\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {3 \,3^{\frac {1}{6}} h \left (\frac {c \,h^{2} \left (\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}-9 b x -9 c \,x^{2}\right )}{\left (-b h +2 c g \right )^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2^{\frac {2}{3}} \left (1-\frac {3 h \left (2 c x +b \right )}{-b h +2 c g}\right )^{\frac {2}{3}} \sqrt {3}}{3 \left (1+\frac {3 h \left (2 c x +b \right )}{-b h +2 c g}\right )^{\frac {1}{3}}}\right )}{f \left (-\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}+9 b x +9 c \,x^{2}\right )^{\frac {1}{3}}}+\frac {3^{\frac {2}{3}} h \left (\frac {c \,h^{2} \left (\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}-9 b x -9 c \,x^{2}\right )}{\left (-b h +2 c g \right )^{2}}\right )^{\frac {1}{3}} \ln \left (\frac {f \left (b^{2} h^{2}-b c g h +c^{2} g^{2}\right )}{3 c^{2} h^{2}}+\frac {b f x}{c}+f \,x^{2}\right )}{2 f \left (-\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}+9 b x +9 c \,x^{2}\right )^{\frac {1}{3}}}-\frac {3 \,3^{\frac {2}{3}} h \left (\frac {c \,h^{2} \left (\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}-9 b x -9 c \,x^{2}\right )}{\left (-b h +2 c g \right )^{2}}\right )^{\frac {1}{3}} \ln \left (\left (1-\frac {3 h \left (2 c x +b \right )}{-b h +2 c g}\right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (1+\frac {3 h \left (2 c x +b \right )}{-b h +2 c g}\right )^{\frac {1}{3}}\right )}{2 f \left (-\frac {\left (-2 b h +c g \right ) \left (b h +c g \right )}{c \,h^{2}}+9 b x +9 c \,x^{2}\right )^{\frac {1}{3}}} \]
command
Integrate[(g + h*x)/(((-(c^2*g^2) + b*c*g*h + 2*b^2*h^2)/(9*c*h^2) + b*x + c*x^2)^(1/3)*((f*(b^2 - (-(c^2*g^2) + b*c*g*h + 2*b^2*h^2)/(3*h^2)))/c^2 + (b*f*x)/c + f*x^2)),x]
Mathematica 13.1 output
\[ \frac {3^{2/3} \sqrt [3]{c} h^{5/3} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}{-4 b h+2 c (g-3 h x)+\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}\right )-2 \log \left (\sqrt {h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )+\log \left (h \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )+2 \log \left (\sqrt {c} \left (c g-2 b h-3 c h x-\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )\right )-\log \left (c \left (4 b^2 h^2-4 b c h (g-3 h x)+c^2 (g-3 h x)^2-2 b \sqrt [3]{c} h^{5/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{4/3} h^{2/3} \sqrt [3]{2 c g-b h} (g-3 h x) \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{2/3} h^{4/3} (2 c g-b h)^{2/3} \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )\right )\right )}{2 f (2 c g-b h)^{2/3}} \]
Mathematica 12.3 output
\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx \]________________________________________________________________________________________