\[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} e^{\frac {5}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{d}+\frac {2 a^{2} e^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{d}-\frac {4 a^{2} e \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a^{2} e \cos \left (d x +c \right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{5 d}+\frac {a^{2} e \sec \left (d x +c \right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{d}+\frac {9 a^{2} e^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \sin \left (d x +c \right )}}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \sqrt {\sin \left (d x +c \right )}} \]
command
integrate((a+a*sec(d*x+c))^2*(e*sin(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-27 i \, \sqrt {2} \sqrt {-i} a^{2} \cos \left (d x + c\right ) e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 27 i \, \sqrt {2} \sqrt {i} a^{2} \cos \left (d x + c\right ) e^{\frac {5}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 30 \, a^{2} \arctan \left (\frac {2 \, {\left (76 \, \cos \left (d x + c\right )^{2} + 425 \, {\left (\sin \left (d x + c\right ) - 1\right )} \sqrt {\sin \left (d x + c\right )} - 152 \, \sin \left (d x + c\right ) - 152\right )}}{361 \, \cos \left (d x + c\right )^{2} + 978 \, \sin \left (d x + c\right ) - 722}\right ) \cos \left (d x + c\right ) e^{\frac {5}{2}} + 15 \, a^{2} \cos \left (d x + c\right ) e^{\frac {5}{2}} \log \left (\frac {\cos \left (d x + c\right )^{2} - 4 \, {\left (\sin \left (d x + c\right ) + 1\right )} \sqrt {\sin \left (d x + c\right )} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) - 2 \, {\left (6 \, a^{2} \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} + 20 \, a^{2} \cos \left (d x + c\right ) e^{\frac {5}{2}} - 15 \, a^{2} e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )^{\frac {3}{2}}}{30 \, d \cos \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-{\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2} + {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )^{2} + 2 \, {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]