\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {i \left (-i d +c \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {-i d +c}\, \sqrt {a +b \tan \left (f x +e \right )}}{\sqrt {-i b +a}\, \sqrt {c +d \tan \left (f x +e \right )}}\right ) \sqrt {-i b +a}}{f}+\frac {i \left (i d +c \right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {i d +c}\, \sqrt {a +b \tan \left (f x +e \right )}}{\sqrt {i b +a}\, \sqrt {c +d \tan \left (f x +e \right )}}\right ) \sqrt {i b +a}}{f}+\frac {\left (a d +3 b c \right ) \arctanh \left (\frac {\sqrt {d}\, \sqrt {a +b \tan \left (f x +e \right )}}{\sqrt {b}\, \sqrt {c +d \tan \left (f x +e \right )}}\right ) \sqrt {d}}{f \sqrt {b}}+\frac {d \sqrt {a +b \tan \left (f x +e \right )}\, \sqrt {c +d \tan \left (f x +e \right )}}{f} \]
command
int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2),x)
Maple 2022.1 output
\[\text {output too large to display}\]
Maple 2021.1 output
\[ \int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}\, dx \]________________________________________________________________________________________