\[ \int x^4 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {304 b^{2} d x}{3675 c^{4}}-\frac {152 b^{2} d \,x^{3}}{11025 c^{2}}+\frac {38 b^{2} d \,x^{5}}{6125}+\frac {2 b^{2} c^{2} d \,x^{7}}{343}-\frac {2 b d \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{21 c^{5}}+\frac {4 b d \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{35 c^{5}}-\frac {2 b d \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{49 c^{5}}+\frac {2 d \,x^{5} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{35}+\frac {d \,x^{5} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{7}-\frac {32 b d \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{525 c^{5}}+\frac {16 b d \,x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{525 c^{3}}-\frac {4 b d \,x^{4} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{175 c} \]
command
int(x^4*(c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{4} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d \,b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{2}}{7}-\frac {3 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {2 \arcsinh \left (c x \right )^{2} c x}{35}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{35}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{49}+\frac {62 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1225}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {37384 c x}{385875}-\frac {484 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}-\frac {3358 c x \left (c^{2} x^{2}+1\right )}{385875}-\frac {4 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{35}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d a b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}+\frac {76 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c^{5}} \]