16.6 Problem number 150

$$\int \frac{c+dx+e{x}^2+f{x}^3}{{(a-b{x}^4)}^3}dx$$

Optimal antiderivative $$\frac{x(5e{x}^2+6dx+7c)}{32a^2(-b{x}^4+a)}+\frac{af+bx(e{x}^2+dx+c)}{8ab{(-b{x}^4+a)}^2}+\frac{3d\mathrm{arctanh}\left(\frac{x^2\sqrt{b}}{\sqrt{a}}\right)}{16a^{\frac{5}{2}}\sqrt{b}}+\frac{\arctan \left(\frac{b^{\frac{1}{4}}x}{a^{\frac{1}{4}}}\right)(-5e\sqrt{a}+21c\sqrt{b})}{64a^{\frac{11}{4}}{b}^{\frac{3}{4}}}+\frac{\mathrm{arctanh}\left(\frac{b^{\frac{1}{4}}x}{a^{\frac{1}{4}}}\right)(5e\sqrt{a}+21c\sqrt{b})}{64a^{\frac{11}{4}}{b}^{\frac{3}{4}}}$$

command

integrate((f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**3,x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$-\mathrm{RootSum}(268435456t^4{a}^{11}{b}^3+t^2(-6881280a^6{b}^{2}ce-4718592a^6{b}^2{d}^2)+t(-153600a^{4}bd{e}^2-2709504a^3{b}^2{c}^{2}d)-625a^2{e}^4+22050ab{c}^2{e}^2-60480abc{d}^{2}e+20736ab{d}^4-194481b^2{c}^4,(t\mapsto t\log (x+\frac{-262144000t^3{a}^{10}{b}^2{e}^3-4624220160t^3{a}^9{b}^3{c}^{2}e+12683575296t^3{a}^9{b}^{3}c{d}^2+309657600t^2{a}^7{b}^{2}cd{e}^2-283115520t^2{a}^7{b}^2{d}^{3}e-1820786688t^2{a}^6{b}^3{c}^{3}d+5040000t{a}^{5}bc{e}^4+6912000t{a}^{5}b{d}^2{e}^3+118540800t{a}^4{b}^2{c}^3{e}^2-365783040t{a}^4{b}^2{c}^2{d}^{2}e-111476736t{a}^4{b}^{2}c{d}^4+522764928t{a}^3{b}^3{c}^5+112500a^{3}d{e}^5-4536000a^{2}bc{d}^3{e}^2+2488320a^{2}b{d}^{5}e+58344300a{b}^2{c}^{4}de-80015040a{b}^2{c}^3{d}^3}{15625a^3{e}^6+275625a^{2}b{c}^2{e}^4-3024000a^{2}bc{d}^2{e}^3+2073600a^{2}b{d}^4{e}^2-4862025a{b}^2{c}^4{e}^2+53343360a{b}^2{c}^3{d}^{2}e-36578304a{b}^2{c}^2{d}^4-85766121b^3{c}^6})))-\frac{-4a^{2}f-11abcx-10abd{x}^2-9abe{x}^3+7b^{2}c{x}^5+6b^{2}d{x}^6+5b^{2}e{x}^7}{32a^{4}b-64a^3{b}^2{x}^4+32a^2{b}^3{x}^8}$$