\[ \int \frac {d+b e \cos (x)+c e \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx \]
Optimal antiderivative \[ \frac {2 e \sqrt {\frac {\cos \left (x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {a +b \cos \left (x \right )+c \sin \left (x \right )}}{\cos \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \sqrt {\frac {a +b \cos \left (x \right )+c \sin \left (x \right )}{a +\sqrt {b^{2}+c^{2}}}}}+\frac {2 \left (-a e +d \right ) \sqrt {\frac {\cos \left (x -\arctan \left (b , c\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ), \sqrt {2}\, \sqrt {\frac {\sqrt {b^{2}+c^{2}}}{a +\sqrt {b^{2}+c^{2}}}}\right ) \sqrt {\frac {a +b \cos \left (x \right )+c \sin \left (x \right )}{a +\sqrt {b^{2}+c^{2}}}}}{\cos \left (\frac {x}{2}-\frac {\arctan \left (b , c\right )}{2}\right ) \sqrt {a +b \cos \left (x \right )+c \sin \left (x \right )}} \]
command
integrate((d+b*e*cos(x)+c*e*sin(x))/(a+b*cos(x)+c*sin(x))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-3 i \, \sqrt {2} {\left (b^{2} + c^{2}\right )} \sqrt {b + i \, c} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} + 6 i \, b c^{3} + 3 \, c^{4} - 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} - 9 i \, a c^{5} + 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} - 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} + 6 i \, b c^{3} + 3 \, c^{4} - 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} - 9 i \, a c^{5} + 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} - 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, \frac {2 \, a b - 2 i \, a c + 3 \, {\left (b^{2} + c^{2}\right )} \cos \left (x\right ) - 3 \, {\left (i \, b^{2} + i \, c^{2}\right )} \sin \left (x\right )}{3 \, {\left (b^{2} + c^{2}\right )}}\right )\right ) + 3 i \, \sqrt {2} {\left (b^{2} + c^{2}\right )} \sqrt {b - i \, c} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} - 6 i \, b c^{3} + 3 \, c^{4} + 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} + 9 i \, a c^{5} - 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} + 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} - 6 i \, b c^{3} + 3 \, c^{4} + 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} + 9 i \, a c^{5} - 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} + 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, \frac {2 \, a b + 2 i \, a c + 3 \, {\left (b^{2} + c^{2}\right )} \cos \left (x\right ) - 3 \, {\left (-i \, b^{2} - i \, c^{2}\right )} \sin \left (x\right )}{3 \, {\left (b^{2} + c^{2}\right )}}\right )\right ) + \sqrt {2} {\left (-2 i \, a b e - 2 \, a c e + 3 i \, b d + 3 \, c d\right )} \sqrt {b + i \, c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} + 6 i \, b c^{3} + 3 \, c^{4} - 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} - 9 i \, a c^{5} + 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} - 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, \frac {2 \, a b - 2 i \, a c + 3 \, {\left (b^{2} + c^{2}\right )} \cos \left (x\right ) - 3 \, {\left (i \, b^{2} + i \, c^{2}\right )} \sin \left (x\right )}{3 \, {\left (b^{2} + c^{2}\right )}}\right ) + \sqrt {2} {\left (2 i \, a b e - 2 \, a c e - 3 i \, b d + 3 \, c d\right )} \sqrt {b - i \, c} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} b^{2} - 3 \, b^{4} - 4 \, a^{2} c^{2} - 6 i \, b c^{3} + 3 \, c^{4} + 2 i \, {\left (4 \, a^{2} b - 3 \, b^{3}\right )} c\right )}}{3 \, {\left (b^{4} + 2 \, b^{2} c^{2} + c^{4}\right )}}, -\frac {8 \, {\left (8 \, a^{3} b^{3} - 9 \, a b^{5} + 27 \, a b c^{4} + 9 i \, a c^{5} - 2 i \, {\left (4 \, a^{3} + 9 \, a b^{2}\right )} c^{3} - 6 \, {\left (4 \, a^{3} b - 3 \, a b^{3}\right )} c^{2} + 3 i \, {\left (8 \, a^{3} b^{2} - 9 \, a b^{4}\right )} c\right )}}{27 \, {\left (b^{6} + 3 \, b^{4} c^{2} + 3 \, b^{2} c^{4} + c^{6}\right )}}, \frac {2 \, a b + 2 i \, a c + 3 \, {\left (b^{2} + c^{2}\right )} \cos \left (x\right ) - 3 \, {\left (-i \, b^{2} - i \, c^{2}\right )} \sin \left (x\right )}{3 \, {\left (b^{2} + c^{2}\right )}}\right )}{3 \, {\left (b^{2} + c^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {b e \cos \left (x\right ) + c e \sin \left (x\right ) + d}{\sqrt {b \cos \left (x\right ) + c \sin \left (x\right ) + a}}, x\right ) \]