I want to calculate both of these integrals:

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


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.


  • $\begingroup$ 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. $\endgroup$ – Thies Heidecke Jun 21 '19 at 12:54
  • 1
    $\begingroup$ 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] $\endgroup$ – EA66224 Jun 21 '19 at 13:00

This is not an answer but may serve as a pointer in the right direction. If you use Rubi to solve the 3rd example and look at the steps it takes to find the solution, you can observe that the first step is crucial:


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]

Maybe this helps you, maybe it doesn't. Depending on how much you desire to solve this integral, it might be worth to talk to @AlbertRich in our Gitter chatroom.

As a reference, I post the complete list of integration steps for the above integral

enter image description here


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