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

Optimal. Leaf size=306 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{18 a^{8/3} b^{2/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

[Out]

-c/(2*a^2*x^2) - d/(a^2*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*a^2*(a + b*x^3)) + ((5*b^(4/
3)*c + 4*a^(1/3)*b*d - 2*a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3
]*a^(8/3)*b^(2/3)) + (e*Log[x])/a^2 - ((b^(1/3)*(5*b*c - 2*a*f) - a^(1/3)*(4*b*d - a*g))*Log[a^(1/3) + b^(1/3)
*x])/(9*a^(8/3)*b^(2/3)) + ((b^(1/3)*(5*b*c - 2*a*f) - a^(1/3)*(4*b*d - a*g))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(18*a^(8/3)*b^(2/3)) - (e*Log[a + b*x^3])/(3*a^2)

________________________________________________________________________________________

Rubi [A]  time = 0.576867, antiderivative size = 304, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}-2 a f+5 b c\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (5 b c-2 a f)-\sqrt [3]{a} (4 b d-a g)\right )}{9 a^{8/3} b^{2/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^{4/3} (-g)+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}-\frac{x \left (x (b d-a g)+x^2 (b e-a h)-a f+b c\right )}{3 a^2 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}+\frac{e \log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(2*a^2*x^2) - d/(a^2*x) - (x*(b*c - a*f + (b*d - a*g)*x + (b*e - a*h)*x^2))/(3*a^2*(a + b*x^3)) + ((5*b^(4/
3)*c + 4*a^(1/3)*b*d - 2*a*b^(1/3)*f - a^(4/3)*g)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3
]*a^(8/3)*b^(2/3)) + (e*Log[x])/a^2 - ((b^(1/3)*(5*b*c - 2*a*f) - a^(1/3)*(4*b*d - a*g))*Log[a^(1/3) + b^(1/3)
*x])/(9*a^(8/3)*b^(2/3)) + ((5*b*c - 2*a*f - (a^(1/3)*(4*b*d - a*g))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x
+ b^(2/3)*x^2])/(18*a^(8/3)*b^(1/3)) - (e*Log[a + b*x^3])/(3*a^2)

Rule 1829

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)*Coeff[R, x, i]*x^(i - m))/a, {i, 0, n - 1}], x], x], x] - S
imp[(x*R*(a + b*x^n)^(p + 1))/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[((c*x)^m*Pq)/(a + b*
x^n), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

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/
b]

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{c+d x+e x^2+f x^3+g x^4+h x^5}{x^3 \left (a+b x^3\right )^2} \, dx &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{\int \frac{-3 b^2 c-3 b^2 d x-3 b^2 e x^2+2 b^2 \left (\frac{b c}{a}-f\right ) x^3+b^2 \left (\frac{b d}{a}-g\right ) x^4}{x^3 \left (a+b x^3\right )} \, dx}{3 a b^2}\\ &=-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac{\int \left (-\frac{3 b^2 c}{a x^3}-\frac{3 b^2 d}{a x^2}-\frac{3 b^2 e}{a x}+\frac{b^2 \left (5 b c-2 a f+(4 b d-a g) x+3 b e x^2\right )}{a \left (a+b x^3\right )}\right ) \, dx}{3 a b^2}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{e \log (x)}{a^2}-\frac{\int \frac{5 b c-2 a f+(4 b d-a g) x+3 b e x^2}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{e \log (x)}{a^2}-\frac{\int \frac{5 b c-2 a f+(4 b d-a g) x}{a+b x^3} \, dx}{3 a^2}-\frac{(b e) \int \frac{x^2}{a+b x^3} \, dx}{a^2}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{e \log (x)}{a^2}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{\int \frac{\sqrt [3]{a} \left (2 \sqrt [3]{b} (5 b c-2 a f)+\sqrt [3]{a} (4 b d-a g)\right )+\sqrt [3]{b} \left (-\sqrt [3]{b} (5 b c-2 a f)+\sqrt [3]{a} (4 b d-a g)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{8/3} \sqrt [3]{b}}-\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{e \log (x)}{a^2}-\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f-a^{4/3} g\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{7/3} \sqrt [3]{b}}+\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\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^{8/3} \sqrt [3]{b}}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{e \log (x)}{a^2}-\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^2}-\frac{\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f-a^{4/3} g\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^{8/3} b^{2/3}}\\ &=-\frac{c}{2 a^2 x^2}-\frac{d}{a^2 x}-\frac{x \left (b c-a f+(b d-a g) x+(b e-a h) x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac{\left (5 b^{4/3} c+4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f-a^{4/3} g\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} b^{2/3}}+\frac{e \log (x)}{a^2}-\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{\left (5 b c-2 a f-\frac{\sqrt [3]{a} (4 b d-a g)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{e \log \left (a+b x^3\right )}{3 a^2}\\ \end{align*}

Mathematica [A]  time = 0.488182, size = 292, normalized size = 0.95 \[ -\frac{\frac{6 a \left (a^2 h-a b (e+x (f+g x))+b^2 x (c+d x)\right )}{b \left (a+b x^3\right )}-\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d-2 a \sqrt [3]{b} f+5 b^{4/3} c\right )}{b^{2/3}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^{4/3} g-4 \sqrt [3]{a} b d+2 a \sqrt [3]{b} f-5 b^{4/3} c\right )}{b^{2/3}}+6 a e \log \left (a+b x^3\right )+\frac{9 a c}{x^2}+\frac{18 a d}{x}-18 a e \log (x)}{18 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((9*a*c)/x^2 + (18*a*d)/x + (6*a*(a^2*h + b^2*x*(c + d*x) - a*b*(e + x*(f + g*x))))/(b*(a + b*x^3)) + (2*Sqrt
[3]*a^(1/3)*(-5*b^(4/3)*c - 4*a^(1/3)*b*d + 2*a*b^(1/3)*f + a^(4/3)*g)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt
[3]])/b^(2/3) - 18*a*e*Log[x] + (2*a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*a*b^(1/3)*f + a^(4/3)*g)*Log[a^(1/
3) + b^(1/3)*x])/b^(2/3) - (a^(1/3)*(5*b^(4/3)*c - 4*a^(1/3)*b*d - 2*a*b^(1/3)*f + a^(4/3)*g)*Log[a^(2/3) - a^
(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) + 6*a*e*Log[a + b*x^3])/(18*a^3)

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Maple [B]  time = 0.013, size = 527, 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((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x^3/(b*x^3+a)^2,x)

[Out]

1/3/a/(b*x^3+a)*x^2*g-1/3/a^2*b*x^2/(b*x^3+a)*d+1/3/a*x/(b*x^3+a)*f-1/3*b/a^2*x/(b*x^3+a)*c-1/3/(b*x^3+a)/b*h+
1/3/a/(b*x^3+a)*e-5/9/a^2*c/(1/b*a)^(2/3)*ln(x+(1/b*a)^(1/3))+5/18/a^2*c/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+
(1/b*a)^(2/3))-5/9/a^2*c/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+2/9/a*f/b/(1/b*a)^(2/
3)*ln(x+(1/b*a)^(1/3))-1/9/a*f/b/(1/b*a)^(2/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))+2/9/a*f/b/(1/b*a)^(2/3)*3
^(1/2)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))+4/9/a^2/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))*d-2/9/a^2/(1/b*a)^(
1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(2/3))*d-4/9/a^2*3^(1/2)/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)
*x-1))*d-1/9/a*g/b/(1/b*a)^(1/3)*ln(x+(1/b*a)^(1/3))+1/18/a*g/b/(1/b*a)^(1/3)*ln(x^2-(1/b*a)^(1/3)*x+(1/b*a)^(
2/3))+1/9/a*g*3^(1/2)/b/(1/b*a)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*x-1))-1/3*e*ln(b*x^3+a)/a^2-d/a^2/x-
1/2*c/a^2/x^2+e*ln(x)/a^2

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 121.35, size = 28467, normalized size = 93.03 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/324*(108*(4*b^2*d - a*b*g)*x^4 + 324*a*b*d*x + 54*(5*b^2*c - 2*a*b*f)*x^3 + 162*a*b*c - 108*(a*b*e - a^2*h)
*x^2 + 2*(a^2*b^2*x^5 + a^3*b*x^2)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5
*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) -
1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*
a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*
e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^
(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^
7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*
g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b
+ 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8
*b^2))^(1/3) + 54*e/a^2)*log(-160*a*b^3*c*d^2 + 75*a*b^3*c^2*e - 36*a^2*b^2*d*e^2 + 12*a^3*b*e*f^2 - 1/324*(4*
a^6*b^2*d - a^7*b*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^
5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^
4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 -
a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f
+ 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*
sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*
(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d
*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 2
4*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) +
54*e/a^2)^2 - 2*(5*a^3*b*c - 2*a^4*f)*g^2 - 1/18*(25*a^3*b^3*c^2 - 24*a^4*b^2*d*e - 20*a^4*b^2*c*f + 4*a^5*b*
f^2 + 6*a^5*b*e*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*
b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*
c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^
4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f +
16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sq
rt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(1
25*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g
^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*
d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 5
4*e/a^2) + 4*(16*a^2*b^2*d^2 - 15*a^2*b^2*c*e)*f + (80*a^2*b^2*c*d + 9*a^3*b*e^2 - 32*a^3*b*d*f)*g - (125*b^4*
c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^
4*g^3)*x) + (162*b^2*e*x^5 + 162*a*b*e*x^2 - (a^2*b^2*x^5 + a^3*b*x^2)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*
c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^
2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 -
8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^
3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*
c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g
+ (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c
*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 -
2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^
3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2) - 3*sqrt(1/3)*(a^2*b^2*x^5 + a^3*b*x^2)*sqrt(-(((
-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 +
1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 -
150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/
1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2
- 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3
/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3
*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^
2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*
(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)^2*a^5*b - 10
8*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^
6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^
3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2)
- 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*
f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27
*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a
*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^
8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g
+ 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)*a^3*b*e
+ 25920*b^2*c*d + 2916*a*b*e^2 - 10368*a*b*d*f - 1296*(5*a*b*c - 2*a^2*f)*g)/(a^5*b)))*log(160*a*b^3*c*d^2 - 7
5*a*b^3*c^2*e + 36*a^2*b^2*d*e^2 - 12*a^3*b*e*f^2 + 1/324*(4*a^6*b^2*d - a^7*b*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4
- (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*
f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b
^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^
3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(3
2*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d +
2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 6
0*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 +
a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2
- 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)^2 + 2*(5*a^3*b*c - 2*a^4*f)*g^2 + 1/18
*(25*a^3*b^3*c^2 - 24*a^4*b^2*d*e - 20*a^4*b^2*c*f + 4*a^5*b*f^2 + 6*a^5*b*e*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 -
(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*
g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2
*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3
- 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*
d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2
*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*
a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a
^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 -
2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2) - 4*(16*a^2*b^2*d^2 - 15*a^2*b^2*c*e)*f
- (80*a^2*b^2*c*d + 9*a^3*b*e^2 - 32*a^3*b*d*f)*g - 2*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b
^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)*x + 1/108*sqrt(1/3)*(450*a^3*b^3*c^2 + 2
16*a^4*b^2*d*e - 360*a^4*b^2*c*f + 72*a^5*b*f^2 - 54*a^5*b*e*g - (4*a^6*b^2*d - a^7*b*g)*((-I*sqrt(3) + 1)*(9*
e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d +
2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 6
0*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 +
a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2
- 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^
2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^
2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^
4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*
a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2))*sqrt(-(((-I*sqrt(3) + 1)*(9*e^
2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2
*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*
a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a
^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 -
2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*
c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*
f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*
c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^
2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)^2*a^5*b - 108*((-I*sqrt(3) + 1)*(
9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d
+ 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f +
60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3
+ a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b
^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*
b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*
c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*
b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c
)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)*a^3*b*e + 25920*b^2*c*d + 291
6*a*b*e^2 - 10368*a*b*d*f - 1296*(5*a*b*c - 2*a^2*f)*g)/(a^5*b))) + (162*b^2*e*x^5 + 162*a*b*e*x^2 - (a^2*b^2*
x^5 + a^3*b*x^2)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b)
)/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^
3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*
g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 1
6*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt
(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125
*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2
- a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*
e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*
e/a^2) + 3*sqrt(1/3)*(a^2*b^2*x^5 + a^3*b*x^2)*sqrt(-(((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g +
(9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g
)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*
a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d
*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)
*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f
- 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^
3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*
g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75
*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)^2*a^5*b - 108*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f
*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5
*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 -
48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g +
6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^
2*f)*a*b^3)/(a^8*b^2))^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*
d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*
b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*
e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e
+ 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 54*e/a^2)*a^3*b*e + 25920*b^2*c*d + 2916*a*b*e^2 - 10368*a*b*d*f - 1296*
(5*a*b*c - 2*a^2*f)*g)/(a^5*b)))*log(160*a*b^3*c*d^2 - 75*a*b^3*c^2*e + 36*a^2*b^2*d*e^2 - 12*a^3*b*e*f^2 + 1/
324*(4*a^6*b^2*d - a^7*b*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a
*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*
(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d
*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 2
4*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) +
81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) -
1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*
a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*
e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^
(1/3) + 54*e/a^2)^2 + 2*(5*a^3*b*c - 2*a^4*f)*g^2 + 1/18*(25*a^3*b^3*c^2 - 24*a^4*b^2*d*e - 20*a^4*b^2*c*f + 4
*a^5*b*f^2 + 6*a^5*b*e*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b
)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/1458*(1
25*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d*g
^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^3 - 24*
d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1/3) + 8
1*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/
1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^
3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^
3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1
/3) + 54*e/a^2) - 4*(16*a^2*b^2*d^2 - 15*a^2*b^2*c*e)*f - (80*a^2*b^2*c*d + 9*a^3*b*e^2 - 32*a^3*b*d*f)*g - 2*
(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^3*b*d
*g^2 - a^4*g^3)*x - 1/108*sqrt(1/3)*(450*a^3*b^3*c^2 + 216*a^4*b^2*d*e - 360*a^4*b^2*c*f + 72*a^5*b*f^2 - 54*a
^5*b*e*g - (4*a^6*b^2*d - a^7*b*g)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5
*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) -
1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*
a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*
e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^
(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^
7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*
g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b
+ 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8
*b^2))^(1/3) + 54*e/a^2))*sqrt(-(((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c
*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b) - 1/
1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 12*a^
3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(9*e^
3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2))^(1
/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*
b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g
+ 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b +
3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b
^2))^(1/3) + 54*e/a^2)^2*a^5*b - 108*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f -
5*c*g)*a*b)/(a^5*b))/(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(a^7*b)
- 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^2*g + 1
2*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*b + 3*(
9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a^8*b^2)
)^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b^2*c*d + 2*a^2*f*g + (9*e^2 - 8*d*f - 5*c*g)*a*b)*e/(
a^7*b) - 1/1458*(125*b^4*c^3 + 64*a*b^3*d^3 - 150*a*b^3*c^2*f + 60*a^2*b^2*c*f^2 - 8*a^3*b*f^3 - 48*a^2*b^2*d^
2*g + 12*a^3*b*d*g^2 - a^4*g^3)/(a^8*b^2) - 1/1458*(125*b^4*c^3 + a^4*g^3 - 2*(4*f^3 - 9*e*f*g + 6*d*g^2)*a^3*
b + 3*(9*e^3 - 24*d*e*f + 16*d^2*g + 5*(4*f^2 - 3*e*g)*c)*a^2*b^2 - 2*(32*d^3 - 90*c*d*e + 75*c^2*f)*a*b^3)/(a
^8*b^2))^(1/3) + 54*e/a^2)*a^3*b*e + 25920*b^2*c*d + 2916*a*b*e^2 - 10368*a*b*d*f - 1296*(5*a*b*c - 2*a^2*f)*g
)/(a^5*b))) - 324*(b^2*e*x^5 + a*b*e*x^2)*log(x))/(a^2*b^2*x^5 + a^3*b*x^2)

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

[Out]

Timed out

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Giac [A]  time = 1.07838, size = 483, normalized size = 1.58 \begin{align*} -\frac{e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac{e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d + \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b^{2} c - 2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b f + 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} b d - \left (-a b^{2}\right )^{\frac{2}{3}} a g\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} + \frac{{\left (4 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} b g \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 5 \, a^{2} b^{2} c - 2 \, a^{3} b f\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{5} b} - \frac{2 \,{\left (4 \, b^{2} d - a b g\right )} x^{4} + 6 \, a b d x +{\left (5 \, b^{2} c - 2 \, a b f\right )} x^{3} + 3 \, a b c + 2 \,{\left (a^{2} h - a b e\right )} x^{2}}{6 \,{\left (b x^{3} + a\right )} a^{2} b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*e*log(abs(b*x^3 + a))/a^2 + e*log(abs(x))/a^2 - 1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*b^2*c - 2*(-a*b^2)^(1/3)*a*
b*f - 4*(-a*b^2)^(2/3)*b*d + (-a*b^2)^(2/3)*a*g)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^3*b^
2) - 1/18*(5*(-a*b^2)^(1/3)*b^2*c - 2*(-a*b^2)^(1/3)*a*b*f + 4*(-a*b^2)^(2/3)*b*d - (-a*b^2)^(2/3)*a*g)*log(x^
2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2) + 1/9*(4*a^2*b^2*d*(-a/b)^(1/3) - a^3*b*g*(-a/b)^(1/3) + 5*a^2*b^
2*c - 2*a^3*b*f)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*b) - 1/6*(2*(4*b^2*d - a*b*g)*x^4 + 6*a*b*d*x +
(5*b^2*c - 2*a*b*f)*x^3 + 3*a*b*c + 2*(a^2*h - a*b*e)*x^2)/((b*x^3 + a)*a^2*b*x^2)