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