12.19 Problem number 207

$$\int x^4{(d+c^{2}d{x}^2)}^2{(a+b{\sinh }^{-1}(cx))}^{2}dx$$

Optimal antiderivative $$\frac{4208b^2{d}^{2}x}{99225c^4}-\frac{2104b^2{d}^2{x}^3}{297675c^2}+\frac{526b^2{d}^2{x}^5}{165375}+\frac{212b^2{c}^2{d}^2{x}^7}{27783}+\frac{2b^2{c}^4{d}^2{x}^9}{729}-\frac{8b{d}^2{(c^2{x}^2+1)}^{\frac{3}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{189c^5}+\frac{2b{d}^2{(c^2{x}^2+1)}^{\frac{5}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{315c^5}+\frac{20b{d}^2{(c^2{x}^2+1)}^{\frac{7}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{441c^5}-\frac{2b{d}^2{(c^2{x}^2+1)}^{\frac{9}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{81c^5}+\frac{8d^2{x}^5{(a+b\mathrm{arcsinh}\left(cx\right))}^2}{315}+\frac{4d^2{x}^5(c^2{x}^2+1){(a+b\mathrm{arcsinh}\left(cx\right))}^2}{63}+\frac{d^2{x}^5{(c^2{x}^2+1)}^2{(a+b\mathrm{arcsinh}\left(cx\right))}^2}{9}-\frac{128b{d}^2(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{4725c^5}+\frac{64b{d}^2{x}^2(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{4725c^3}-\frac{16b{d}^2{x}^4(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{1575c}$$

command

int(x^4*(c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x)

Maple 2022.1 output

$$\int x^4{(c^{2}d{x}^2+d)}^2{(a+b\mathrm{arcsinh}\left(cx\right))}^{2}dx$$

Maple 2021.1 output

$$\frac{d^2{a}^2(\frac{1}{9}{c}^9{x}^9+\frac{2}{7}{c}^7{x}^7+\frac{1}{5}{c}^5{x}^5)+d^2{b}^2(\frac{\mathrm{arcsinh}{\left(cx\right)}^2{c}^3{x}^3{(c^2{x}^2+1)}^3}{9}-\frac{\mathrm{arcsinh}{\left(cx\right)}^{2}cx{(c^2{x}^2+1)}^3}{21}+\frac{8\mathrm{arcsinh}{\left(cx\right)}^{2}cx}{315}+\frac{\mathrm{arcsinh}{\left(cx\right)}^{2}cx{(c^2{x}^2+1)}^2}{105}+\frac{4\mathrm{arcsinh}{\left(cx\right)}^{2}cx(c^2{x}^2+1)}{315}-\frac{2\mathrm{arcsinh}\left(cx\right)c^2{x}^2{(c^2{x}^2+1)}^{\frac{7}{2}}}{81}+\frac{82\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{7}{2}}}{3969}+\frac{2cx{(c^2{x}^2+1)}^4}{729}+\frac{1493104cx}{31255875}-\frac{836cx{(c^2{x}^2+1)}^3}{250047}-\frac{33862cx{(c^2{x}^2+1)}^2}{10418625}-\frac{47248cx(c^2{x}^2+1)}{31255875}-\frac{16\mathrm{arcsinh}\left(cx\right)\sqrt{c^2{x}^2+1}}{315}-\frac{2\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{5}{2}}}{525}-\frac{8\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{3}{2}}}{945})+2d^{2}ab(\frac{\mathrm{arcsinh}\left(cx\right)c^9{x}^9}{9}+\frac{2\mathrm{arcsinh}\left(cx\right)c^7{x}^7}{7}+\frac{\mathrm{arcsinh}\left(cx\right)c^5{x}^5}{5}-\frac{c^8{x}^8\sqrt{c^2{x}^2+1}}{81}-\frac{106c^6{x}^6\sqrt{c^2{x}^2+1}}{3969}-\frac{263c^4{x}^4\sqrt{c^2{x}^2+1}}{33075}+\frac{1052c^2{x}^2\sqrt{c^2{x}^2+1}}{99225}-\frac{2104\sqrt{c^2{x}^2+1}}{99225})}{c^5}$$