\[ \int x^4 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {4208 b^{2} d^{2} x}{99225 c^{4}}-\frac {2104 b^{2} d^{2} x^{3}}{297675 c^{2}}+\frac {526 b^{2} d^{2} x^{5}}{165375}+\frac {212 b^{2} c^{2} d^{2} x^{7}}{27783}+\frac {2 b^{2} c^{4} d^{2} x^{9}}{729}-\frac {8 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{189 c^{5}}+\frac {2 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{315 c^{5}}+\frac {20 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{441 c^{5}}-\frac {2 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{81 c^{5}}+\frac {8 d^{2} x^{5} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{315}+\frac {4 d^{2} x^{5} \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{63}+\frac {d^{2} x^{5} \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{9}-\frac {128 b \,d^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{4725 c^{5}}+\frac {64 b \,d^{2} x^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{4725 c^{3}}-\frac {16 b \,d^{2} x^{4} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{1575 c} \]
command
int(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{2} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {2}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{2} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{3}}{9}-\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{21}+\frac {8 \arcsinh \left (c x \right )^{2} c x}{315}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}+\frac {4 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \arcsinh \left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{81}+\frac {82 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{3969}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {1493104 c x}{31255875}-\frac {836 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}-\frac {33862 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {47248 c x \left (c^{2} x^{2}+1\right )}{31255875}-\frac {16 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}-\frac {8 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{9}+\frac {2 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {106 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {263 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}+\frac {1052 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}-\frac {2104 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{5}} \]