\[ \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \arcsinh \left (d x +c \right )\right )}{5 d e \left (e \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {4 b \sqrt {1+\left (d x +c \right )^{2}}}{15 d \,e^{2} \left (e \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 b \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \,e^{\frac {7}{2}} \sqrt {1+\left (d x +c \right )^{2}}} \]
command
integrate((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} b d^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) + {\left (3 \, a d^{2} + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{15 \, {\left ({\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (1\right )^{4} + 4 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (1\right )^{3} \sinh \left (1\right ) + 6 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{2} + 4 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{3} + {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )} \sinh \left (1\right )^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}}{d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}, x\right ) \]