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