<s>[INST] Help with unsolved indefinite integral ! $\int \!r\ln \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$ I really need to solve this indefinite integral: 
$\int \!r\ln  \left( r \right) \sqrt {ar+{b}^{2}+{r}^{2}}\,{\rm d}r$
It seems much more complicated than it looks. I have found a integral table with
a integral resembles my integral, and the solution involves Polylogs. The link of the intgral table is below:
http://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html
Anyone willing to take this challenge and help with this integral ? 
Thank you very much in advance 
 [/INST] The integral can be constructed from the following generic cases
$$
\int x \log(x)\sqrt{a^2+x^2} dx=
\frac{1}{3} \left(-\frac{1}{3} \sqrt{a^2+x^2} \left(4 a^2+x^2\right)-a^3 \log(x)+\left(a^2+x^2\right)^{3/2} \log(x)+a^3 \log\left[a
\left(a+\sqrt{a^2+x^2}\right)\right]\right)
$$

$$
\int x \log(x+b)\sqrt{a^2+x^2}dx
=\frac{1}{6} \left(-\frac{1}{3} \sqrt{a^2+x^2} \left(8 a^2+6 b^2-3 b x+2 x^2\right)-2 \left(a^2+b^2\right)^{3/2} \log(b+x)+2 \left(a^2+x^2\right)^{3/2}
\log(b+x)+b \left(3 a^2+2 b^2\right) \log\left[x+\sqrt{a^2+x^2}\right]+2 \left(a^2+b^2\right)^{3/2} \log\left[a^2-b x+\sqrt{a^2+b^2}
\sqrt{a^2+x^2}\right]\right)
$$

and 
$$
\int x \log(c x+b)\sqrt{a^2+x^2}dx=\frac{1}{18 c^3}\left(-c \sqrt{a^2+x^2} \left(6 b^2-3 b c x+2 c^2 \left(4 a^2+x^2\right)\right)-6 \left(b^2+a^2 c^2\right)^{3/2} \log(b+cx)+6 c^3 \left(a^2+x^2\right)^{3/2} \log(b+c x)+3 b \left(2 b^2+3 a^2 c^2\right) \log\left[x+\sqrt{a^2+x^2}\right]+6 \left(b^2+a^2 c^2\right)^{3/2}
\log\left[a^2 c-b x+\sqrt{b^2+a^2 c^2} \sqrt{a^2+x^2}\right]\right)
$$
Finally, we consider a form that yields result in terms of dilogarithm function (setting $y(x)=\sqrt{1+x^2}$)
$$
\int \log(x)\sqrt{1+x^2}dx=\frac{1}{24} (-\pi^2-6 x y+12 x y \log(x)+6 \text{arcsinh}(x) \Big(\text{arcsinh}(x)+2 \log(x)-2 \log(1+x+y)-1-12 \text{Li}_2(-x-y)+12 \text{Li}_2(1-x-y)\Big)
$$
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