## Does Mathematica implement Risch Algorithm? If it does, in which cases?

I am asking this questions because when trying to evaluate the Integrals: $ $ (1)\ \int \frac{x}{\sqrt{x^4+10x^2-96×-71}}\ dx$ $ $ $ (2)\ \int \frac{x^2+2x+1+(3x+1)\sqrt{x+\ln(x)}}{x(x+\sqrt{x+\ln(x)})\sqrt{x+\ln(x)}}\ dx$ $

$ \textbf{Mathematica 11}$ : For $ (1)$ I obtain an evaluation involving non-elementary functions, while $ (2)$ is not even evaluated.

$ $ \$ $ $ \textbf{Maple 2016}$ : I get the same results as in Mathematica.

$ $ \$ $ $ \textbf{Rubi 4.11}$ neither are evaluated.

$ $ \$ $ The two indefinite integrals can be expressed with elementary functions. $ $ $ $ $ \textbf{Code:}\ (Mathematica)$ $ $ \$ $

$ (1)$ `Integrate[x/Sqrt[x^4 + 10 x^2 - 96 x - 71], x]`

$ (2)$ `Integrate[(x^2 + 2 x +`

1 + (3 x + 1) Sqrt[x + Log[x]])/(x (x + Sqrt[x + Log[x]]) Sqrt[

x + Log[x]]), x]

$ $ $ $ *ADDENDUM:* The antiderivative of the integrand $ (1)$ is correctly expressed with elementary functions using $ \textbf{Axiom}$ . Why can’t the two powerful Computer Algebra Systems listed above, obtain the same result as Axiom?