10.11 Problem number 539

$$\int x^4{(a+b{x}^2)}^{5/2}(A+B{x}^2)dx$$

Optimal antiderivative $$\frac{a(12Ab-5aB)x^5{(b{x}^2+a)}^{\frac{3}{2}}}{192b}+\frac{(12Ab-5aB)x^5{(b{x}^2+a)}^{\frac{5}{2}}}{120b}+\frac{B{x}^5{(b{x}^2+a)}^{\frac{7}{2}}}{12b}+\frac{a^5(12Ab-5aB)\mathrm{arctanh}\left(\frac{x\sqrt{b}}{\sqrt{b{x}^2+a}}\right)}{1024b^{\frac{7}{2}}}-\frac{a^4(12Ab-5aB)x\sqrt{b{x}^2+a}}{1024b^3}+\frac{a^3(12Ab-5aB)x^3\sqrt{b{x}^2+a}}{1536b^2}+\frac{a^2(12Ab-5aB)x^5\sqrt{b{x}^2+a}}{384b}$$

command

integrate(x**4*(b*x**2+a)**(5/2)*(B*x**2+A),x)

Sympy 1.10.1 under Python 3.10.4 output

$$\text{Timed out}$$

Sympy 1.8 under Python 3.8.8 output

$$-\frac{3A{a}^{\frac{9}{2}}x}{256b^2\sqrt{1+\frac{b{x}^2}{a}}}-\frac{A{a}^{\frac{7}{2}}{x}^3}{256b\sqrt{1+\frac{b{x}^2}{a}}}+\frac{129A{a}^{\frac{5}{2}}{x}^5}{640\sqrt{1+\frac{b{x}^2}{a}}}+\frac{73A{a}^{\frac{3}{2}}b{x}^7}{160\sqrt{1+\frac{b{x}^2}{a}}}+\frac{29A\sqrt{a}{b}^2{x}^9}{80\sqrt{1+\frac{b{x}^2}{a}}}+\frac{3A{a}^5\mathrm{asinh}\left(\frac{\sqrt{b}x}{\sqrt{a}}\right)}{256b^{\frac{5}{2}}}+\frac{A{b}^3{x}^{11}}{10\sqrt{a}\sqrt{1+\frac{b{x}^2}{a}}}+\frac{5B{a}^{\frac{11}{2}}x}{1024b^3\sqrt{1+\frac{b{x}^2}{a}}}+\frac{5B{a}^{\frac{9}{2}}{x}^3}{3072b^2\sqrt{1+\frac{b{x}^2}{a}}}-\frac{B{a}^{\frac{7}{2}}{x}^5}{1536b\sqrt{1+\frac{b{x}^2}{a}}}+\frac{55B{a}^{\frac{5}{2}}{x}^7}{384\sqrt{1+\frac{b{x}^2}{a}}}+\frac{67B{a}^{\frac{3}{2}}b{x}^9}{192\sqrt{1+\frac{b{x}^2}{a}}}+\frac{7B\sqrt{a}{b}^2{x}^{11}}{24\sqrt{1+\frac{b{x}^2}{a}}}-\frac{5B{a}^6\mathrm{asinh}\left(\frac{\sqrt{b}x}{\sqrt{a}}\right)}{1024b^{\frac{7}{2}}}+\frac{B{b}^3{x}^{13}}{12\sqrt{a}\sqrt{1+\frac{b{x}^2}{a}}}$$