\[ \int \frac {(a+a \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (c -d \right ) \tan \left (f x +e \right )}{c \left (c +d \right ) f \left (c +d \sec \left (f x +e \right )\right ) \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{c^{2} f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}}+\frac {a^{\frac {5}{2}} \left (c^{2}-3 c d -2 d^{2}\right ) \arctanh \left (\frac {\sqrt {d}\, \sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}\, \sqrt {c +d}}\right ) \tan \left (f x +e \right )}{c^{2} \left (c +d \right )^{\frac {3}{2}} f \sqrt {d}\, \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {\sqrt {-a} a^{2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c^{2} {\left | a \right |}} - \frac {\sqrt {2} {\left (\sqrt {-a} a^{2} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 3 \, \sqrt {-a} a^{2} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 2 \, \sqrt {-a} a^{2} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} d + a c + 3 \, a d\right )}}{4 \, \sqrt {-c d - d^{2}} a}\right )}{{\left (\sqrt {2} c^{3} + \sqrt {2} c^{2} d\right )} \sqrt {-c d - d^{2}} a} + \frac {4 \, {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{2} c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + \sqrt {-a} a^{3} c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - \sqrt {-a} a^{3} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{4} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{4} d + 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} a c + 6 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} a d + a^{2} c - a^{2} d\right )} {\left (\sqrt {2} c^{2} + \sqrt {2} c d\right )}}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________