\[ \int \cosh ^{\frac {7}{2}}(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {10 i \sqrt {\frac {\cosh \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right )}{21 \cosh \left (\frac {a}{2}+\frac {b x}{2}\right ) b}+\frac {2 \left (\cosh ^{\frac {5}{2}}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{7 b}+\frac {10 \sinh \left (b x +a \right ) \left (\sqrt {\cosh }\left (b x +a \right )\right )}{21 b} \]
command
integrate(cosh(b*x+a)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {40 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{6} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + {\left (45 \, \cosh \left (b x + a\right )^{2} + 23\right )} \sinh \left (b x + a\right )^{4} + 23 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{3} + 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (45 \, \cosh \left (b x + a\right )^{4} + 138 \, \cosh \left (b x + a\right )^{2} - 23\right )} \sinh \left (b x + a\right )^{2} - 23 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (9 \, \cosh \left (b x + a\right )^{5} + 46 \, \cosh \left (b x + a\right )^{3} - 23 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 3\right )} \sqrt {\cosh \left (b x + a\right )}}{84 \, {\left (b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\cosh \left (b x + a\right )^{\frac {7}{2}}, x\right ) \]