves, Jul 31 '15 at 10:08
I don't think you will get anything better than in the answer to your previous question. Especially for non-integer $b$. So what's the point of this question? — Yuriy S, Mar 16 '16 at 12:23

Can $\int_{0}^{1}\frac{x^{p}\ln^{q}(x+a)}{(x+a)^{b}}dx$ be expressed in a simple form?

0 Answers0