\[ \int x \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {175 b^{2} d^{3} x^{2}}{3072}+\frac {35 b^{2} c^{2} d^{3} x^{4}}{3072}+\frac {7 b^{2} d^{3} \left (c^{2} x^{2}+1\right )^{3}}{1152 c^{2}}+\frac {b^{2} d^{3} \left (c^{2} x^{2}+1\right )^{4}}{256 c^{2}}-\frac {35 b \,d^{3} x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{768 c}-\frac {7 b \,d^{3} x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{192 c}-\frac {b \,d^{3} x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{32 c}-\frac {35 d^{3} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{1024 c^{2}}+\frac {d^{3} \left (c^{2} x^{2}+1\right )^{4} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{8 c^{2}}-\frac {35 b \,d^{3} x \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{512 c} \]
command
int(x*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{3} a^{2} \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}+\frac {3}{4} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}-\frac {7 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{192}-\frac {35 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{768}-\frac {35 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{512}-\frac {35 \arcsinh \left (c x \right )^{2}}{1024}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}+1\right )^{3}}{1152}+\frac {35 \left (c^{2} x^{2}+1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}+\frac {35}{1024}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {\arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{384}-\frac {163 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {c^{2} x^{2}+1}}{1024}+\frac {93 \arcsinh \left (c x \right )}{1024}\right )}{c^{2}} \]