I'm working my way through a series of ever tougher integrals. I'm stuck at no. 283:

$\displaystyle \int \frac {dx} {x \sqrt {a x^2 + b x + c} }$

... and there are some further even tougher ones.

I understand that it is supposed to evaluate to:

$\displaystyle - \frac 1 {\sqrt c} \ln \left({\frac {2 \sqrt c \sqrt {a x^2 + b x + c} + bx + 2c} x} }\right)$

or:

$\displaystyle - \frac 1 {\sqrt c} \sinh^{-1} \left({ \frac {bx + 2 c} {|x| \sqrt {4 a c - b^2} } }\right)$

depending on the sign of $\displaystyle b^2 - 4 a c$

I have already tried:

a) completing the square on the expression under the square root

b) substituting $\displaystyle z = 2 a x + b$ and $\displaystyle z = (2 a x + b)^2$

c) integrating by parts with $\displaystyle u = \sqrt {a x^2 + b x + c}$ and $\displaystyle dv = x$ etc.

I've also tried working backward, differentiating the $\displaystyle \sinh^{-1}$ expression, to see where it gets me, but the penny is not dropping.

Any hints?

The good cause this is for is ProofWiki which now has over 10000 proofs up.

Thanks, guys.