Problem in Integration

I want to calculate both of these integrals:

Integrate[1/Sqrt[c x +  Sqrt[ b x + x^2] + a], x]


and

Integrate[1/Sqrt[ x + c Sqrt[ b x + x^2] + a], x]


But Mathematica cannot calculate them while it can calculate

Integrate[1/Sqrt[ x +  Sqrt[ b x + x^2] + a], x].


Do you have any suggestions on how to calculate these?

Even if I assign real numbers instead of $$a,b$$, and $$c,$$ Mathematica is not able to compute these integrals.

Thanks

• Which values can a, b and c assume? If they are not arbitrary complex values, you can use the Assumptions option to give Integrate a hint, which might make the integral solvable. – Thies Heidecke Jun 21 '19 at 12:54
• even the Assumption option does not help. Mathematica cannot calculate the Integrate[1/Sqrt[c x + Sqrt[ b x + x^2] + a], x, Assumptions -> {a, b, c} [Element] Reals] – EA66224 Jun 21 '19 at 13:00

To make it short, the condition that e^2-c*f^2==0 is the problem for your first two examples. Look carefully at the expression in the transformation rule and then compare the coefficients. You will find that while the rule cannot be applied to c x + Sqrt[...] or x + c Sqrt[...], the condition will hold for c x + c Sqrt[...] and therefore, both Mathematica and Rubi can solve
Integrate[1/Sqrt[c x + c Sqrt[b x + x^2] + a], x]