Fricas, an open-source clone of Axiom, implements a considerable chunk of Risch, see http://fricas-wiki.math.uni.wroc.pl/RischImplementationStatus
Fricas is also available as a optional package of SageMath open-source system.
Edit: here how it goes in SageMath with Fricas as backend.
sage: r=integrate((29*x^2+18*x-3)/(x^6+4*x^5+6*x^4-12*x^3+33*x^2-16*x)^(1/2),x,algorithm="fricas") sage: r log(x^29 + 40*x^28 + 776*x^27 + 9648*x^26 + 85820*x^25 + 578480*x^24 + 3058536*x^23 + 12979632*x^22 + 45004902*x^21 + 129708992*x^20 + 317208072*x^19 + 675607056*x^18 + 1288213884*x^17 + 2238714832*x^16 + 3548250712*x^15 + 5097069328*x^14 + 6677210721*x^13 + 8106250392*x^12 + 9056612528*x^11 + 8991685504*x^10 + 7944578304*x^9 + 6614046720*x^8 + 4834279424*x^7 + 2374631424*x^6 + 916848640*x^5 + 638582784*x^4 - 279969792*x^3 - 528482304*x^2 + (x^26 + 38*x^25 + 699*x^24 + 8220*x^23 + 68953*x^22 + 436794*x^21 + 2161755*x^20 + 8550024*x^19 + 27506475*x^18 + 73265978*x^17 + 165196041*x^16 + 324386076*x^15 + 570906027*x^14 + 914354726*x^13 + 1326830817*x^12 + 1731692416*x^11 + 2055647184*x^10 + 2257532160*x^9 + 2246693120*x^8 + 1939619840*x^7 + 1494073344*x^6 + 1097859072*x^5 + 640024576*x^4 + 207618048*x^3 + 95420416*x^2 + 50331648*x - 50331648)*sqrt(x^6 + 4*x^5 + 6*x^4 - 12*x^3 + 33*x^2 - 16*x) + 150994944*x - 134217728)