99.7 Problem number 234

\[ \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{5/2}} \, dx \]

Optimal antiderivative \[ -\frac {2 \left (a +b \arcsinh \left (d x +c \right )\right )}{3 d e \left (e \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {4 b \sqrt {1+\left (d x +c \right )^{2}}}{3 d \,e^{2} \sqrt {e \left (d x +c \right )}}+\frac {4 b \sqrt {e \left (d x +c \right )}\, \sqrt {1+\left (d x +c \right )^{2}}}{3 d \,e^{3} \left (d x +c +1\right )}-\frac {4 b \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \,e^{\frac {5}{2}} \sqrt {1+\left (d x +c \right )^{2}}}+\frac {2 b \left (d x +c +1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1+\left (d x +c \right )^{2}}{\left (d x +c +1\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {\sqrt {e \left (d x +c \right )}}{\sqrt {e}}\right )\right ) d \,e^{\frac {5}{2}} \sqrt {1+\left (d x +c \right )^{2}}} \]

command

integrate((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ -\frac {2 \, {\left (\sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {d^{3} \cosh \left (1\right ) + d^{3} \sinh \left (1\right )} {\rm weierstrassZeta}\left (-\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + {\left (a d + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \sqrt {{\left (d x + c\right )} \cosh \left (1\right ) + {\left (d x + c\right )} \sinh \left (1\right )}\right )}}{3 \, {\left ({\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \cosh \left (1\right )^{3} + 3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )} \sinh \left (1\right )^{3}\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]