\[ \int (a+a \sec (e+f x))^{5/2} (c+d \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {2 a^{3} \left (3 c^{3}+12 c^{2} d +12 c \,d^{2}+4 d^{3}\right ) \tan \left (f x +e \right )}{f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a d \left (3 c^{2}+15 c d +13 d^{2}\right ) \left (a -a \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{5 f \sqrt {a +a \sec \left (f x +e \right )}}-\frac {6 d^{2} \left (c +2 d \right ) \left (a -a \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{7 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 d^{3} \left (a -a \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{9 a f \sqrt {a +a \sec \left (f x +e \right )}}-\frac {2 \left (c^{3}+12 c^{2} d +24 c \,d^{2}+12 d^{3}\right ) \left (a^{3}-a^{3} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {7}{2}} c^{3} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(5/2)*(c+d*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {315 \, \sqrt {-a} a^{3} c^{3} \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 (945 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3780 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3780 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1260 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (3570 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 12600 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 10080 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2520 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (5040 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 15876 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 11340 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3276 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (3150 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 9072 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 6480 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1872 \, \sqrt {2} a^{7} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (735 \, \sqrt {2} a^{7} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2016 \, \sqrt {2} a^{7} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1440 \, \sqrt {2} a^{7} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 416 \, \sqrt {2} a^{7} d^{3} \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 )^{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 )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{315 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________