∫ x(c+dx+ex2+fx3+gx4+hx5)____________________

3.415 \(\int \frac{x (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=289 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]

[Out]

(h*x)/b^2 - (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a*b^2*(a + b*x^3)) - ((b^(5/3)*c + a^

(2/3)*b*e + 2*a*b^(2/3)*f - 4*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)

*b^(7/3)) - ((b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^(7/3)) + (

(b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^

(7/3)) + (g*Log[a + b*x^3])/(3*b^2)

________________________________________________________________________________________

Rubi [A] time = 0.505355, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1828, 1887, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{18 a^{4/3} b^{7/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (b^{2/3} (2 a f+b c)-a^{2/3} (b e-4 a h)\right )}{9 a^{4/3} b^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{2/3} b e-4 a^{5/3} h+2 a b^{2/3} f+b^{5/3} c\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{x \left (-b x (b c-a f)-b x^2 (b d-a g)+a (b e-a h)\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{h x}{b^2} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(h*x)/b^2 - (x*(a*(b*e - a*h) - b*(b*c - a*f)*x - b*(b*d - a*g)*x^2))/(3*a*b^2*(a + b*x^3)) - ((b^(5/3)*c + a^

(2/3)*b*e + 2*a*b^(2/3)*f - 4*a^(5/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(4/3)

*b^(7/3)) - ((b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(4/3)*b^(7/3)) + (

(b^(2/3)*(b*c + 2*a*f) - a^(2/3)*(b*e - 4*a*h))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b^

(7/3)) + (g*Log[a + b*x^3])/(3*b^2)

Rule 1828

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = m + Expon[Pq, x]}, Module[{Q = Pol

ynomialQuotient[b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] +

1)*x^m*Pq, a + b*x^n, x]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[

a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q

- 1)/n] + 1)), x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 0]

Rule 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x

] && PolyQ[Pq, x] && IntegerQ[n]

Rule 1871

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,

2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] || !Ration

alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1860

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R

t[a/b, 3]]}, -Dist[(r*(B*r - A*s))/(3*a*s), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) + s

*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[a/

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub

st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;

FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /

; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ

[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{x \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac{\int \frac{-a (b e-a h)-b (b c+2 a f) x-3 a b g x^2-3 a b h x^3}{a+b x^3} \, dx}{3 a b^2}\\ &=-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac{\int \left (-3 a h-\frac{a (b e-4 a h)+b (b c+2 a f) x+3 a b g x^2}{a+b x^3}\right ) \, dx}{3 a b^2}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{\int \frac{a (b e-4 a h)+b (b c+2 a f) x+3 a b g x^2}{a+b x^3} \, dx}{3 a b^2}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{\int \frac{a (b e-4 a h)+b (b c+2 a f) x}{a+b x^3} \, dx}{3 a b^2}+\frac{g \int \frac{x^2}{a+b x^3} \, dx}{b}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{\int \frac{\sqrt [3]{a} \left (\sqrt [3]{a} b (b c+2 a f)+2 a \sqrt [3]{b} (b e-4 a h)\right )+\sqrt [3]{b} \left (\sqrt [3]{a} b (b c+2 a f)-a \sqrt [3]{b} (b e-4 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{5/3} b^{7/3}}-\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} b^2}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a b^2}+\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{4/3} b^{7/3}}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{7/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^2}+\frac{\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{4/3} b^{7/3}}\\ &=\frac{h x}{b^2}-\frac{x \left (a (b e-a h)-b (b c-a f) x-b (b d-a g) x^2\right )}{3 a b^2 \left (a+b x^3\right )}-\frac{\left (b^{5/3} c+a^{2/3} b e+2 a b^{2/3} f-4 a^{5/3} h\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{7/3}}-\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{7/3}}+\frac{\left (b^{2/3} (b c+2 a f)-a^{2/3} (b e-4 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{7/3}}+\frac{g \log \left (a+b x^3\right )}{3 b^2}\\ \end{align*}

Mathematica [A] time = 0.196368, size = 285, normalized size = 0.99 \[ \frac{\frac{6 b^{2/3} \left (a^2 (g+h x)-a b (d+x (e+f x))+b^2 c x^2\right )}{a \left (a+b x^3\right )}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-a^{2/3} b^{4/3} e+4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{2/3} b^{4/3} e-4 a^{5/3} \sqrt [3]{b} h+2 a b f+b^2 c\right )}{a^{4/3}}+6 b^{2/3} g \log \left (a+b x^3\right )+18 b^{2/3} h x}{18 b^{8/3}} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(x*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(18*b^(2/3)*h*x + (6*b^(2/3)*(b^2*c*x^2 + a^2*(g + h*x) - a*b*(d + x*(e + f*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]

*(b^2*c + a^(2/3)*b^(4/3)*e + 2*a*b*f - 4*a^(5/3)*b^(1/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4

/3) - (2*(b^2*c - a^(2/3)*b^(4/3)*e + 2*a*b*f + 4*a^(5/3)*b^(1/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + ((b^2

*c - a^(2/3)*b^(4/3)*e + 2*a*b*f + 4*a^(5/3)*b^(1/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3

) + 6*b^(2/3)*g*Log[a + b*x^3])/(18*b^(8/3))

________________________________________________________________________________________

Maple [B] time = 0.01, size = 502, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x)

[Out]

h*x/b^2-1/3/b*x^2/(b*x^3+a)*f+1/3*x^2/a/(b*x^3+a)*c+1/3/b^2/(b*x^3+a)*a*h*x-1/3/b*x/(b*x^3+a)*e+1/3/b^2/(b*x^3

+a)*a*g-1/3/b/(b*x^3+a)*d-4/9/b^3*a/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))*h+1/9/b^2*e/(1/b*a)^(2/3)*ln(x+(1/b*a)^(

1/3))+2/9/b^3*a/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*h-1/18/b^2*e/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^

(1/3)*x+(1/b*a)^(2/3))-4/9/b^3*a/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*h+1/9/b^2*e/(

1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-2/9/b^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*f-1/9

/b/a/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*c+1/9/b^2/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*f+1/18/b/

a/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*c+2/9/b^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1

/b*a)^(1/3)*x-1))*f+1/9/b/a*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))*c+1/3*g*ln(b*x^3+a

)/b^2

________________________________________________________________________________________

Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C] time = 13.0278, size = 27753, normalized size = 96.03 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(36*a*b*h*x^4 - 12*a*b*d + 12*a^2*g + 12*(b^2*c - a*b*f)*x^2 - 2*(a*b^3*x^3 + a^2*b^2)*(2*(1/2)^(2/3)*(-I

*sqrt(3) + 1)*(9*g^2/b^4 - (b^2*c*e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)/(a^2*b^4))/(54*g^3/b^6 - 9*(b

^2*c*e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*

a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 12*a^3*b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4

*c^2*f + 64*a^5*h^3 - 3*(9*g^3 - 24*f*g*h + 16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2

- (e^3 - 3*(4*f^2 - 3*e*g)*c)*a^2*b^3)/(a^4*b^7))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*g^3/b^6 - 9*(b^2*c*

e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*a^2*b

^3*c*f^2 + 8*a^3*b^2*f^3 - 12*a^3*b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4*c^2*

f + 64*a^5*h^3 - 3*(9*g^3 - 24*f*g*h + 16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2 - (e

^3 - 3*(4*f^2 - 3*e*g)*c)*a^2*b^3)/(a^4*b^7))^(1/3) - 6*g/b^2)*log(-2*a*b^4*c^2*e - 8*a^2*b^3*c*e*f - 8*a^3*b^

2*e*f^2 + 3*a^3*b^2*e^2*g + 48*a^5*g*h^2 - 1/4*(a^3*b^6*c + 2*a^4*b^5*f)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*g^

2/b^4 - (b^2*c*e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)/(a^2*b^4))/(54*g^3/b^6 - 9*(b^2*c*e + (9*g^2 - 8

*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a

^3*b^2*f^3 - 12*a^3*b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4*c^2*f + 64*a^5*h^3

- 3*(9*g^3 - 24*f*g*h + 16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2 - (e^3 - 3*(4*f^2

- 3*e*g)*c)*a^2*b^3)/(a^4*b^7))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*g^3/b^6 - 9*(b^2*c*e + (9*g^2 - 8*f*h)

*a^2 + 2*(e*f - 2*c*h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^

2*f^3 - 12*a^3*b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4*c^2*f + 64*a^5*h^3 - 3*

(9*g^3 - 24*f*g*h + 16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2 - (e^3 - 3*(4*f^2 - 3*e

*g)*c)*a^2*b^3)/(a^4*b^7))^(1/3) - 6*g/b^2)^2 - 9*(a^3*b^2*c + 2*a^4*b*f)*g^2 + 1/2*(a^3*b^4*e^2 - 8*a^4*b^3*e

*h + 16*a^5*b^2*h^2 - 6*(a^3*b^4*c + 2*a^4*b^3*f)*g)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*g^2/b^4 - (b^2*c*e + (

9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*h)*a*b)/(a^2*b^4))/(54*g^3/b^6 - 9*(b^2*c*e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f -

2*c*h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 12*a^3*

b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4*c^2*f + 64*a^5*h^3 - 3*(9*g^3 - 24*f*g

*h + 16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2 - (e^3 - 3*(4*f^2 - 3*e*g)*c)*a^2*b^3)

/(a^4*b^7))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*g^3/b^6 - 9*(b^2*c*e + (9*g^2 - 8*f*h)*a^2 + 2*(e*f - 2*c*

h)*a*b)*g/(a^2*b^6) - (b^5*c^3 + a^2*b^3*e^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 12*a^3*b^2*e

^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)/(a^4*b^7) - (b^5*c^3 + 6*a*b^4*c^2*f + 64*a^5*h^3 - 3*(9*g^3 - 24*f*g*h +

16*e*h^2)*a^4*b + 2*(4*f^3 - 9*e*f*g + 6*e^2*h + 18*c*g*h)*a^3*b^2 - (e^3 - 3*(4*f^2 - 3*e*g)*c)*a^2*b^3)/(a^4

*b^7))^(1/3) - 6*g/b^2) + 8*(a^2*b^3*c^2 + 4*a^3*b^2*c*f + 4*a^4*b*f^2 - 3*a^4*b*e*g)*h - (b^5*c^3 + a^2*b^3*e

^3 + 6*a*b^4*c^2*f + 12*a^2*b^3*c*f^2 + 8*a^3*b^2*f^3 - 12*a^3*b^2*e^2*h + 48*a^4*b*e*h^2 - 64*a^5*h^3)*x) - 1

2*(a*b*e - 4*a^2*h)*x + (18*a*b*g*x^3 + 18*a^2*g + (a*b^3*x^3 + a^2*b^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9*g^