\[ \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^2 \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{5 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{2} \left (12 c^{2}+50 c d +18 d^{2}+d \left (4 c +9 d \right ) \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{15 f \sqrt {a +a \sec \left (f x +e \right )}}+\frac {2 a^{\frac {5}{2}} c^{2} \arctanh \left (\frac {\sqrt {a -a \sec \left (f x +e \right )}}{\sqrt {a}}\right ) \tan \left (f x +e \right )}{f \sqrt {a -a \sec \left (f x +e \right )}\, \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((a+a*sec(f*x+e))^(3/2)*(c+d*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {15 \, \sqrt {-a} a^{2} c^{2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}} - \frac {2 \, {\left ({\left (\sqrt {2} {\left (15 \, a^{4} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 40 \, a^{4} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 12 \, a^{4} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, \sqrt {2} {\left (3 \, a^{4} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 10 \, a^{4} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 3 \, a^{4} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, \sqrt {2} {\left (a^{4} c^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 4 \, a^{4} c d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 2 \, a^{4} d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{15 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________