Doing the integral, possible using some elliptic functions/integrals.

Question

I have the following integral $$I=\int_{[0, t]}\sqrt{a+b\sin{\tau}+c\sin^2{\tau}}d\tau$$ By virtue of the transformation $\xi=\sin{\tau}$ the integral becomes $$I=\int_{[0, \sin{t}]}\sqrt{(a+b\xi+c\xi^2)(1-\xi^2)}d\xi$$ I.e. has the form $$\int_{[0, x_{1}]}\sqrt{c_{1}+c_{2}x+c_{3}x^2+c_{4}x^3+c_{5}x^4}dx$$ The similar integral, however with a cubic under the square root, may be reduced by a bunch of clever transformations to the complete elliptic integrals, I wonder if someone can see any suitable transformation in this case. Pre-thanks