30.1 Problem number 38

$$\int \frac{1}{{(4ac+4c^2{x}^2+4cd{x}^3+d^2{x}^4)}^2}dx$$

Optimal antiderivative $$-\frac{(\frac{c}{d}+x)(c^3-4a{d}^2-c{d}^2{(\frac{c}{d}+x)}^2)}{16ac(4a{d}^2+c^3)(d^2{x}^4+4cd{x}^3+4c^2{x}^2+4ac)}-\frac{d\mathrm{arctanh}\left(\frac{c\sqrt{2}+dx\sqrt{2}+c^{\frac{1}{4}}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}}}{c^{\frac{1}{4}}\sqrt{c^{\frac{3}{2}}-\sqrt{4a{d}^2+c^3}}}\right)(c^3+12a{d}^2+c^{\frac{3}{2}}\sqrt{4a{d}^2+c^3})\sqrt{2}}{64a{c}^{\frac{7}{4}}{(4a{d}^2+c^3)}^{\frac{3}{2}}\sqrt{c^{\frac{3}{2}}-\sqrt{4a{d}^2+c^3}}}+\frac{d\mathrm{arctanh}\left(\frac{-(dx+c)\sqrt{2}+c^{\frac{1}{4}}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}}}{c^{\frac{1}{4}}\sqrt{c^{\frac{3}{2}}-\sqrt{4a{d}^2+c^3}}}\right)(c^3+12a{d}^2+c^{\frac{3}{2}}\sqrt{4a{d}^2+c^3})\sqrt{2}}{64a{c}^{\frac{7}{4}}{(4a{d}^2+c^3)}^{\frac{3}{2}}\sqrt{c^{\frac{3}{2}}-\sqrt{4a{d}^2+c^3}}}-\frac{d\ln (d^2{(\frac{c}{d}+x)}^2+\sqrt{c}\sqrt{4a{d}^2+c^3}-c^{\frac{1}{4}}d(\frac{c}{d}+x)\sqrt{2}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}})(c^3+12a{d}^2-c^{\frac{3}{2}}\sqrt{4a{d}^2+c^3})\sqrt{2}}{128a{c}^{\frac{7}{4}}{(4a{d}^2+c^3)}^{\frac{3}{2}}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}}}+\frac{d\ln (d^2{(\frac{c}{d}+x)}^2+\sqrt{c}\sqrt{4a{d}^2+c^3}+c^{\frac{1}{4}}d(\frac{c}{d}+x)\sqrt{2}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}})(c^3+12a{d}^2-c^{\frac{3}{2}}\sqrt{4a{d}^2+c^3})\sqrt{2}}{128a{c}^{\frac{7}{4}}{(4a{d}^2+c^3)}^{\frac{3}{2}}\sqrt{c^{\frac{3}{2}}+\sqrt{4a{d}^2+c^3}}}$$

command

integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**2,x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$\frac{4acd+3c^{2}d{x}^2+c{d}^2{x}^3+x(4a{d}^2+2c^3)}{256a^3{c}^2{d}^2+64a^2{c}^5+x^4(64a^{2}c{d}^4+16a{c}^4{d}^2)+x^3(256a^2{c}^2{d}^3+64a{c}^{5}d)+x^2(256a^2{c}^3{d}^2+64a{c}^6)}+\mathrm{RootSum}(t^4(1073741824a^9{c}^7{d}^6+805306368a^8{c}^{10}{d}^4+201326592a^7{c}^{13}{d}^2+16777216a^6{c}^{16})+t^2(491520a^5{c}^5{d}^4+122880a^4{c}^8{d}^2+8192a^3{c}^{11})+81a^2{d}^4+18a{c}^3{d}^2+c^6,(t\mapsto t\log (x+\frac{-67108864t^3{a}^7{c}^7{d}^8-58720256t^3{a}^6{c}^{10}{d}^6-18874368t^3{a}^5{c}^{13}{d}^4-2621440t^3{a}^4{c}^{16}{d}^2-131072t^3{a}^3{c}^{19}+27648t{a}^4{c}^2{d}^8-9216t{a}^3{c}^5{d}^6-5440t{a}^2{c}^8{d}^4-736ta{c}^{11}{d}^2-32t{c}^{14}+324a^{2}c{d}^7+81a{c}^4{d}^5+5c^7{d}^3}{324a^2{d}^8+81a{c}^3{d}^6+5c^6{d}^4})))$$