\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{c+d \sec (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {2 a^{3} \tan \left (f x +e \right )}{d f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{c f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}}-\frac {2 a^{\frac {7}{2}} \left (c -d \right )^{2} \arctanh \left (\frac {\sqrt {d}\, \sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}\, \sqrt {c +d}}\right ) \tan \left (f x +e \right )}{c \,d^{\frac {3}{2}} f \sqrt {c +d}\, \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(5/2)/(c+d*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {2 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} a^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} d} + \frac {\sqrt {-a} a^{2} \log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c} - \frac {\sqrt {-a} a^{2} \log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{c} + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-a} a^{3} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - 2 \, \sqrt {2} \sqrt {-a} a^{3} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + \sqrt {2} \sqrt {-a} a^{3} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} d + a c + 3 \, a d\right )}}{4 \, \sqrt {-c d - d^{2}} a}\right )}{\sqrt {-c d - d^{2}} a c d}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________