\[ \int x^2 \sinh ^{-1}(a+b x)^2 \, dx \]
Optimal antiderivative \[ -\frac {4 x}{9 b^{2}}+\frac {2 a^{2} x}{b^{2}}-\frac {a \left (b x +a \right )^{2}}{2 b^{3}}+\frac {2 \left (b x +a \right )^{3}}{27 b^{3}}-\frac {a \arcsinh \! \left (b x +a \right )^{2}}{2 b^{3}}+\frac {a^{3} \arcsinh \! \left (b x +a \right )^{2}}{3 b^{3}}+\frac {x^{3} \arcsinh \! \left (b x +a \right )^{2}}{3}+\frac {4 \arcsinh \! \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9 b^{3}}-\frac {2 a^{2} \arcsinh \! \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{b^{3}}+\frac {a \left (b x +a \right ) \arcsinh \! \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{b^{3}}-\frac {2 \left (b x +a \right )^{2} \arcsinh \! \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{9 b^{3}} \]
command
int(x^2*arcsinh(b*x+a)^2,x)
Maple 2022.1 output
\[\int x^{2} \arcsinh \left (b x +a \right )^{2}\, dx\]
Maple 2021.1 output
\[ \frac {-\frac {a \left (2 \arcsinh \left (b x +a \right )^{2} \left (b x +a \right )^{2}-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+\arcsinh \left (b x +a \right )^{2}+\left (b x +a \right )^{2}+1\right )}{2}-\frac {\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )}{3}+\frac {\arcsinh \left (b x +a \right )^{2} \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{3}+\frac {2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}}{3}-\frac {14 b x}{27}-\frac {14 a}{27}-\frac {2 \arcsinh \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{9}+\frac {2 \left (1+\left (b x +a \right )^{2}\right ) \left (b x +a \right )}{27}+a^{2} \left (\arcsinh \left (b x +a \right )^{2} \left (b x +a \right )-2 \arcsinh \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+2 b x +2 a \right )}{b^{3}} \]