\[ \int (a+i a \tan (e+f x))^{5/2} (c+d \tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-1\right )^{\frac {1}{4}} a^{\frac {5}{2}} \left (c -3 \,\mathrm {I} d \right ) \left (c^{2}+18 \,\mathrm {I} c d +15 d^{2}\right ) \arctanh \! \left (\frac {\left (-1\right )^{\frac {3}{4}} \sqrt {d}\, \sqrt {a +\mathrm {I} a \tan \left (f x +e \right )}}{\sqrt {a}\, \sqrt {c +d \tan \left (f x +e \right )}}\right )}{8 d^{\frac {3}{2}} f}-\frac {4 \,\mathrm {I} a^{\frac {5}{2}} \left (c -\mathrm {I} d \right )^{\frac {3}{2}} \arctanh \! \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {c -\mathrm {I} d}\, \sqrt {a +\mathrm {I} a \tan \left (f x +e \right )}}\right ) \sqrt {2}}{f}+\frac {a^{2} \left (c^{2}+14 \,\mathrm {I} c d +19 d^{2}\right ) \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}\, \sqrt {c +d \tan \! \left (f x +e \right )}}{8 d f}+\frac {a^{2} \left (c +13 \,\mathrm {I} d \right ) \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}\, \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {3}{2}}}{12 d f}-\frac {a^{2} \sqrt {a +\mathrm {I} a \tan \! \left (f x +e \right )}\, \left (c +d \tan \! \left (f x +e \right )\right )^{\frac {5}{2}}}{3 d f} \]
command
integrate((a+I*a*tan(f*x+e))^(5/2)*(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Exception raised: TypeError} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {{\left (2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{3} a^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} c - 2 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} d\right )} \sqrt {2 \, a d^{2} + 2 \, \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} c + c^{2} + d^{2}} a d} {\left (\frac {i \, {\left (d \tan \left (f x + e\right ) + c\right )} a d - i \, a c d}{a d^{2} + \sqrt {{\left (d \tan \left (f x + e\right ) + c\right )}^{2} a^{2} d^{2} - 2 \, {\left (d \tan \left (f x + e\right ) + c\right )} a^{2} c d^{2} + a^{2} c^{2} d^{2} + a^{2} d^{4}}} + 1\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{4 \, {\left ({\left (d \tan \left (f x + e\right ) + c\right )} d^{2} - c d^{2} + i \, d^{3}\right )}} \]