\[ \int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {2 a^{\frac {5}{2}} c \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right )}{f}+\frac {2 a d \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}} \tan \left (f x +e \right )}{5 f}+\frac {2 a^{3} \left (35 c +32 d \right ) \tan \left (f x +e \right )}{15 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} \left (5 c +8 d \right ) \sqrt {a +a \sec \left (f x +e \right )}\, \tan \left (f x +e \right )}{15 f} \]
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
\[ -\frac {\frac {15 \, \sqrt {-a} a^{3} c \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}} - \frac {2 \, {\left (45 \, \sqrt {2} a^{5} c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 60 \, \sqrt {2} a^{5} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (80 \, \sqrt {2} a^{5} c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 80 \, \sqrt {2} a^{5} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (35 \, \sqrt {2} a^{5} c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 32 \, \sqrt {2} a^{5} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{15 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________