\[ \int (a \cosh (x)+b \sinh (x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \cosh \left (x \right )+a \sinh \left (x \right )\right ) \left (a \cosh \left (x \right )+b \sinh \left (x \right )\right )^{\frac {3}{2}}}{5}-\frac {6 i \left (a^{2}-b^{2}\right ) \sqrt {\frac {\cos \left (i x -\arctan \left (a , -i b \right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {i x}{2}-\frac {\arctan \left (a , -i b \right )}{2}\right ), \sqrt {2}\right ) \sqrt {a \cosh \left (x \right )+b \sinh \left (x \right )}}{5 \cos \left (\frac {i x}{2}-\frac {\arctan \left (a , -i b \right )}{2}\right ) \sqrt {\frac {a \cosh \left (x \right )+b \sinh \left (x \right )}{\sqrt {a^{2}-b^{2}}}}} \]
command
integrate((a*cosh(x)+b*sinh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {12 \, {\left (\sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {a + b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (a - b\right )}}{a + b}, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{4} - 12 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 6 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} - 2 \, a^{2} + 2 \, b^{2}\right )} \sinh \left (x\right )^{2} - a^{2} + 2 \, a b - b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{3} - 6 \, {\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}}{10 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )} \sqrt {a \cosh \left (x\right ) + b \sinh \left (x\right )}, x\right ) \]