\[ \int (d \cos (a+b x))^{3/2} \sin ^4(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {12 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{77 b d}-\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}} \left (\sin ^{3}\left (b x +a \right )\right )}{11 b d}+\frac {8 d^{2} \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right )}{77 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \sqrt {d \cos \left (b x +a \right )}}+\frac {8 d \sin \left (b x +a \right ) \sqrt {d \cos \left (b x +a \right )}}{77 b} \]
command
integrate((d*cos(b*x+a))^(3/2)*sin(b*x+a)^4,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 i \, \sqrt {2} d^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - 2 i \, \sqrt {2} d^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - {\left (7 \, d \cos \left (b x + a\right )^{4} - 13 \, d \cos \left (b x + a\right )^{2} + 4 \, d\right )} \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{77 \, b} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (d \cos \left (b x + a\right )^{5} - 2 \, d \cos \left (b x + a\right )^{3} + d \cos \left (b x + a\right )\right )} \sqrt {d \cos \left (b x + a\right )}, x\right ) \]