12.24 Problem number 216

$$\int x^4{(d+c^{2}d{x}^2)}^3{(a+b{\sinh }^{-1}(cx))}^{2}dx$$

Optimal antiderivative $$\frac{100976b^2{d}^{3}x}{4002075c^4}-\frac{50488b^2{d}^3{x}^3}{12006225c^2}+\frac{12622b^2{d}^3{x}^5}{6670125}+\frac{9410b^2{c}^2{d}^3{x}^7}{1120581}+\frac{182b^2{c}^4{d}^3{x}^9}{29403}+\frac{2b^2{c}^6{d}^3{x}^{11}}{1331}-\frac{16b{d}^3{(c^2{x}^2+1)}^{\frac{3}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{693c^5}+\frac{4b{d}^3{(c^2{x}^2+1)}^{\frac{5}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{1155c^5}-\frac{2b{d}^3{(c^2{x}^2+1)}^{\frac{7}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{1617c^5}+\frac{8b{d}^3{(c^2{x}^2+1)}^{\frac{9}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{297c^5}-\frac{2b{d}^3{(c^2{x}^2+1)}^{\frac{11}{2}}(a+b\mathrm{arcsinh}\left(cx\right))}{121c^5}+\frac{16d^3{x}^5{(a+b\mathrm{arcsinh}\left(cx\right))}^2}{1155}+\frac{8d^3{x}^5(c^2{x}^2+1){(a+b\mathrm{arcsinh}\left(cx\right))}^2}{231}+\frac{2d^3{x}^5{(c^2{x}^2+1)}^2{(a+b\mathrm{arcsinh}\left(cx\right))}^2}{33}+\frac{d^3{x}^5{(c^2{x}^2+1)}^3{(a+b\mathrm{arcsinh}\left(cx\right))}^2}{11}-\frac{256b{d}^3(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{17325c^5}+\frac{128b{d}^3{x}^2(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{17325c^3}-\frac{32b{d}^3{x}^4(a+b\mathrm{arcsinh}\left(cx\right))\sqrt{c^2{x}^2+1}}{5775c}$$

command

int(x^4*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x)

Maple 2022.1 output

$$\int x^4{(c^{2}d{x}^2+d)}^3{(a+b\mathrm{arcsinh}\left(cx\right))}^{2}dx$$

Maple 2021.1 output

$$\frac{d^3{a}^2(\frac{1}{11}{c}^{11}{x}^{11}+\frac{1}{3}{c}^9{x}^9+\frac{3}{7}{c}^7{x}^7+\frac{1}{5}{c}^5{x}^5)+d^3{b}^2(\frac{\mathrm{arcsinh}{\left(cx\right)}^2{c}^3{x}^3{(c^2{x}^2+1)}^4}{11}-\frac{\mathrm{arcsinh}{\left(cx\right)}^{2}cx{(c^2{x}^2+1)}^4}{33}+\frac{16\mathrm{arcsinh}{\left(cx\right)}^{2}cx}{1155}+\frac{\mathrm{arcsinh}{\left(cx\right)}^{2}cx{(c^2{x}^2+1)}^3}{231}+\frac{2\mathrm{arcsinh}{\left(cx\right)}^{2}cx{(c^2{x}^2+1)}^2}{385}+\frac{8\mathrm{arcsinh}{\left(cx\right)}^{2}cx(c^2{x}^2+1)}{1155}-\frac{5487704cx{(c^2{x}^2+1)}^2}{4622396625}-\frac{16\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{3}{2}}}{3465}-\frac{606416cx(c^2{x}^2+1)}{13867189875}-\frac{428cx{(c^2{x}^2+1)}^4}{323433}-\frac{2\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{7}{2}}}{1617}-\frac{4\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{5}{2}}}{1925}-\frac{148174cx{(c^2{x}^2+1)}^3}{110937519}+\frac{34\mathrm{arcsinh}\left(cx\right){(c^2{x}^2+1)}^{\frac{9}{2}}}{3267}+\frac{2cx{(c^2{x}^2+1)}^5}{1331}-\frac{32\mathrm{arcsinh}\left(cx\right)\sqrt{c^2{x}^2+1}}{1155}-\frac{2\mathrm{arcsinh}\left(cx\right)c^2{x}^2{(c^2{x}^2+1)}^{\frac{9}{2}}}{121}+\frac{382986368cx}{13867189875})+2d^{3}ab(\frac{\mathrm{arcsinh}\left(cx\right)c^{11}{x}^{11}}{11}+\frac{\mathrm{arcsinh}\left(cx\right)c^9{x}^9}{3}+\frac{3\mathrm{arcsinh}\left(cx\right)c^7{x}^7}{7}+\frac{\mathrm{arcsinh}\left(cx\right)c^5{x}^5}{5}-\frac{c^{10}{x}^{10}\sqrt{c^2{x}^2+1}}{121}-\frac{91c^8{x}^8\sqrt{c^2{x}^2+1}}{3267}-\frac{4705c^6{x}^6\sqrt{c^2{x}^2+1}}{160083}-\frac{6311c^4{x}^4\sqrt{c^2{x}^2+1}}{1334025}+\frac{25244c^2{x}^2\sqrt{c^2{x}^2+1}}{4002075}-\frac{50488\sqrt{c^2{x}^2+1}}{4002075})}{c^5}$$