\[ \int (c e+d e x)^{5/2} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (e \left (d x +c \right )\right )^{\frac {7}{2}} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}{7 d e}-\frac {20 b \,e^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}, i\right ) \sqrt {-d x -c +1}}{147 d \sqrt {d x +c -1}}-\frac {4 b \left (e \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{49 d}-\frac {20 b \,e^{2} \sqrt {d x +c -1}\, \sqrt {e \left (d x +c \right )}\, \sqrt {d x +c +1}}{147 d} \]
command
integrate((d*e*x+c*e)^(5/2)*(a+b*arccosh(d*x+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (21 \, {\left ({\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b d^{5} x^{3} + 3 \, b c d^{4} x^{2} + 3 \, b c^{2} d^{3} x + b c^{3} d^{2}\right )} \sinh \left (1\right )^{2}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 10 \, \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\left (b \cosh \left (1\right )^{2} + 2 \, b \cosh \left (1\right ) \sinh \left (1\right ) + b \sinh \left (1\right )^{2}\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (21 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \cosh \left (1\right )^{2} + 42 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 21 \, {\left (a d^{5} x^{3} + 3 \, a c d^{4} x^{2} + 3 \, a c^{2} d^{3} x + a c^{3} d^{2}\right )} \sinh \left (1\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (3 \, b d^{4} x^{2} + 6 \, b c d^{3} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{147 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a d^{2} e^{2} x^{2} + 2 \, a c d e^{2} x + a c^{2} e^{2} + {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + b c^{2} e^{2}\right )} \operatorname {arcosh}\left (d x + c\right )\right )} \sqrt {d e x + c e}, x\right ) \]