\[ \int \sec (c+d x) (a+b \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (a -b \right )^{\frac {5}{2}} \arctanh \! \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a -b}}\right )}{d}+\frac {\left (a +b \right )^{\frac {5}{2}} \arctanh \! \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{d}-\frac {2 b \left (a +b \sin \! \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {4 a b \sqrt {a +b \sin \! \left (d x +c \right )}}{d} \]
command
integrate(sec(d*x+c)*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {could not integrate} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {output too large to display} \]