22.167 Problem number 1409

\[ \int \frac {\sqrt {c e+d e x}}{\sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx \]

Optimal antiderivative \[ \frac {2 \EllipticE \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right ) \sqrt {e}}{d}-\frac {2 \EllipticF \left (\frac {\sqrt {d e x +c e}}{\sqrt {e}}, i\right ) \sqrt {e}}{d} \]

command

integrate((d*e*x+c*e)^(1/2)/(-d^2*x^2-2*c*d*x-c^2+1)^(1/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \frac {2 \, \sqrt {-d^{3} e} {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}{d^{2}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \]