3.143 \(\int \frac{g+h x}{\sqrt [3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} (\frac{f (b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2})}{c^2}+\frac{b f x}{c}+f x^2)} \, dx\)

Optimal. Leaf size=488 \[ \frac{3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac{f \left (b^2 h^2-b c g h+c^2 g^2\right )}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt [6]{3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}} \]

[Out]

(3*3^(1/6)*h*((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)*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*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) + (3^(2/3)*h*((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)*Log[(f*(c^2*g^2 - b*c*g*h + b^2*h^2))/(3
*c^2*h^2) + (b*f*x)/c + f*x^2])/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) - (3*3^
(2/3)*h*((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)*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))

________________________________________________________________________________________

Rubi [A]  time = 0.360419, antiderivative size = 488, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 104, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1041, 1040} \[ \frac{3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac{f \left (b^2 h^2-b c g h+c^2 g^2\right )}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt [6]{3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}} \]

Antiderivative was successfully verified.

[In]

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]

[Out]

(3*3^(1/6)*h*((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)*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*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) + (3^(2/3)*h*((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)*Log[(f*(c^2*g^2 - b*c*g*h + b^2*h^2))/(3
*c^2*h^2) + (b*f*x)/c + f*x^2])/(2*f*(-(((c*g - 2*b*h)*(c*g + b*h))/(c*h^2)) + 9*b*x + 9*c*x^2)^(1/3)) - (3*3^
(2/3)*h*((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)*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))

Rule 1041

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)), x_Sy
mbol] :> With[{q = -(c/(b^2 - 4*a*c))}, Dist[(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]

Rule 1040

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)), x_Sy
mbol] :> 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]

Rubi steps

\begin{align*} \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{\sqrt [3]{-\frac{c \left (\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2\right )}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}} \int \frac{g+h x}{\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 ) \sqrt [3]{-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 h^2 \left (b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}\right )}-\frac{b c x}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}-\frac{c^2 x^2}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}}} \, dx}{\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}}\\ &=\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}} \tan ^{-1}\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}}\\ \end{align*}

Mathematica [F]  time = 0.549227, size = 0, normalized size = 0. \[ \int \frac{g+h x}{\sqrt [3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac{f \left (b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac{b f x}{c}+f x^2\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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]

[Out]

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]

________________________________________________________________________________________

Maple [F]  time = 3.165, size = 0, normalized size = 0. \begin{align*} \int{(hx+g){\frac{1}{\sqrt [3]{{\frac{2\,{b}^{2}{h}^{2}+bcgh-{c}^{2}{g}^{2}}{9\,c{h}^{2}}}+bx+c{x}^{2}}}} \left ({\frac{f}{{c}^{2}} \left ({b}^{2}+{\frac{-2\,{b}^{2}{h}^{2}-bcgh+{c}^{2}{g}^{2}}{3\,{h}^{2}}} \right ) }+{\frac{bfx}{c}}+f{x}^{2} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 3 \, \int \frac{h x + g}{{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="maxima")

[Out]

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)

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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, algorithm="giac")

[Out]

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)