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