\[ \int (a+b \cosh (x))^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (a +b \cosh \left (x \right )\right )^{\frac {3}{2}} \sinh \left (x \right )}{5}+\frac {16 a b \sinh \left (x \right ) \sqrt {a +b \cosh \left (x \right )}}{15}-\frac {2 i \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {a +b \cosh \left (x \right )}}{15 \cosh \left (\frac {x}{2}\right ) \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}+\frac {16 i a \left (a^{2}-b^{2}\right ) \sqrt {\frac {\cosh \left (x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {x}{2}\right ), \sqrt {2}\, \sqrt {\frac {b}{a +b}}\right ) \sqrt {\frac {a +b \cosh \left (x \right )}{a +b}}}{15 \cosh \left (\frac {x}{2}\right ) \sqrt {a +b \cosh \left (x \right )}} \]
command
integrate((a+b*cosh(x))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {4 \, {\left (\sqrt {2} {\left (a^{3} - 33 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} - 33 \, a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} - 33 \, a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, {\left (\sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, b^{3} \cosh \left (x\right )^{4} + 3 \, b^{3} \sinh \left (x\right )^{4} + 22 \, a b^{2} \cosh \left (x\right )^{3} - 22 \, a b^{2} \cosh \left (x\right ) + 2 \, {\left (6 \, b^{3} \cosh \left (x\right ) + 11 \, a b^{2}\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 33 \, a b^{2} \cosh \left (x\right ) - 46 \, a^{2} b - 18 \, b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, b^{3} \cosh \left (x\right )^{3} + 33 \, a b^{2} \cosh \left (x\right )^{2} - 11 \, a b^{2} - 4 \, {\left (23 \, a^{2} b + 9 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}}{90 \, {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (b^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2}\right )} \sqrt {b \cosh \left (x\right ) + a}, x\right ) \]