\[ \int \frac {(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (c -d \right ) \tan \left (f x +e \right )}{2 c \left (c +d \right ) f \left (c +d \sec \left (f x +e \right )\right )^{2} \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{2} \left (3 c^{2}-7 c d -4 d^{2}\right ) \tan \left (f x +e \right )}{4 c^{2} \left (c +d \right )^{2} f \left (c +d \sec \left (f x +e \right )\right ) \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{c^{3} f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{\frac {5}{2}} \left (3 c^{3}-15 c^{2} d -20 c \,d^{2}-8 d^{3}\right ) \arctanh \left (\frac {\sqrt {d}\, \sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}\, \sqrt {c +d}}\right ) \tan \left (f x +e \right )}{4 c^{3} \left (c +d \right )^{\frac {5}{2}} f \sqrt {d}\, \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________