\[ \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {3+2 \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {\sqrt {30}}{5}\right ) \sqrt {2+3 \cos \left (d x +c \right )}\, \left (\sqrt {\sec }\left (d x +c \right )\right ) \sqrt {5}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {3+2 \sec \left (d x +c \right )}} \]
command
integrate(sec(d*x+c)^(1/2)/(3+2*sec(d*x+c))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + \frac {4}{9}\right ) + i \, \sqrt {6} {\rm weierstrassPInverse}\left (-\frac {44}{27}, \frac {784}{729}, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + \frac {4}{9}\right )}{3 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {\sec \left (d x + c\right )}}{\sqrt {2 \, \sec \left (d x + c\right ) + 3}}, x\right ) \]