\[ \int x^4 \left (a+b \text {ArcSin}\left (c+d x^2\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {x^{5} \left (a +b \arcsin \left (d \,x^{2}+c \right )\right )}{5}-\frac {2 b \left (c +1\right ) \left (23 c^{2}+9\right ) \EllipticE \left (\frac {x \sqrt {d}}{\sqrt {1-c}}, \sqrt {\frac {-1+c}{c +1}}\right ) \sqrt {1-c}\, \sqrt {1-\frac {d \,x^{2}}{1-c}}\, \sqrt {1+\frac {d \,x^{2}}{c +1}}}{75 d^{\frac {5}{2}} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}+\frac {2 b \left (c +1\right ) \left (15 c^{2}+8 c +9\right ) \EllipticF \left (\frac {x \sqrt {d}}{\sqrt {1-c}}, \sqrt {\frac {-1+c}{c +1}}\right ) \sqrt {1-c}\, \sqrt {1-\frac {d \,x^{2}}{1-c}}\, \sqrt {1+\frac {d \,x^{2}}{c +1}}}{75 d^{\frac {5}{2}} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}-\frac {16 b c x \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{75 d^{2}}+\frac {2 b \,x^{3} \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{25 d} \]
command
integrate(x^4*(a+b*arcsin(d*x^2+c)),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 \, b d^{3} x^{6} \arcsin \left (d x^{2} + c\right ) + 15 \, a d^{3} x^{6} + 2 \, {\left (3 \, b d^{2} x^{4} - 8 \, b c d x^{2} + 23 \, b c^{2} + 9 \, b\right )} \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}}{75 \, d^{3} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (b x^{4} \arcsin \left (d x^{2} + c\right ) + a x^{4}, x\right ) \]