Elliptic Integrals of the First Kind
B. C. Carlson · SIAM Journal on Mathematical Analysis · 1977
The reciprocal square root of any real polynomial with known zeros and degree not exceeding four is integrated in terms of a standard integral by a new quadratic transformation which preserves symmetry in the zeros. If at least one zero is real, this method, unlike earlier methods, leads to a single standard integral instead of a difference of two standard integrals even when neither limit of integration is a zero. If no zero is real, a particular point on the real line has special significance. Formulas listed in integral tables are unified and generalized.