\[ \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {298 b^{2} d^{2} x}{225}+\frac {76 b^{2} c^{2} d^{2} x^{3}}{675}+\frac {2 b^{2} c^{4} d^{2} x^{5}}{125}-\frac {8 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{45 c}-\frac {2 b \,d^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \arcsinh \! \left (c x \right )\right )}{25 c}+\frac {8 d^{2} x \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{15}+\frac {4 d^{2} x \left (c^{2} x^{2}+1\right ) \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{15}+\frac {d^{2} x \left (c^{2} x^{2}+1\right )^{2} \left (a +b \arcsinh \! \left (c x \right )\right )^{2}}{5}-\frac {16 b \,d^{2} \left (a +b \arcsinh \! \left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}{15 c} \]
command
int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)
Maple 2022.1 output
\[\int \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \arcsinh \left (c x \right )^{2} c x}{15}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \arcsinh \left (c x \right ) c^{3} x^{3}}{3}+\arcsinh \left (c x \right ) c x -\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c} \]