Integrating the logarithm of a function including a square root of a second degree polynomial

Question

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without success. For example, the latest try used the fact that: $$\frac{d}{dx}\left(x\cdot \left(k+\sqrt{ax^2+bx+c}\right)\right)=\ln\left(k+\sqrt{ax^2+bx+c}\right)+\frac{x(b+2ax)}{2(k+\sqrt{ax^2+bx+c})\sqrt{ax^2+bx+c}}$$ Hence $$\int \ln(k+\sqrt{ax^2+bx+c}) dx=x \ln(k+\sqrt{ax^2+bx+c})-\int \frac{x(b+2ax)}{2(k+\sqrt{ax^2+bx+c})\sqrt{ax^2+bx+c}}$$ but then I run into trouble with the integral of the last term. Any ideas?

Answer 1

http://www.wolframalpha.com/input/?i=integrate+ln%28k+%2B+sqrt%28ax%5E2+%2B+bx+%2Bc%29%29

(Select the whole link; for some reason a part of it isn't part of the active link.)

One look at the answer Wolfram Alpha gives you, tells you that there is a lot more to this integral than meets the eye... The main problem with your latest method, is the fact that the integral you get while trying to solve it using integration by parts method has too large a denominator for practical human computation/evaluation.

The only idea I can give you, is to try solving smaller cases of this integral (Some examples, like giving values to your constants...Or making the quadratic a linear...Or removing the constant k), & see where that takes you. This way, you get an idea of what needs to be done for solving the original one.

Still, unlikely that this can be solved without computer aid. Where did you come across this? It's a good question!