15.3 Problem number 295

$$\int (g+hx{)}^2(A+B\log (e(a+bx{)}^n(c+dx{)}^{-n}))dx$$

Optimal antiderivative $$-\frac{B(-ad+bc)h(-adh-bch+3bdg)nx}{3b^2{d}^2}-\frac{B(-ad+bc)h^{2}n{x}^2}{6bd}-\frac{B{(-ah+bg)}^{3}n\ln (bx+a)}{3b^{3}h}+\frac{B{(-ch+dg)}^{3}n\ln (dx+c)}{3d^{3}h}+\frac{{(hx+g)}^3(A+B\ln \left(e{(bx+a)}^n{(dx+c)}^{-n}\right))}{3h}$$

command

integrate((h*x+g)^2*(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

$$\text{Timed out}$$

Giac 1.7.0 via sagemath 9.3 output

$$\frac{1}{3}(A{h}^2+B{h}^2)x^3+\frac{1}{3}(B{h}^{2}n{x}^3+3Bghn{x}^2+3B{g}^{2}nx)\log (bx+a)-\frac{1}{3}(B{h}^{2}n{x}^3+3Bghn{x}^2+3B{g}^{2}nx)\log (dx+c)-\frac{(Bbc{h}^{2}n-Bad{h}^{2}n-6Abdgh-6Bbdgh)x^2}{6bd}+\frac{(3Ba{b}^2{g}^{2}n-3B{a}^{2}bghn+B{a}^3{h}^{2}n)\log (bx+a)}{3b^3}-\frac{(3Bc{d}^2{g}^{2}n-3B{c}^{2}dghn+B{c}^3{h}^{2}n)\log (-dx-c)}{3d^3}-\frac{(3B{b}^{2}cdghn-3Bab{d}^{2}ghn-B{b}^2{c}^2{h}^{2}n+B{a}^2{d}^2{h}^{2}n-3A{b}^2{d}^2{g}^2-3B{b}^2{d}^2{g}^2)x}{3b^2{d}^2}$$