Integrand size = 104, antiderivative size = 488 \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3 \sqrt [6]{3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1+\frac {3 h (b+2 c x)}{2 c g-b h}}}\right )}{f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}+\frac {3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac {f \left (c^2 g^2-b c g h+b^2 h^2\right )}{3 c^2 h^2}+\frac {b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}-\frac {3\ 3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\frac {3 h (b+2 c x)}{2 c g-b h}}\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}} \]
-3*3^(1/6)*h*(c*h^2*((-2*b*h+c*g)*(b*h+c*g)/c/h^2-9*b*x-9*c*x^2)/(-b*h+2*c *g)^2)^(1/3)*arctan(-1/3*3^(1/2)+1/3*2^(2/3)*(1-3*h*(2*c*x+b)/(-b*h+2*c*g) )^(2/3)/(1+3*h*(2*c*x+b)/(-b*h+2*c*g))^(1/3)*3^(1/2))/f/(-(-2*b*h+c*g)*(b* h+c*g)/c/h^2+9*b*x+9*c*x^2)^(1/3)+1/2*3^(2/3)*h*(c*h^2*((-2*b*h+c*g)*(b*h+ c*g)/c/h^2-9*b*x-9*c*x^2)/(-b*h+2*c*g)^2)^(1/3)*ln(1/3*f*(b^2*h^2-b*c*g*h+ c^2*g^2)/c^2/h^2+b*f*x/c+f*x^2)/f/(-(-2*b*h+c*g)*(b*h+c*g)/c/h^2+9*b*x+9*c *x^2)^(1/3)-3/2*3^(2/3)*h*(c*h^2*((-2*b*h+c*g)*(b*h+c*g)/c/h^2-9*b*x-9*c*x ^2)/(-b*h+2*c*g)^2)^(1/3)*ln((1-3*h*(2*c*x+b)/(-b*h+2*c*g))^(2/3)+2^(1/3)* (1+3*h*(2*c*x+b)/(-b*h+2*c*g))^(1/3))/f/(-(-2*b*h+c*g)*(b*h+c*g)/c/h^2+9*b *x+9*c*x^2)^(1/3)
Time = 2.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.22 \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3^{2/3} \sqrt [3]{c} h^{5/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}{-4 b h+2 c (g-3 h x)+\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}\right )-2 \log \left (\sqrt {h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )+\log \left (h \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )+2 \log \left (\sqrt {c} \left (c g-2 b h-3 c h x-\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )\right )-\log \left (c \left (4 b^2 h^2-4 b c h (g-3 h x)+c^2 (g-3 h x)^2-2 b \sqrt [3]{c} h^{5/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{4/3} h^{2/3} \sqrt [3]{2 c g-b h} (g-3 h x) \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{2/3} h^{4/3} (2 c g-b h)^{2/3} \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )\right )\right )}{2 f (2 c g-b h)^{2/3}} \]
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]
(3^(2/3)*c^(1/3)*h^(5/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*c^(1/3)*h^(2/3)*(2*c*g - b*h)^(1/3)*((2*b^2)/c - (c*g^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(1/3)) /(-4*b*h + 2*c*(g - 3*h*x) + c^(1/3)*h^(2/3)*(2*c*g - b*h)^(1/3)*((2*b^2)/ c - (c*g^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(1/3))] - 2*Log[Sqrt[h]*((2*b ^2)/c - (c*g^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(1/3)] + Log[h*((2*b^2)/c - (c*g^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(2/3)] + 2*Log[Sqrt[c]*(c*g - 2*b*h - 3*c*h*x - c^(1/3)*h^(2/3)*(2*c*g - b*h)^(1/3)*((2*b^2)/c - (c*g^2) /h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(1/3))] - Log[c*(4*b^2*h^2 - 4*b*c*h*(g - 3*h*x) + c^2*(g - 3*h*x)^2 - 2*b*c^(1/3)*h^(5/3)*(2*c*g - b*h)^(1/3)*((2 *b^2)/c - (c*g^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(1/3) + c^(4/3)*h^(2/3) *(2*c*g - b*h)^(1/3)*(g - 3*h*x)*((2*b^2)/c - (c*g^2)/h^2 + (b*g)/h + 9*b* x + 9*c*x^2)^(1/3) + c^(2/3)*h^(4/3)*(2*c*g - b*h)^(2/3)*((2*b^2)/c - (c*g ^2)/h^2 + (b*g)/h + 9*b*x + 9*c*x^2)^(2/3))]))/(2*f*(2*c*g - b*h)^(2/3))
Time = 0.56 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1374, 27, 1373}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {g+h x}{\sqrt [3]{\frac {2 b^2 h^2+b c g h-c^2 g^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {2 b^2 h^2+b c g h-c^2 g^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 1374 |
| \(\displaystyle \frac {3^{2/3} \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \int \frac {3 (g+h x)}{\left (3 f x^2+\frac {3 b f x}{c}+\frac {f \left (c^2 g^2-b c h g+b^2 h^2\right )}{c^2 h^2}\right ) \sqrt [3]{-\frac {9 c^2 x^2 h^2}{(2 c g-b h)^2}-\frac {9 b c x h^2}{(2 c g-b h)^2}+\frac {(c g-2 b h) (c g+b h)}{(2 c g-b h)^2}}}dx}{\sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3\ 3^{2/3} \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \int \frac {g+h x}{\left (3 f x^2+\frac {3 b f x}{c}+\frac {f \left (c^2 g^2-b c h g+b^2 h^2\right )}{c^2 h^2}\right ) \sqrt [3]{-\frac {9 c^2 x^2 h^2}{(2 c g-b h)^2}-\frac {9 b c x h^2}{(2 c g-b h)^2}+\frac {(c g-2 b h) (c g+b h)}{(2 c g-b h)^2}}}dx}{\sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}\) |
| \(\Big \downarrow \) 1373 |
| \(\displaystyle \frac {3\ 3^{2/3} \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \left (\frac {h \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{\sqrt {3} f}+\frac {h \log \left (\frac {f \left (b^2 h^2-b c g h+c^2 g^2\right )}{c^2 h^2}+\frac {3 b f x}{c}+3 f x^2\right )}{6 f}-\frac {h \log \left (\left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f}\right )}{\sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}\) |
Int[(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]
(3*3^(2/3)*((c*h^2*(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2) - 9*b*x - 9*c*x^2) )/(2*c*g - b*h)^2)^(1/3)*((h*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 - (3*h*(b + 2* c*x))/(2*c*g - b*h))^(2/3))/(Sqrt[3]*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h)) ^(1/3))])/(Sqrt[3]*f) + (h*Log[(f*(c^2*g^2 - b*c*g*h + b^2*h^2))/(c^2*h^2) + (3*b*f*x)/c + 3*f*x^2])/(6*f) - (h*Log[(1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3) + 2^(1/3)*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h))^(1/3)])/(2*f)) )/(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)
3.2.43.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q = (-9*c*(h^2/(2*c*g - b *h)^2))^(1/3)}, Simp[Sqrt[3]*h*q*(ArcTan[1/Sqrt[3] - 2^(2/3)*((1 - (3*h*(b + 2*c*x))/(2*c*g - b*h))^(2/3)/(Sqrt[3]*(1 + (3*h*(b + 2*c*x))/(2*c*g - b*h ))^(1/3)))]/f), x] + (-Simp[3*h*q*(Log[(1 - 3*h*((b + 2*c*x)/(2*c*g - b*h)) )^(2/3) + 2^(1/3)*(1 + 3*h*((b + 2*c*x)/(2*c*g - b*h)))^(1/3)]/(2*f)), x] + Simp[h*q*(Log[d + e*x + f*x^2]/(2*f)), x])] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] && EqQ[c^2* g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && GtQ[-9*c*(h^2/(2*c*g - b*h)^2) , 0]
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)), x_Symbol] :> With[{q = -c/(b^2 - 4*a*c)}, Si mp[(q*(a + b*x + c*x^2))^(1/3)/(a + b*x + c*x^2)^(1/3) Int[(g + h*x)/((q* a + b*q*x + c*q*x^2)^(1/3)*(d + e*x + f*x^2)), x], x]] /; FreeQ[{a, b, c, d , e, f, g, h}, x] && EqQ[c*e - b*f, 0] && EqQ[c^2*d - f*(b^2 - 3*a*c), 0] & & EqQ[c^2*g^2 - b*c*g*h - 2*b^2*h^2 + 9*a*c*h^2, 0] && !GtQ[4*a - b^2/c, 0 ]
\[\int \frac {h x +g}{\left (\frac {2 b^{2} h^{2}+b c g h -c^{2} g^{2}}{9 c \,h^{2}}+b x +c \,x^{2}\right )^{\frac {1}{3}} \left (\frac {f \left (b^{2}+\frac {-2 b^{2} h^{2}-b c g h +c^{2} g^{2}}{3 h^{2}}\right )}{c^{2}}+\frac {b f x}{c}+f \,x^{2}\right )}d x\]
int((h*x+g)/(1/9*(2*b^2*h^2+b*c*g*h-c^2*g^2)/c/h^2+b*x+c*x^2)^(1/3)/(f*(b^ 2+1/3*(-2*b^2*h^2-b*c*g*h+c^2*g^2)/h^2)/c^2+b*f*x/c+f*x^2),x)
int((h*x+g)/(1/9*(2*b^2*h^2+b*c*g*h-c^2*g^2)/c/h^2+b*x+c*x^2)^(1/3)/(f*(b^ 2+1/3*(-2*b^2*h^2-b*c*g*h+c^2*g^2)/h^2)/c^2+b*f*x/c+f*x^2),x)
Timed out. \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\text {Timed out} \]
integrate((h*x+g)/(1/9*(2*b^2*h^2+b*c*g*h-c^2*g^2)/c/h^2+b*x+c*x^2)^(1/3)/ (f*(b^2+1/3*(-2*b^2*h^2-b*c*g*h+c^2*g^2)/h^2)/c^2+b*f*x/c+f*x^2),x, algori thm="fricas")
\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3 \cdot 3^{\frac {2}{3}} c^{2} h^{2} \left (\int \frac {g}{b^{2} h^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} - b c g h \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 b c h^{2} x \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + c^{2} g^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 c^{2} h^{2} x^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}}}\, dx + \int \frac {h x}{b^{2} h^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} - b c g h \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 b c h^{2} x \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + c^{2} g^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 c^{2} h^{2} x^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}}}\, dx\right )}{f} \]
integrate((h*x+g)/(1/9*(2*b**2*h**2+b*c*g*h-c**2*g**2)/c/h**2+b*x+c*x**2)* *(1/3)/(f*(b**2+1/3*(-2*b**2*h**2-b*c*g*h+c**2*g**2)/h**2)/c**2+b*f*x/c+f* x**2),x)
3*3**(2/3)*c**2*h**2*(Integral(g/(b**2*h**2*(2*b**2/c + b*g/h + 9*b*x - c* g**2/h**2 + 9*c*x**2)**(1/3) - b*c*g*h*(2*b**2/c + b*g/h + 9*b*x - c*g**2/ h**2 + 9*c*x**2)**(1/3) + 3*b*c*h**2*x*(2*b**2/c + b*g/h + 9*b*x - c*g**2/ h**2 + 9*c*x**2)**(1/3) + c**2*g**2*(2*b**2/c + b*g/h + 9*b*x - c*g**2/h** 2 + 9*c*x**2)**(1/3) + 3*c**2*h**2*x**2*(2*b**2/c + b*g/h + 9*b*x - c*g**2 /h**2 + 9*c*x**2)**(1/3)), x) + Integral(h*x/(b**2*h**2*(2*b**2/c + b*g/h + 9*b*x - c*g**2/h**2 + 9*c*x**2)**(1/3) - b*c*g*h*(2*b**2/c + b*g/h + 9*b *x - c*g**2/h**2 + 9*c*x**2)**(1/3) + 3*b*c*h**2*x*(2*b**2/c + b*g/h + 9*b *x - c*g**2/h**2 + 9*c*x**2)**(1/3) + c**2*g**2*(2*b**2/c + b*g/h + 9*b*x - c*g**2/h**2 + 9*c*x**2)**(1/3) + 3*c**2*h**2*x**2*(2*b**2/c + b*g/h + 9* b*x - c*g**2/h**2 + 9*c*x**2)**(1/3)), x))/f
\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int { \frac {3 \, {\left (h x + g\right )}}{{\left (c x^{2} + b x - \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac {1}{3}} {\left (3 \, f x^{2} + \frac {3 \, b f x}{c} + \frac {{\left (3 \, b^{2} + \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}} \,d x } \]
integrate((h*x+g)/(1/9*(2*b^2*h^2+b*c*g*h-c^2*g^2)/c/h^2+b*x+c*x^2)^(1/3)/ (f*(b^2+1/3*(-2*b^2*h^2-b*c*g*h+c^2*g^2)/h^2)/c^2+b*f*x/c+f*x^2),x, algori thm="maxima")
3*integrate((h*x + g)/((c*x^2 + b*x - 1/9*(c^2*g^2 - b*c*g*h - 2*b^2*h^2)/ (c*h^2))^(1/3)*(3*f*x^2 + 3*b*f*x/c + (3*b^2 + (c^2*g^2 - b*c*g*h - 2*b^2* h^2)/h^2)*f/c^2)), x)
\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int { \frac {3 \, {\left (h x + g\right )}}{{\left (c x^{2} + b x - \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac {1}{3}} {\left (3 \, f x^{2} + \frac {3 \, b f x}{c} + \frac {{\left (3 \, b^{2} + \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}} \,d x } \]
integrate((h*x+g)/(1/9*(2*b^2*h^2+b*c*g*h-c^2*g^2)/c/h^2+b*x+c*x^2)^(1/3)/ (f*(b^2+1/3*(-2*b^2*h^2-b*c*g*h+c^2*g^2)/h^2)/c^2+b*f*x/c+f*x^2),x, algori thm="giac")
integrate(3*(h*x + g)/((c*x^2 + b*x - 1/9*(c^2*g^2 - b*c*g*h - 2*b^2*h^2)/ (c*h^2))^(1/3)*(3*f*x^2 + 3*b*f*x/c + (3*b^2 + (c^2*g^2 - b*c*g*h - 2*b^2* h^2)/h^2)*f/c^2)), x)
Timed out. \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int \frac {g+h\,x}{{\left (b\,x+c\,x^2+\frac {\frac {2\,b^2\,h^2}{9}+\frac {b\,c\,g\,h}{9}-\frac {c^2\,g^2}{9}}{c\,h^2}\right )}^{1/3}\,\left (f\,x^2-\frac {f\,\left (\frac {\frac {2\,b^2\,h^2}{3}+\frac {b\,c\,g\,h}{3}-\frac {c^2\,g^2}{3}}{h^2}-b^2\right )}{c^2}+\frac {b\,f\,x}{c}\right )} \,d x \]
int((g + h*x)/((b*x + c*x^2 + ((2*b^2*h^2)/9 - (c^2*g^2)/9 + (b*c*g*h)/9)/ (c*h^2))^(1/3)*(f*x^2 - (f*(((2*b^2*h^2)/3 - (c^2*g^2)/3 + (b*c*g*h)/3)/h^ 2 - b^2))/c^2 + (b*f*x)/c)),x)