\[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \left (6 c +13 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{35 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{7 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} \left (72 c^{3}+486 c^{2} d +378 c \,d^{2}+104 d^{3}+d \left (24 c^{2}+111 c d +52 d^{2}\right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{105 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {5}{2}} c^{3} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(3/2)*(c+d*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {105 \, \sqrt {-a} a^{2} 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 (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 210 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1680 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1260 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 280 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1470 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 882 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 266 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 252 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 76 \, \sqrt {2} a^{5} 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 )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{105 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________