\(\int \frac {f+g x+h x^2}{(d+e x)^2 (a+b x+c x^2)^2} \, dx\) [159]

Optimal result

Integrand size = 30, antiderivative size = 673 \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]

[Out]

-e*(d^2*h-d*e*g+e^2*f)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)+(-b^3*e^2*f+b^2*e*(a*e*g+2*c*d*f)-2*a*c*(c*d*(-d*g+2*e*f)
+a*e*(-2*d*h+e*g))-b*(c^2*d^2*f+a^2*e^2*h-a*c*(-d^2*h-2*d*e*g+3*e^2*f))-c*(2*c^2*d^2*f+2*a^2*e^2*h-a*b*e*(2*d*
h+e*g)+b^2*(d^2*h+e^2*f)-c*(b*d*(d*g+2*e*f)+2*a*(d^2*h-2*d*e*g+e^2*f)))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/
(c*x^2+b*x+a)+(4*c^4*d^4*f-b^3*e^3*(2*a*d*h-a*e*g-b*d*g+2*b*e*f)-2*c^3*d^2*(b*d*(d*g+4*e*f)-2*a*(d^2*h-2*d*e*g
+6*e^2*f))-6*c^2*e*(4*a*b*d*e^2*f-b^2*d^3*g+2*a^2*e*(2*d^2*h-2*d*e*g+e^2*f))-c*e*(6*a^2*b*e^3*g-4*a^3*e^3*h-b^
3*d*(-2*d^2*h-3*d*e*g+4*e^2*f)-6*a*b^2*e*(2*d^2*h-d*e*g+2*e^2*f)))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a
*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)^3-e*(e^2*(2*a*d*h-a*e*g-b*d*g+2*b*e*f)-c*d*(2*d^2*h-3*d*e*g+4*e^2*f))*ln(e*x
+d)/(a*e^2-b*d*e+c*d^2)^3+1/2*e*(e^2*(2*a*d*h-a*e*g-b*d*g+2*b*e*f)-c*d*(2*d^2*h-3*d*e*g+4*e^2*f))*ln(c*x^2+b*x
+a)/(a*e^2-b*d*e+c*d^2)^3

Rubi [A] (verified)

Time = 1.57 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1660, 1642, 648, 632, 212, 642} \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {c x \left (2 a^2 e^2 h-c \left (2 a \left (d^2 h-2 d e g+e^2 f\right )+b d (d g+2 e f)\right )-a b e (2 d h+e g)+b^2 \left (d^2 h+e^2 f\right )+2 c^2 d^2 f\right )+b \left (a^2 e^2 h-a c \left (d^2 (-h)-2 d e g+3 e^2 f\right )+c^2 d^2 f\right )-b^2 e (a e g+2 c d f)+2 a c (a e (e g-2 d h)+c d (2 e f-d g))+b^3 e^2 f}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-6 c^2 e \left (2 a^2 e \left (2 d^2 h-2 d e g+e^2 f\right )+4 a b d e^2 f-b^2 d^3 g\right )-c e \left (-4 a^3 e^3 h+6 a^2 b e^3 g-6 a b^2 e \left (2 d^2 h-d e g+2 e^2 f\right )+b^3 (-d) \left (-2 d^2 h-3 d e g+4 e^2 f\right )\right )-b^3 e^3 (2 a d h-a e g-b d g+2 b e f)-2 c^3 d^2 \left (b d (d g+4 e f)-2 a \left (d^2 h-2 d e g+6 e^2 f\right )\right )+4 c^4 d^4 f\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}+\frac {e \log \left (a+b x+c x^2\right ) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{2 \left (a e^2-b d e+c d^2\right )^3}-\frac {e \left (d^2 h-d e g+e^2 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac {e \log (d+e x) \left (e^2 (2 a d h-a e g-b d g+2 b e f)-c d \left (2 d^2 h-3 d e g+4 e^2 f\right )\right )}{\left (a e^2-b d e+c d^2\right )^3} \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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 648

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 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1660

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*
c*f - b*g)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {\frac {2 c^3 d^4 f-b^2 e^2 \left (a e^2 f-b d (2 e f-d g)-a d^2 h\right )-c^2 d^2 \left (b d (2 e f+d g)-2 a \left (5 e^2 f-2 d e g+d^2 h\right )\right )-c e \left (2 b^2 d^2 (e f-d g)-2 a^2 e \left (2 e^2 f-d^2 h\right )+a b d \left (8 e^2 f-3 d e g+2 d^2 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^2}+\frac {e \left (4 c^3 d^3 f-c \left (2 a b e^2 (2 e f+d g)-4 a^2 e^2 (e g-d h)-b^2 d^2 (e g+2 d h)\right )-b^2 e \left (a e (e g-2 d h)-b \left (e^2 f-d^2 h\right )\right )-2 c^2 d \left (b d (2 e f+d g)-2 a \left (e^2 f+d e g-d^2 h\right )\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^2}+\frac {c e^2 \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x^2}{\left (c d^2-b d e+a e^2\right )^2}}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{-b^2+4 a c} \\ & = -\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (-\frac {\left (b^2-4 a c\right ) e^2 \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (b^2-4 a c\right ) e^2 \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right )}{\left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {2 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )+c^2 e \left (3 b^2 d^3 g-2 a b d \left (10 e^2 f-3 d e g+2 d^2 h\right )-6 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )+c e^2 \left (b^3 d (4 e f-3 d g)+2 a^3 e^2 h-a^2 b e (5 e g-4 d h)+a b^2 \left (10 e^2 f-5 d e g+6 d^2 h\right )\right )-c \left (b^2-4 a c\right ) e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) x}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x+c x^2\right )}\right ) \, dx}{-b^2+4 a c} \\ & = -\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\int \frac {2 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )+c^2 e \left (3 b^2 d^3 g-2 a b d \left (10 e^2 f-3 d e g+2 d^2 h\right )-6 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )+c e^2 \left (b^3 d (4 e f-3 d g)+2 a^3 e^2 h-a^2 b e (5 e g-4 d h)+a b^2 \left (10 e^2 f-5 d e g+6 d^2 h\right )\right )-c \left (b^2-4 a c\right ) e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\left (e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {b^3 e^2 f-b^2 e (2 c d f+a e g)+2 a c (c d (2 e f-d g)+a e (e g-2 d h))+b \left (c^2 d^2 f+a^2 e^2 h-a c \left (3 e^2 f-2 d e g-d^2 h\right )\right )+c \left (2 c^2 d^2 f+2 a^2 e^2 h-a b e (e g+2 d h)+b^2 \left (e^2 f+d^2 h\right )-c \left (b d (2 e f+d g)+2 a \left (e^2 f-2 d e g+d^2 h\right )\right )\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (4 c^4 d^4 f-b^3 e^3 (2 b e f-b d g-a e g+2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )-c e \left (6 a^2 b e^3 g-4 a^3 e^3 h-b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )-6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}-\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e \left (e^2 (2 b e f-b d g-a e g+2 a d h)-c d \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.97 \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=-\frac {e \left (e^2 f-d e g+d^2 h\right )}{\left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {-b^3 e^2 f+b^2 \left (a e^2 g-c \left (-2 d e f+e^2 f x+d^2 h x\right )\right )+b \left (-a^2 e^2 h+c^2 d (-d f+2 e f x+d g x)+a c \left (-d^2 h+e^2 (3 f+g x)-2 d e (g-h x)\right )\right )+2 c \left (-c^2 d^2 f x+a c \left (e^2 f x-2 d e (f+g x)+d^2 (g+h x)\right )-a^2 e (-2 d h+e (g+h x))\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {\left (4 c^4 d^4 f+b^3 e^3 (-2 b e f+b d g+a e g-2 a d h)-2 c^3 d^2 \left (b d (4 e f+d g)-2 a \left (6 e^2 f-2 d e g+d^2 h\right )\right )-6 c^2 e \left (4 a b d e^2 f-b^2 d^3 g+2 a^2 e \left (e^2 f-2 d e g+2 d^2 h\right )\right )+c e \left (-6 a^2 b e^3 g+4 a^3 e^3 h+b^3 d \left (4 e^2 f-3 d e g-2 d^2 h\right )+6 a b^2 e \left (2 e^2 f-d e g+2 d^2 h\right )\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {\left (e^3 (-2 b e f+b d g+a e g-2 a d h)+c d e \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}-\frac {\left (e^3 (-2 b e f+b d g+a e g-2 a d h)+c d e \left (4 e^2 f-3 d e g+2 d^2 h\right )\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^3} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1344\) vs. \(2(668)=1336\).

Time = 0.84 (sec) , antiderivative size = 1345, normalized size of antiderivative = 2.00

method	result	size
default	\(\text {Expression too large to display}\)	\(1345\)
risch	\(\text {Expression too large to display}\)	\(8771\)

[In]

int((h*x^2+g*x+f)/(e*x+d)^2/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-e*(d^2*h-d*e*g+e^2*f)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-e*(2*a*d*e^2*h-a*e^3*g-b*d*e^2*g+2*b*e^3*f-2*c*d^3*h+3*c*
d^2*e*g-4*c*d*e^2*f)/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)+1/(a*e^2-b*d*e+c*d^2)^3*((c*(2*a^3*e^4*h-4*a^2*b*d*e^3*h-
a^2*b*e^4*g+4*a^2*c*d*e^3*g-2*a^2*c*e^4*f+3*a*b^2*d^2*e^2*h+a*b^2*d*e^3*g+a*b^2*e^4*f-6*a*b*c*d^2*e^2*g-2*a*c^
2*d^4*h+4*a*c^2*d^3*e*g-b^3*d^3*e*h-b^3*d*e^3*f+b^2*c*d^4*h+b^2*c*d^3*e*g+3*b^2*c*d^2*e^2*f-b*c^2*d^4*g-4*b*c^
2*d^3*e*f+2*c^3*d^4*f)/(4*a*c-b^2)*x+(a^3*b*e^4*h-4*a^3*c*d*e^3*h+2*a^3*c*e^4*g-a^2*b^2*d*e^3*h-a^2*b^2*e^4*g+
6*a^2*b*c*d^2*e^2*h-3*a^2*b*c*e^4*f-4*a^2*c^2*d^3*e*h+4*a^2*c^2*d*e^3*f+a*b^3*d*e^3*g+a*b^3*e^4*f-a*b^2*c*d^3*
e*h-3*a*b^2*c*d^2*e^2*g+a*b^2*c*d*e^3*f+a*b*c^2*d^4*h+4*a*b*c^2*d^3*e*g-6*a*b*c^2*d^2*e^2*f-2*a*c^3*d^4*g+4*a*
c^3*d^3*e*f-b^4*d*e^3*f+3*b^3*c*d^2*e^2*f-3*b^2*c^2*d^3*e*f+b*c^3*d^4*f)/(4*a*c-b^2))/(c*x^2+b*x+a)+1/(4*a*c-b
^2)*(1/2*(8*a^2*c^2*d*e^3*h-4*a^2*c^2*e^4*g-2*a*b^2*c*d*e^3*h+a*b^2*c*e^4*g-4*a*b*c^2*d*e^3*g+8*a*b*c^2*e^4*f-
8*a*c^3*d^3*e*h+12*a*c^3*d^2*e^2*g-16*a*c^3*d*e^3*f+b^3*c*d*e^3*g-2*b^3*c*e^4*f+2*b^2*c^2*d^3*e*h-3*b^2*c^2*d^
2*e^2*g+4*b^2*c^2*d*e^3*f)/c*ln(c*x^2+b*x+a)+2*(a*b^3*e^4*g+b^4*d*e^3*g-b*c^3*d^4*g+2*a^3*c*e^4*h-6*a^2*c^2*e^
4*f+2*a*c^3*d^4*h-4*a*c^3*d^3*e*g+4*b^3*c*d*e^3*f-4*b*c^3*d^3*e*f-5*a^2*b*c*e^4*g+12*a^2*c^2*d*e^3*g+10*a*b^2*
c*e^4*f-1/2*(8*a^2*c^2*d*e^3*h-4*a^2*c^2*e^4*g-2*a*b^2*c*d*e^3*h+a*b^2*c*e^4*g-4*a*b*c^2*d*e^3*g+8*a*b*c^2*e^4
*f-8*a*c^3*d^3*e*h+12*a*c^3*d^2*e^2*g-16*a*c^3*d*e^3*f+b^3*c*d*e^3*g-2*b^3*c*e^4*f+2*b^2*c^2*d^3*e*h-3*b^2*c^2
*d^2*e^2*g+4*b^2*c^2*d*e^3*f)*b/c+4*a^2*b*c*d*e^3*h-5*a*b^2*c*d*e^3*g-4*a*b*c^2*d^3*e*h+6*a*b*c^2*d^2*e^2*g-20
*a*b*c^2*d*e^3*f+6*a*b^2*c*d^2*e^2*h+12*a*c^3*d^2*e^2*f-3*b^3*c*d^2*e^2*g+3*b^2*c^2*d^3*e*g-12*a^2*c^2*d^2*e^2
*h-2*a*b^3*d*e^3*h-2*b^4*e^4*f+2*c^4*d^4*f)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

Fricas [F(-1)]

Timed out. \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((h*x**2+g*x+f)/(e*x+d)**2/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1506 vs. \(2 (666) = 1332\).

Time = 0.29 (sec) , antiderivative size = 1506, normalized size of antiderivative = 2.24 \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(4*c*d*e^3*f - 2*b*e^4*f - 3*c*d^2*e^2*g + b*d*e^3*g + a*e^4*g + 2*c*d^3*e*h - 2*a*d*e^3*h)*log(c - 2*c*d
/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)^2)/(c^3*d^6 - 3*b*c^2*d^5
*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3
*a^2*b*d*e^5 + a^3*e^6) - (e^7*f/(e*x + d) - d*e^6*g/(e*x + d) + d^2*e^5*h/(e*x + d))/(c^2*d^4*e^4 - 2*b*c*d^3
*e^5 + b^2*d^2*e^6 + 2*a*c*d^2*e^6 - 2*a*b*d*e^7 + a^2*e^8) - (4*c^4*d^4*e^2*f - 8*b*c^3*d^3*e^3*f + 24*a*c^3*
d^2*e^4*f + 4*b^3*c*d*e^5*f - 24*a*b*c^2*d*e^5*f - 2*b^4*e^6*f + 12*a*b^2*c*e^6*f - 12*a^2*c^2*e^6*f - 2*b*c^3
*d^4*e^2*g + 6*b^2*c^2*d^3*e^3*g - 8*a*c^3*d^3*e^3*g - 3*b^3*c*d^2*e^4*g + b^4*d*e^5*g - 6*a*b^2*c*d*e^5*g + 2
4*a^2*c^2*d*e^5*g + a*b^3*e^6*g - 6*a^2*b*c*e^6*g + 4*a*c^3*d^4*e^2*h - 2*b^3*c*d^3*e^3*h + 12*a*b^2*c*d^2*e^4
*h - 24*a^2*c^2*d^2*e^4*h - 2*a*b^3*d*e^5*h + 4*a^3*c*e^6*h)*arctan((2*c*d - 2*c*d^2/(e*x + d) - b*e + 2*b*d*e
/(e*x + d) - 2*a*e^2/(e*x + d))/(sqrt(-b^2 + 4*a*c)*e))/((b^2*c^3*d^6 - 4*a*c^4*d^6 - 3*b^3*c^2*d^5*e + 12*a*b
*c^3*d^5*e + 3*b^4*c*d^4*e^2 - 9*a*b^2*c^2*d^4*e^2 - 12*a^2*c^3*d^4*e^2 - b^5*d^3*e^3 - 2*a*b^3*c*d^3*e^3 + 24
*a^2*b*c^2*d^3*e^3 + 3*a*b^4*d^2*e^4 - 9*a^2*b^2*c*d^2*e^4 - 12*a^3*c^2*d^2*e^4 - 3*a^2*b^3*d*e^5 + 12*a^3*b*c
*d*e^5 + a^3*b^2*e^6 - 4*a^4*c*e^6)*sqrt(-b^2 + 4*a*c)*e^2) - ((2*c^4*d^3*e*f - 3*b*c^3*d^2*e^2*f + 3*b^2*c^2*
d*e^3*f - 6*a*c^3*d*e^3*f - b^3*c*e^4*f + 3*a*b*c^2*e^4*f - b*c^3*d^3*e*g + 6*a*c^3*d^2*e^2*g - 3*a*b*c^2*d*e^
3*g + a*b^2*c*e^4*g - 2*a^2*c^2*e^4*g + b^2*c^2*d^3*e*h - 2*a*c^3*d^3*e*h - 3*a*b*c^2*d^2*e^2*h + 6*a^2*c^2*d*
e^3*h - a^2*b*c*e^4*h)/(c*d^2 - b*d*e + a*e^2) - (2*c^4*d^4*e^2*f - 4*b*c^3*d^3*e^3*f + 6*b^2*c^2*d^2*e^4*f -
12*a*c^3*d^2*e^4*f - 4*b^3*c*d*e^5*f + 12*a*b*c^2*d*e^5*f + b^4*e^6*f - 4*a*b^2*c*e^6*f + 2*a^2*c^2*e^6*f - b*
c^3*d^4*e^2*g + 8*a*c^3*d^3*e^3*g - 6*a*b*c^2*d^2*e^4*g + 4*a*b^2*c*d*e^5*g - 8*a^2*c^2*d*e^5*g - a*b^3*e^6*g
+ 3*a^2*b*c*e^6*g + b^2*c^2*d^4*e^2*h - 2*a*c^3*d^4*e^2*h - 4*a*b*c^2*d^3*e^3*h + 12*a^2*c^2*d^2*e^4*h - 4*a^2
*b*c*d*e^5*h + a^2*b^2*e^6*h - 2*a^3*c*e^6*h)/((c*d^2 - b*d*e + a*e^2)*(e*x + d)*e))/((c*d^2 - b*d*e + a*e^2)^
2*(b^2 - 4*a*c)*(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2 + a*e^2/(e*x + d)
^2))

Mupad [B] (verification not implemented)

Time = 21.29 (sec) , antiderivative size = 26278, normalized size of antiderivative = 39.05 \[ \int \frac {f+g x+h x^2}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((a*b^2*e^3*f - 2*a*c^2*d^3*g + b*c^2*d^3*f - 4*a^2*c*e^3*f + b^3*d*e^2*f - 2*a*b^2*d*e^2*g + 4*a*c^2*d^2*e*f
+ a*b^2*d^2*e*h + a^2*b*d*e^2*h + 6*a^2*c*d*e^2*g - 2*b^2*c*d^2*e*f - 8*a^2*c*d^2*e*h + a*b*c*d^3*h - 3*a*b*c*
d*e^2*f + 2*a*b*c*d^2*e*g)/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^
2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2) + (x*(2*b^3*e^3
*f + 2*c^3*d^3*f - a*b^2*e^3*g - 2*a*c^2*d^3*h - b*c^2*d^3*g + a^2*b*e^3*h + 2*a^2*c*e^3*g + b^2*c*d^3*h - b^3
*d*e^2*g + b^3*d^2*e*h + 2*a*c^2*d*e^2*f + 2*a*c^2*d^2*e*g - b*c^2*d^2*e*f - b^2*c*d*e^2*f - 2*a^2*c*d*e^2*h -
7*a*b*c*e^3*f + 5*a*b*c*d*e^2*g - 5*a*b*c*d^2*e*h))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 -
b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*
c*d^2*e^2) - (x^2*(6*a*c^2*e^3*f - 2*b^2*c*e^3*f - 2*a^2*c*e^3*h - 2*c^3*d^2*e*f - 8*a*c^2*d*e^2*g + 2*b*c^2*d
*e^2*f + 6*a*c^2*d^2*e*h + b*c^2*d^2*e*g + b^2*c*d*e^2*g - 2*b^2*c*d^2*e*h + a*b*c*e^3*g + 2*a*b*c*d*e^2*h))/(
4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e^3 + 2*b^
3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2))/(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c
*e*x^3) + symsum(log((x*(36*a^2*c^5*e^7*f^2 + 4*b^4*c^3*e^7*f^2 + 4*a^4*c^3*e^7*h^2 + 4*c^7*d^4*e^3*f^2 + a^2*
b^2*c^3*e^7*g^2 + 64*a^2*c^5*d^2*e^5*g^2 + 12*b^2*c^5*d^2*e^5*f^2 + 36*a^2*c^5*d^4*e^3*h^2 - 24*a^3*c^4*d^2*e^
5*h^2 + b^2*c^5*d^4*e^3*g^2 + 2*b^3*c^4*d^3*e^4*g^2 + b^4*c^3*d^2*e^5*g^2 + 4*b^4*c^3*d^4*e^3*h^2 - 24*a^3*c^4
*e^7*f*h - 24*a*b^2*c^4*e^7*f^2 - 24*a*c^6*d^2*e^5*f^2 - 8*b*c^6*d^3*e^4*f^2 - 8*b^3*c^4*d*e^6*f^2 - 16*a*b*c^
5*d^3*e^4*g^2 + 2*a*b^3*c^3*d*e^6*g^2 - 16*a^2*b*c^4*d*e^6*g^2 - 8*a^3*b*c^3*d*e^6*h^2 + 8*a^2*b^2*c^3*e^7*f*h
+ 80*a^2*c^5*d^2*e^5*f*h - 96*a^2*c^5*d^3*e^4*g*h + 8*b^2*c^5*d^4*e^3*f*h - 8*b^3*c^4*d^3*e^4*f*h + 8*b^4*c^3
*d^2*e^5*f*h - 4*b^3*c^4*d^4*e^3*g*h - 4*b^4*c^3*d^3*e^4*g*h - 14*a*b^2*c^4*d^2*e^5*g^2 - 24*a*b^2*c^4*d^4*e^3
*h^2 - 8*a*b^3*c^3*d^3*e^4*h^2 + 24*a^2*b*c^4*d^3*e^4*h^2 + 24*a*b*c^5*d*e^6*f^2 - 4*a*b^3*c^3*e^7*f*g + 12*a^
2*b*c^4*e^7*f*g + 32*a*c^6*d^3*e^4*f*g - 96*a^2*c^5*d*e^6*f*g - 4*a^3*b*c^3*e^7*g*h - 24*a*c^6*d^4*e^3*f*h - 4
*b*c^6*d^4*e^3*f*g - 4*b^4*c^3*d*e^6*f*g + 32*a^3*c^4*d*e^6*g*h + 12*a^2*b^2*c^3*d^2*e^5*h^2 - 24*a*b*c^5*d^2*
e^5*f*g + 48*a*b^2*c^4*d*e^6*f*g + 16*a*b*c^5*d^3*e^4*f*h - 8*a*b^3*c^3*d*e^6*f*h + 16*a^2*b*c^4*d*e^6*f*h + 1
2*a*b*c^5*d^4*e^3*g*h - 40*a*b^2*c^4*d^2*e^5*f*h + 48*a*b^2*c^4*d^3*e^4*g*h - 24*a^2*b*c^4*d^2*e^5*g*h))/(16*a
^2*c^6*d^8 + a^4*b^4*e^8 + 16*a^6*c^2*e^8 + b^4*c^4*d^8 + b^8*d^4*e^4 - 8*a*b^2*c^5*d^8 - 8*a^5*b^2*c*e^8 - 4*
a*b^7*d^3*e^5 - 4*a^3*b^5*d*e^7 - 4*b^5*c^3*d^7*e - 4*b^7*c*d^5*e^3 + 6*a^2*b^6*d^2*e^6 + 64*a^3*c^5*d^6*e^2 +
96*a^4*c^4*d^4*e^4 + 64*a^5*c^3*d^2*e^6 + 6*b^6*c^2*d^6*e^2 + 64*a^2*b^2*c^4*d^6*e^2 + 32*a^2*b^3*c^3*d^5*e^3
- 74*a^2*b^4*c^2*d^4*e^4 + 144*a^3*b^2*c^3*d^4*e^4 + 32*a^3*b^3*c^2*d^3*e^5 + 64*a^4*b^2*c^2*d^2*e^6 + 32*a*b
^3*c^4*d^7*e + 4*a*b^6*c*d^4*e^4 - 64*a^2*b*c^5*d^7*e + 32*a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 - 44*a*b^4*c^3
*d^6*e^2 + 20*a*b^5*c^2*d^5*e^3 + 20*a^2*b^5*c*d^3*e^5 - 192*a^3*b*c^4*d^5*e^3 - 44*a^3*b^4*c*d^2*e^6 - 192*a^
4*b*c^3*d^3*e^5) - root(3840*a^6*b*c^5*d^5*e^7*z^3 + 3840*a^5*b*c^6*d^7*e^5*z^3 + 1920*a^7*b*c^4*d^3*e^9*z^3 +
1920*a^4*b*c^7*d^9*e^3*z^3 - 288*a^7*b^3*c^2*d*e^11*z^3 - 288*a^2*b^3*c^7*d^11*e*z^3 + 210*a^4*b^7*c*d^3*e^9*
z^3 + 210*a*b^7*c^4*d^9*e^3*z^3 - 174*a^5*b^6*c*d^2*e^10*z^3 - 174*a*b^6*c^5*d^10*e^2*z^3 - 120*a^3*b^8*c*d^4*
e^8*z^3 - 120*a*b^8*c^3*d^8*e^4*z^3 + 12*a^2*b^9*c*d^5*e^7*z^3 + 12*a*b^9*c^2*d^7*e^5*z^3 + 384*a^8*b*c^3*d*e^
11*z^3 + 384*a^3*b*c^8*d^11*e*z^3 + 72*a^6*b^5*c*d*e^11*z^3 + 72*a*b^5*c^6*d^11*e*z^3 + 18*a*b^10*c*d^6*e^6*z^
3 - 4800*a^5*b^2*c^5*d^6*e^6*z^3 - 3120*a^6*b^2*c^4*d^4*e^8*z^3 - 3120*a^4*b^2*c^6*d^8*e^4*z^3 + 2160*a^4*b^4*
c^4*d^6*e^6*z^3 - 1776*a^4*b^5*c^3*d^5*e^7*z^3 - 1776*a^3*b^5*c^4*d^7*e^5*z^3 + 1740*a^5*b^4*c^3*d^4*e^8*z^3 +
1740*a^3*b^4*c^5*d^8*e^4*z^3 + 960*a^5*b^3*c^4*d^5*e^7*z^3 + 960*a^4*b^3*c^5*d^7*e^5*z^3 - 672*a^7*b^2*c^3*d^
2*e^10*z^3 - 672*a^3*b^2*c^7*d^10*e^2*z^3 + 648*a^6*b^4*c^2*d^2*e^10*z^3 + 648*a^2*b^4*c^6*d^10*e^2*z^3 - 600*
a^5*b^5*c^2*d^3*e^9*z^3 - 600*a^2*b^5*c^5*d^9*e^3*z^3 + 372*a^3*b^7*c^2*d^5*e^7*z^3 + 372*a^2*b^7*c^3*d^7*e^5*
z^3 + 316*a^3*b^6*c^3*d^6*e^6*z^3 - 222*a^2*b^8*c^2*d^6*e^6*z^3 - 160*a^6*b^3*c^3*d^3*e^9*z^3 - 160*a^3*b^3*c^
6*d^9*e^3*z^3 + 15*a^4*b^6*c^2*d^4*e^8*z^3 + 15*a^2*b^6*c^4*d^8*e^4*z^3 - 6*b^11*c*d^7*e^5*z^3 - 6*b^7*c^5*d^1
1*e*z^3 - 6*a^5*b^7*d*e^11*z^3 - 6*a*b^11*d^5*e^7*z^3 - 12*a^7*b^4*c*e^12*z^3 - 12*a*b^4*c^7*d^12*z^3 - 20*b^9
*c^3*d^9*e^3*z^3 + 15*b^10*c^2*d^8*e^4*z^3 + 15*b^8*c^4*d^10*e^2*z^3 - 1280*a^6*c^6*d^6*e^6*z^3 - 960*a^7*c^5*
d^4*e^8*z^3 - 960*a^5*c^7*d^8*e^4*z^3 - 384*a^8*c^4*d^2*e^10*z^3 - 384*a^4*c^8*d^10*e^2*z^3 - 20*a^3*b^9*d^3*e
^9*z^3 + 15*a^4*b^8*d^2*e^10*z^3 + 15*a^2*b^10*d^4*e^8*z^3 + 48*a^8*b^2*c^2*e^12*z^3 + 48*a^2*b^2*c^8*d^12*z^3
- 64*a^9*c^3*e^12*z^3 - 64*a^3*c^9*d^12*z^3 + b^12*d^6*e^6*z^3 + b^6*c^6*d^12*z^3 + a^6*b^6*e^12*z^3 - 44*a^3
*b^4*c*d*e^7*g*h*z - 20*a*b^6*c*d^3*e^5*g*h*z - 12*a*b^2*c^5*d^7*e*g*h*z + 432*a^4*b*c^3*d*e^7*f*h*z + 84*a^2*
b^5*c*d*e^7*f*h*z + 28*a*b^6*c*d^2*e^6*f*h*z - 8*a*b*c^6*d^6*e^2*f*g*z - 804*a^3*b^2*c^3*d^3*e^5*g*h*z + 564*a
^2*b^2*c^4*d^5*e^3*g*h*z + 222*a^3*b^3*c^2*d^2*e^6*g*h*z + 186*a^2*b^4*c^2*d^3*e^5*g*h*z - 166*a^2*b^3*c^3*d^4
*e^4*g*h*z + 792*a^3*b^2*c^3*d^2*e^6*f*h*z - 744*a^2*b^2*c^4*d^4*e^4*f*h*z + 492*a^2*b^3*c^3*d^3*e^5*f*h*z - 2
64*a^2*b^4*c^2*d^2*e^6*f*h*z + 996*a^2*b^2*c^4*d^3*e^5*f*g*z - 870*a^2*b^3*c^3*d^2*e^6*f*g*z + 16*a*b*c^6*d^7*
e*f*h*z - 56*a*b^6*c*d*e^7*f*g*z - 264*a^4*b*c^3*d^2*e^6*g*h*z + 208*a^3*b*c^4*d^4*e^4*g*h*z + 156*a^4*b^2*c^2
*d*e^7*g*h*z - 148*a*b^4*c^3*d^5*e^3*g*h*z + 54*a*b^5*c^2*d^4*e^4*g*h*z - 48*a^2*b^5*c*d^2*e^6*g*h*z - 24*a^2*
b*c^5*d^6*e^2*g*h*z + 10*a*b^3*c^4*d^6*e^2*g*h*z - 656*a^3*b*c^4*d^3*e^5*f*h*z - 308*a^3*b^3*c^2*d*e^7*f*h*z +
116*a*b^4*c^3*d^4*e^4*f*h*z - 84*a*b^5*c^2*d^3*e^5*f*h*z + 68*a*b^3*c^4*d^5*e^3*f*h*z - 48*a^2*b*c^5*d^5*e^3*
f*h*z - 24*a*b^2*c^5*d^6*e^2*f*h*z + 1320*a^3*b*c^4*d^2*e^6*f*g*z - 732*a^3*b^2*c^3*d*e^7*f*g*z + 306*a^2*b^4*
c^2*d*e^7*f*g*z - 304*a*b^4*c^3*d^3*e^5*f*g*z + 222*a*b^5*c^2*d^2*e^6*f*g*z + 110*a*b^3*c^4*d^4*e^4*f*g*z - 84
*a*b^2*c^5*d^5*e^3*f*g*z + 16*a*c^7*d^7*e*f*g*z - 8*a*b^7*d*e^7*f*h*z + 4*a*b*c^6*d^8*g*h*z + 6*b^6*c^2*d^5*e^
3*g*h*z + 6*b^5*c^3*d^6*e^2*g*h*z + 1072*a^4*c^4*d^3*e^5*g*h*z - 720*a^3*c^5*d^5*e^3*g*h*z - 8*b^6*c^2*d^4*e^4
*f*h*z - 8*b^4*c^4*d^6*e^2*f*h*z + 1072*a^3*c^5*d^4*e^4*f*h*z - 960*a^4*c^4*d^2*e^6*f*h*z + 30*b^6*c^2*d^3*e^5
*f*g*z + 30*b^3*c^5*d^6*e^2*f*g*z - 10*b^5*c^3*d^4*e^4*f*g*z - 10*b^4*c^4*d^5*e^3*f*g*z - 1488*a^3*c^5*d^3*e^5
*f*g*z + 48*a^2*c^6*d^5*e^3*f*g*z - 24*a^4*b^2*c^2*e^8*f*h*z + 186*a^3*b^3*c^2*e^8*f*g*z + 4*a^4*b^3*c*d*e^7*h
^2*z + 4*a*b^6*c*d^4*e^4*h^2*z + 4*a*b^3*c^4*d^7*e*h^2*z + 168*a^4*b*c^3*d*e^7*g^2*z + 24*a^2*b^5*c*d*e^7*g^2*
z + 18*a*b^6*c*d^2*e^6*g^2*z - 912*a^3*b*c^4*d*e^7*f^2*z - 192*a*b^5*c^2*d*e^7*f^2*z + 144*a*b*c^6*d^5*e^3*f^2
*z + 432*a^3*b^2*c^3*d^4*e^4*h^2*z - 168*a^4*b^2*c^2*d^2*e^6*h^2*z - 168*a^2*b^2*c^4*d^6*e^2*h^2*z - 108*a^2*b
^4*c^2*d^4*e^4*h^2*z - 20*a^3*b^3*c^2*d^3*e^5*h^2*z - 20*a^2*b^3*c^3*d^5*e^3*h^2*z - 426*a^2*b^2*c^4*d^4*e^4*g
^2*z + 336*a^3*b^2*c^3*d^2*e^6*g^2*z + 274*a^2*b^3*c^3*d^3*e^5*g^2*z - 120*a^2*b^4*c^2*d^2*e^6*g^2*z - 864*a^2
*b^2*c^4*d^2*e^6*f^2*z - 2*b^7*c*d^4*e^4*g*h*z - 2*b^4*c^4*d^7*e*g*h*z - 240*a^5*c^3*d*e^7*g*h*z + 16*a^2*c^6*
d^7*e*g*h*z + 4*b^7*c*d^3*e^5*f*h*z + 4*b^3*c^5*d^7*e*f*h*z - 20*b^7*c*d^2*e^6*f*g*z - 20*b^2*c^6*d^7*e*f*g*z
+ 4*a^2*b^6*d*e^7*g*h*z + 4*a*b^7*d^2*e^6*g*h*z + 528*a^4*c^4*d*e^7*f*g*z + 12*a^5*b*c^2*e^8*g*h*z - 2*a^4*b^3
*c*e^8*g*h*z + 4*a^3*b^4*c*e^8*f*h*z - 228*a^4*b*c^3*e^8*f*g*z - 48*a^2*b^5*c*e^8*f*g*z - 8*a*b*c^6*d^7*e*g^2*
z + 36*a^3*b^4*c*d^2*e^6*h^2*z + 36*a*b^4*c^3*d^6*e^2*h^2*z + 12*a^2*b^5*c*d^3*e^5*h^2*z + 12*a*b^5*c^2*d^5*e^
3*h^2*z - 312*a^3*b*c^4*d^3*e^5*g^2*z + 104*a*b^4*c^3*d^4*e^4*g^2*z - 102*a^3*b^3*c^2*d*e^7*g^2*z - 66*a*b^5*c
^2*d^3*e^5*g^2*z + 24*a^2*b*c^5*d^5*e^3*g^2*z + 24*a*b^2*c^5*d^6*e^2*g^2*z - 18*a*b^3*c^4*d^5*e^3*g^2*z + 744*
a^2*b^3*c^3*d*e^7*f^2*z + 240*a^2*b*c^5*d^3*e^5*f^2*z + 216*a*b^4*c^3*d^2*e^6*f^2*z - 120*a*b^2*c^5*d^4*e^4*f^
2*z + 24*a^5*c^3*e^8*f*h*z + 16*b^7*c*d*e^7*f^2*z + 16*b*c^7*d^7*e*f^2*z - 2*a*b^7*d*e^7*g^2*z + 48*a*b^6*c*e^
8*f^2*z - 4*b^6*c^2*d^6*e^2*h^2*z - 536*a^4*c^4*d^4*e^4*h^2*z + 240*a^5*c^3*d^2*e^6*h^2*z + 240*a^3*c^5*d^6*e^
2*h^2*z - 12*b^6*c^2*d^4*e^4*g^2*z - 12*b^4*c^4*d^6*e^2*g^2*z + 10*b^5*c^3*d^5*e^3*g^2*z + 528*a^3*c^5*d^4*e^4
*g^2*z - 432*a^4*c^4*d^2*e^6*g^2*z + 20*b^4*c^4*d^4*e^4*f^2*z - 16*b^6*c^2*d^2*e^6*f^2*z - 16*b^2*c^6*d^6*e^2*
f^2*z - 16*a^2*c^6*d^6*e^2*g^2*z - 8*b^5*c^3*d^3*e^5*f^2*z - 8*b^3*c^5*d^5*e^3*f^2*z - 4*a^2*b^6*d^2*e^6*h^2*z
+ 912*a^3*c^5*d^2*e^6*f^2*z - 120*a^2*c^6*d^4*e^4*f^2*z - 45*a^4*b^2*c^2*e^8*g^2*z + 264*a^3*b^2*c^3*e^8*f^2*
z - 192*a^2*b^4*c^2*e^8*f^2*z + 4*b^8*d*e^7*f*g*z - 8*a*c^7*d^8*f*h*z + 4*b*c^7*d^8*f*g*z + 4*a*b^7*e^8*f*g*z
+ 6*b^7*c*d^3*e^5*g^2*z + 6*b^3*c^5*d^7*e*g^2*z - 48*a*c^7*d^6*e^2*f^2*z + 12*a^3*b^4*c*e^8*g^2*z - b^8*d^2*e^
6*g^2*z - 4*a^6*c^2*e^8*h^2*z + 48*a^5*c^3*e^8*g^2*z - 4*a^2*c^6*d^8*h^2*z - b^2*c^6*d^8*g^2*z - 36*a^4*c^4*e^
8*f^2*z - a^2*b^6*e^8*g^2*z - 4*c^8*d^8*f^2*z - 4*b^8*e^8*f^2*z - 80*a*b*c^4*d^3*e^3*f*g*h + 24*a^2*b*c^3*d*e^
5*f*g*h + 16*a*b^3*c^2*d*e^5*f*g*h - 72*a*b^2*c^3*d^2*e^4*f*g*h - 48*a^2*b*c^3*d^3*e^3*g*h^2 + 16*a*b^3*c^2*d^
3*e^3*g*h^2 - 12*a*b^2*c^3*d^3*e^3*g^2*h - 6*a^2*b^2*c^2*d*e^5*g^2*h - 72*a^2*b^2*c^2*d*e^5*f*h^2 + 48*a*b^2*c
^3*d^3*e^3*f*h^2 + 24*a^2*b*c^3*d^2*e^4*f*h^2 - 8*a*b^3*c^2*d^2*e^4*f*h^2 - 8*b^5*c*d*e^5*f*g*h - 8*b*c^5*d^5*
e*f*g*h - 8*a*b^4*c*e^6*f*g*h + 24*b^3*c^3*d^3*e^3*f*g*h + 16*b^4*c^2*d^2*e^4*f*g*h + 16*b^2*c^4*d^4*e^2*f*g*h
+ 48*a^2*c^4*d^2*e^4*f*g*h + 48*a^2*b^2*c^2*e^6*f*g*h + 40*a^3*b*c^2*d*e^5*g*h^2 + 28*a*b*c^4*d^4*e^2*g^2*h -
8*a^2*b^3*c*d*e^5*g*h^2 - 8*a*b^4*c*d^2*e^4*g*h^2 + 96*a*b^2*c^3*d*e^5*f^2*h + 24*a*b*c^4*d^2*e^4*f^2*h + 16*
a*b*c^4*d^4*e^2*f*h^2 + 96*a*b*c^4*d^2*e^4*f*g^2 - 48*a*b^2*c^3*d*e^5*f*g^2 + 12*a^2*b^2*c^2*d^2*e^4*g*h^2 - 5
6*a*c^5*d^4*e^2*f*g*h - 8*a*b*c^4*d^5*e*g*h^2 + 4*a*b^4*c*d*e^5*g^2*h + 16*a*b^4*c*d*e^5*f*h^2 - 48*a*b*c^4*d*
e^5*f^2*g - 24*a^3*c^3*e^6*f*g*h + 16*a*c^5*d^5*e*f*h^2 - 6*b^4*c^2*d^3*e^3*g^2*h - 6*b^3*c^3*d^4*e^2*g^2*h +
4*b^4*c^2*d^4*e^2*g*h^2 + 80*a^2*c^4*d^3*e^3*g^2*h - 44*a^2*c^4*d^4*e^2*g*h^2 + 24*a^3*c^3*d^2*e^4*g*h^2 - 16*
b^3*c^3*d^2*e^4*f^2*h - 16*b^2*c^4*d^3*e^3*f^2*h - 8*b^4*c^2*d^3*e^3*f*h^2 - 8*b^3*c^3*d^4*e^2*f*h^2 + 60*b^2*
c^4*d^2*e^4*f^2*g - 48*a^2*c^4*d^3*e^3*f*h^2 - 24*b^3*c^3*d^2*e^4*f*g^2 - 24*b^2*c^4*d^3*e^3*f*g^2 - 24*a^3*b*
c^2*d^2*e^4*h^3 + 24*a^2*b*c^3*d^4*e^2*h^3 + 8*a^2*b^3*c*d^2*e^4*h^3 - 8*a*b^3*c^2*d^4*e^2*h^3 + 18*a*b^2*c^3*
d^2*e^4*g^3 + 2*b^5*c*d^2*e^4*g^2*h + 2*b^2*c^4*d^5*e*g^2*h - 48*a^3*c^3*d*e^5*g^2*h - 8*b^4*c^2*d*e^5*f^2*h -
8*b*c^5*d^4*e^2*f^2*h - 168*a^2*c^4*d*e^5*f^2*h + 96*a*c^5*d^3*e^3*f^2*h + 64*a^3*c^3*d*e^5*f*h^2 + 12*b^4*c^
2*d*e^5*f*g^2 + 12*b*c^5*d^4*e^2*f*g^2 - 168*a*c^5*d^2*e^4*f^2*g + 48*a^2*c^4*d*e^5*f*g^2 + 48*a*c^5*d^3*e^3*f
*g^2 - 12*a^3*b*c^2*e^6*g^2*h + 2*a^2*b^3*c*e^6*g^2*h + 48*a^2*b*c^3*e^6*f^2*h - 48*a*b^3*c^2*e^6*f^2*h - 8*a^
3*b*c^2*e^6*f*h^2 - 60*a^2*b*c^3*e^6*f*g^2 + 48*a*b^2*c^3*e^6*f^2*g + 12*a*b^3*c^2*e^6*f*g^2 + 24*a^2*b*c^3*d*
e^5*g^3 - 24*a*b*c^4*d^3*e^3*g^3 - 6*a*b^3*c^2*d*e^5*g^3 - 12*c^6*d^4*e^2*f^2*g + 4*a^4*c^2*e^6*g*h^2 - 12*b^4
*c^2*e^6*f^2*g + 36*a^2*c^4*e^6*f^2*g - 8*a^4*c^2*d*e^5*h^3 + 8*a^2*c^4*d^5*e*h^3 - 24*b^2*c^4*d*e^5*f^3 - 24*
b*c^5*d^2*e^4*f^3 + 8*c^6*d^5*e*f^2*h + 8*b^5*c*e^6*f^2*h + 144*a*c^5*d*e^5*f^3 - 72*a*b*c^4*e^6*f^3 + 10*b^3*
c^3*d^3*e^3*g^3 - 3*b^4*c^2*d^2*e^4*g^3 - 3*b^2*c^4*d^4*e^2*g^3 - 48*a^2*c^4*d^2*e^4*g^3 - 3*a^2*b^2*c^2*e^6*g
^3 + 16*c^6*d^3*e^3*f^3 + 16*b^3*c^3*e^6*f^3 + 16*a^3*c^3*e^6*g^3, z, k)*((8*a^6*c^3*e^9*h - 24*a^5*c^4*e^9*f
- 8*a*c^8*d^8*e*f + 2*a^2*b^6*c*e^9*f - a^3*b^5*c*e^9*g - 20*a^5*b*c^3*e^9*g + 16*a^5*c^4*d*e^8*g + 2*b^2*c^7*
d^8*e*f + 2*b^8*c*d^2*e^7*f - 8*a^2*c^7*d^8*e*h - b^3*c^6*d^8*e*g - b^8*c*d^3*e^6*g - 18*a^3*b^4*c^2*e^9*f + 4
6*a^4*b^2*c^3*e^9*f + 9*a^4*b^3*c^2*e^9*g - 48*a^2*c^7*d^6*e^3*f - 96*a^3*c^6*d^4*e^5*f - 80*a^4*c^5*d^2*e^7*f
- 2*a^5*b^2*c^2*e^9*h + 16*a^2*c^7*d^7*e^2*g + 48*a^3*c^6*d^5*e^4*g + 48*a^4*c^5*d^3*e^6*g - 6*b^3*c^6*d^7*e^
2*f + 4*b^4*c^5*d^6*e^3*f + 4*b^6*c^3*d^4*e^5*f - 6*b^7*c^2*d^3*e^6*f - 16*a^3*c^6*d^6*e^3*h + 16*a^5*c^4*d^2*
e^7*h + 4*b^4*c^5*d^7*e^2*g - 3*b^5*c^4*d^6*e^3*g - 3*b^6*c^3*d^5*e^4*g + 4*b^7*c^2*d^4*e^5*g - 2*b^5*c^4*d^7*
e^2*h + 4*b^6*c^3*d^6*e^3*h - 2*b^7*c^2*d^5*e^4*h - 4*a*b^2*c^6*d^6*e^3*f - 14*a*b^3*c^5*d^5*e^4*f - 38*a*b^4*
c^4*d^4*e^5*f + 54*a*b^5*c^3*d^3*e^6*f - 10*a*b^6*c^2*d^2*e^7*f + 56*a^2*b*c^6*d^5*e^4*f + 34*a^2*b^5*c^2*d*e^
8*f + 40*a^3*b*c^5*d^3*e^6*f - 74*a^3*b^3*c^3*d*e^8*f - 20*a*b^2*c^6*d^7*e^2*g + 10*a*b^3*c^5*d^6*e^3*g + 34*a
*b^4*c^4*d^5*e^4*g - 33*a*b^5*c^3*d^4*e^5*g + 4*a*b^6*c^2*d^3*e^6*g + 8*a^2*b*c^6*d^6*e^3*g - 16*a^3*b*c^5*d^4
*e^5*g - 10*a^3*b^4*c^2*d*e^8*g - 40*a^4*b*c^4*d^2*e^7*g + 20*a^4*b^2*c^3*d*e^8*g + 10*a*b^3*c^5*d^7*e^2*h - 2
6*a*b^4*c^4*d^6*e^3*h + 12*a*b^5*c^3*d^5*e^4*h - 8*a^2*b*c^6*d^7*e^2*h - 4*a^2*b^6*c*d^2*e^7*h - 8*a^3*b*c^5*d
^5*e^4*h + 8*a^4*b*c^4*d^3*e^6*h - 10*a^4*b^3*c^2*d*e^8*h - 4*a*b^7*c*d*e^8*f + 4*a*b*c^7*d^8*e*g + 112*a^2*b^
2*c^5*d^4*e^5*f - 130*a^2*b^3*c^4*d^3*e^6*f - 28*a^2*b^4*c^3*d^2*e^7*f + 164*a^3*b^2*c^4*d^2*e^7*f - 100*a^2*b
^2*c^5*d^5*e^4*g + 72*a^2*b^3*c^4*d^4*e^5*g + 12*a^2*b^4*c^3*d^3*e^6*g - 7*a^2*b^5*c^2*d^2*e^7*g - 60*a^3*b^2*
c^4*d^3*e^6*g + 22*a^3*b^3*c^3*d^2*e^7*g + 44*a^2*b^2*c^5*d^6*e^3*h - 14*a^2*b^3*c^4*d^5*e^4*h - 12*a^2*b^5*c^
2*d^3*e^6*h + 14*a^3*b^3*c^3*d^3*e^6*h + 26*a^3*b^4*c^2*d^2*e^7*h - 44*a^4*b^2*c^3*d^2*e^7*h + 24*a*b*c^7*d^7*
e^2*f + 8*a^4*b*c^4*d*e^8*f + a*b^7*c*d^2*e^7*g + a^2*b^6*c*d*e^8*g + 2*a*b^2*c^6*d^8*e*h + 2*a*b^7*c*d^3*e^6*
h + 2*a^3*b^5*c*d*e^8*h + 8*a^5*b*c^3*d*e^8*h)/(16*a^2*c^6*d^8 + a^4*b^4*e^8 + 16*a^6*c^2*e^8 + b^4*c^4*d^8 +
b^8*d^4*e^4 - 8*a*b^2*c^5*d^8 - 8*a^5*b^2*c*e^8 - 4*a*b^7*d^3*e^5 - 4*a^3*b^5*d*e^7 - 4*b^5*c^3*d^7*e - 4*b^7*
c*d^5*e^3 + 6*a^2*b^6*d^2*e^6 + 64*a^3*c^5*d^6*e^2 + 96*a^4*c^4*d^4*e^4 + 64*a^5*c^3*d^2*e^6 + 6*b^6*c^2*d^6*e
^2 + 64*a^2*b^2*c^4*d^6*e^2 + 32*a^2*b^3*c^3*d^5*e^3 - 74*a^2*b^4*c^2*d^4*e^4 + 144*a^3*b^2*c^3*d^4*e^4 + 32*a
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92*a^3*b*c^4*d^5*e^3 - 44*a^3*b^4*c*d^2*e^6 - 192*a^4*b*c^3*d^3*e^5) + root(3840*a^6*b*c^5*d^5*e^7*z^3 + 3840*
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288*a^2*b^3*c^7*d^11*e*z^3 + 210*a^4*b^7*c*d^3*e^9*z^3 + 210*a*b^7*c^4*d^9*e^3*z^3 - 174*a^5*b^6*c*d^2*e^10*z
^3 - 174*a*b^6*c^5*d^10*e^2*z^3 - 120*a^3*b^8*c*d^4*e^8*z^3 - 120*a*b^8*c^3*d^8*e^4*z^3 + 12*a^2*b^9*c*d^5*e^7
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3 + 72*a*b^5*c^6*d^11*e*z^3 + 18*a*b^10*c*d^6*e^6*z^3 - 4800*a^5*b^2*c^5*d^6*e^6*z^3 - 3120*a^6*b^2*c^4*d^4*e^
8*z^3 - 3120*a^4*b^2*c^6*d^8*e^4*z^3 + 2160*a^4*b^4*c^4*d^6*e^6*z^3 - 1776*a^4*b^5*c^3*d^5*e^7*z^3 - 1776*a^3*
b^5*c^4*d^7*e^5*z^3 + 1740*a^5*b^4*c^3*d^4*e^8*z^3 + 1740*a^3*b^4*c^5*d^8*e^4*z^3 + 960*a^5*b^3*c^4*d^5*e^7*z^
3 + 960*a^4*b^3*c^5*d^7*e^5*z^3 - 672*a^7*b^2*c^3*d^2*e^10*z^3 - 672*a^3*b^2*c^7*d^10*e^2*z^3 + 648*a^6*b^4*c^
2*d^2*e^10*z^3 + 648*a^2*b^4*c^6*d^10*e^2*z^3 - 600*a^5*b^5*c^2*d^3*e^9*z^3 - 600*a^2*b^5*c^5*d^9*e^3*z^3 + 37
2*a^3*b^7*c^2*d^5*e^7*z^3 + 372*a^2*b^7*c^3*d^7*e^5*z^3 + 316*a^3*b^6*c^3*d^6*e^6*z^3 - 222*a^2*b^8*c^2*d^6*e^
6*z^3 - 160*a^6*b^3*c^3*d^3*e^9*z^3 - 160*a^3*b^3*c^6*d^9*e^3*z^3 + 15*a^4*b^6*c^2*d^4*e^8*z^3 + 15*a^2*b^6*c^
4*d^8*e^4*z^3 - 6*b^11*c*d^7*e^5*z^3 - 6*b^7*c^5*d^11*e*z^3 - 6*a^5*b^7*d*e^11*z^3 - 6*a*b^11*d^5*e^7*z^3 - 12
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10*e^2*z^3 - 1280*a^6*c^6*d^6*e^6*z^3 - 960*a^7*c^5*d^4*e^8*z^3 - 960*a^5*c^7*d^8*e^4*z^3 - 384*a^8*c^4*d^2*e^
10*z^3 - 384*a^4*c^8*d^10*e^2*z^3 - 20*a^3*b^9*d^3*e^9*z^3 + 15*a^4*b^8*d^2*e^10*z^3 + 15*a^2*b^10*d^4*e^8*z^3
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6*z^3 + b^6*c^6*d^12*z^3 + a^6*b^6*e^12*z^3 - 44*a^3*b^4*c*d*e^7*g*h*z - 20*a*b^6*c*d^3*e^5*g*h*z - 12*a*b^2*c
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186*a^2*b^4*c^2*d^3*e^5*g*h*z - 166*a^2*b^3*c^3*d^4*e^4*g*h*z + 792*a^3*b^2*c^3*d^2*e^6*f*h*z - 744*a^2*b^2*c
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6*f*g*z - 732*a^3*b^2*c^3*d*e^7*f*g*z + 306*a^2*b^4*c^2*d*e^7*f*g*z - 304*a*b^4*c^3*d^3*e^5*f*g*z + 222*a*b^5*
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h*z - 960*a^4*c^4*d^2*e^6*f*h*z + 30*b^6*c^2*d^3*e^5*f*g*z + 30*b^3*c^5*d^6*e^2*f*g*z - 10*b^5*c^3*d^4*e^4*f*g
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192*a*b^5*c^2*d*e^7*f^2*z + 144*a*b*c^6*d^5*e^3*f^2*z + 432*a^3*b^2*c^3*d^4*e^4*h^2*z - 168*a^4*b^2*c^2*d^2*e
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z - 20*b^7*c*d^2*e^6*f*g*z - 20*b^2*c^6*d^7*e*f*g*z + 4*a^2*b^6*d*e^7*g*h*z + 4*a*b^7*d^2*e^6*g*h*z + 528*a^4*
c^4*d*e^7*f*g*z + 12*a^5*b*c^2*e^8*g*h*z - 2*a^4*b^3*c*e^8*g*h*z + 4*a^3*b^4*c*e^8*f*h*z - 228*a^4*b*c^3*e^8*f
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z + 12*a^2*b^5*c*d^3*e^5*h^2*z + 12*a*b^5*c^2*d^5*e^3*h^2*z - 312*a^3*b*c^4*d^3*e^5*g^2*z + 104*a*b^4*c^3*d^4*
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6*a*b^4*c^3*d^2*e^6*f^2*z - 120*a*b^2*c^5*d^4*e^4*f^2*z + 24*a^5*c^3*e^8*f*h*z + 16*b^7*c*d*e^7*f^2*z + 16*b*c
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2*z + 240*a^5*c^3*d^2*e^6*h^2*z + 240*a^3*c^5*d^6*e^2*h^2*z - 12*b^6*c^2*d^4*e^4*g^2*z - 12*b^4*c^4*d^6*e^2*g^
2*z + 10*b^5*c^3*d^5*e^3*g^2*z + 528*a^3*c^5*d^4*e^4*g^2*z - 432*a^4*c^4*d^2*e^6*g^2*z + 20*b^4*c^4*d^4*e^4*f^
2*z - 16*b^6*c^2*d^2*e^6*f^2*z - 16*b^2*c^6*d^6*e^2*f^2*z - 16*a^2*c^6*d^6*e^2*g^2*z - 8*b^5*c^3*d^3*e^5*f^2*z
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45*a^4*b^2*c^2*e^8*g^2*z + 264*a^3*b^2*c^3*e^8*f^2*z - 192*a^2*b^4*c^2*e^8*f^2*z + 4*b^8*d*e^7*f*g*z - 8*a*c^
7*d^8*f*h*z + 4*b*c^7*d^8*f*g*z + 4*a*b^7*e^8*f*g*z + 6*b^7*c*d^3*e^5*g^2*z + 6*b^3*c^5*d^7*e*g^2*z - 48*a*c^7
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f^2*z - 80*a*b*c^4*d^3*e^3*f*g*h + 24*a^2*b*c^3*d*e^5*f*g*h + 16*a*b^3*c^2*d*e^5*f*g*h - 72*a*b^2*c^3*d^2*e^4*
f*g*h - 48*a^2*b*c^3*d^3*e^3*g*h^2 + 16*a*b^3*c^2*d^3*e^3*g*h^2 - 12*a*b^2*c^3*d^3*e^3*g^2*h - 6*a^2*b^2*c^2*d
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2*d^2*e^4*f*h^2 - 8*b^5*c*d*e^5*f*g*h - 8*b*c^5*d^5*e*f*g*h - 8*a*b^4*c*e^6*f*g*h + 24*b^3*c^3*d^3*e^3*f*g*h +
16*b^4*c^2*d^2*e^4*f*g*h + 16*b^2*c^4*d^4*e^2*f*g*h + 48*a^2*c^4*d^2*e^4*f*g*h + 48*a^2*b^2*c^2*e^6*f*g*h + 4
0*a^3*b*c^2*d*e^5*g*h^2 + 28*a*b*c^4*d^4*e^2*g^2*h - 8*a^2*b^3*c*d*e^5*g*h^2 - 8*a*b^4*c*d^2*e^4*g*h^2 + 96*a*
b^2*c^3*d*e^5*f^2*h + 24*a*b*c^4*d^2*e^4*f^2*h + 16*a*b*c^4*d^4*e^2*f*h^2 + 96*a*b*c^4*d^2*e^4*f*g^2 - 48*a*b^
2*c^3*d*e^5*f*g^2 + 12*a^2*b^2*c^2*d^2*e^4*g*h^2 - 56*a*c^5*d^4*e^2*f*g*h - 8*a*b*c^4*d^5*e*g*h^2 + 4*a*b^4*c*
d*e^5*g^2*h + 16*a*b^4*c*d*e^5*f*h^2 - 48*a*b*c^4*d*e^5*f^2*g - 24*a^3*c^3*e^6*f*g*h + 16*a*c^5*d^5*e*f*h^2 -
6*b^4*c^2*d^3*e^3*g^2*h - 6*b^3*c^3*d^4*e^2*g^2*h + 4*b^4*c^2*d^4*e^2*g*h^2 + 80*a^2*c^4*d^3*e^3*g^2*h - 44*a^
2*c^4*d^4*e^2*g*h^2 + 24*a^3*c^3*d^2*e^4*g*h^2 - 16*b^3*c^3*d^2*e^4*f^2*h - 16*b^2*c^4*d^3*e^3*f^2*h - 8*b^4*c
^2*d^3*e^3*f*h^2 - 8*b^3*c^3*d^4*e^2*f*h^2 + 60*b^2*c^4*d^2*e^4*f^2*g - 48*a^2*c^4*d^3*e^3*f*h^2 - 24*b^3*c^3*
d^2*e^4*f*g^2 - 24*b^2*c^4*d^3*e^3*f*g^2 - 24*a^3*b*c^2*d^2*e^4*h^3 + 24*a^2*b*c^3*d^4*e^2*h^3 + 8*a^2*b^3*c*d
^2*e^4*h^3 - 8*a*b^3*c^2*d^4*e^2*h^3 + 18*a*b^2*c^3*d^2*e^4*g^3 + 2*b^5*c*d^2*e^4*g^2*h + 2*b^2*c^4*d^5*e*g^2*
h - 48*a^3*c^3*d*e^5*g^2*h - 8*b^4*c^2*d*e^5*f^2*h - 8*b*c^5*d^4*e^2*f^2*h - 168*a^2*c^4*d*e^5*f^2*h + 96*a*c^
5*d^3*e^3*f^2*h + 64*a^3*c^3*d*e^5*f*h^2 + 12*b^4*c^2*d*e^5*f*g^2 + 12*b*c^5*d^4*e^2*f*g^2 - 168*a*c^5*d^2*e^4
*f^2*g + 48*a^2*c^4*d*e^5*f*g^2 + 48*a*c^5*d^3*e^3*f*g^2 - 12*a^3*b*c^2*e^6*g^2*h + 2*a^2*b^3*c*e^6*g^2*h + 48
*a^2*b*c^3*e^6*f^2*h - 48*a*b^3*c^2*e^6*f^2*h - 8*a^3*b*c^2*e^6*f*h^2 - 60*a^2*b*c^3*e^6*f*g^2 + 48*a*b^2*c^3*
e^6*f^2*g + 12*a*b^3*c^2*e^6*f*g^2 + 24*a^2*b*c^3*d*e^5*g^3 - 24*a*b*c^4*d^3*e^3*g^3 - 6*a*b^3*c^2*d*e^5*g^3 -
12*c^6*d^4*e^2*f^2*g + 4*a^4*c^2*e^6*g*h^2 - 12*b^4*c^2*e^6*f^2*g + 36*a^2*c^4*e^6*f^2*g - 8*a^4*c^2*d*e^5*h^
3 + 8*a^2*c^4*d^5*e*h^3 - 24*b^2*c^4*d*e^5*f^3 - 24*b*c^5*d^2*e^4*f^3 + 8*c^6*d^5*e*f^2*h + 8*b^5*c*e^6*f^2*h
+ 144*a*c^5*d*e^5*f^3 - 72*a*b*c^4*e^6*f^3 + 10*b^3*c^3*d^3*e^3*g^3 - 3*b^4*c^2*d^2*e^4*g^3 - 3*b^2*c^4*d^4*e^
2*g^3 - 48*a^2*c^4*d^2*e^4*g^3 - 3*a^2*b^2*c^2*e^6*g^3 + 16*c^6*d^3*e^3*f^3 + 16*b^3*c^3*e^6*f^3 + 16*a^3*c^3*
e^6*g^3, z, k)*((a^5*b^5*c*e^11 + 16*a^7*b*c^3*e^11 - 128*a^7*c^4*d*e^10 + b^5*c^6*d^10*e + b^10*c*d^5*e^6 - 8
*a^6*b^3*c^2*e^11 - 128*a^3*c^8*d^9*e^2 - 512*a^4*c^7*d^7*e^4 - 768*a^5*c^6*d^5*e^6 - 512*a^6*c^5*d^3*e^8 - 3*
b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*e^5 + 16*a^2*b^2*c^7*d^9*e^2 - 264*a^2
*b^3*c^6*d^8*e^3 + 480*a^2*b^4*c^5*d^7*e^4 - 246*a^2*b^5*c^4*d^6*e^5 - 66*a^2*b^6*c^3*d^5*e^6 + 62*a^2*b^7*c^2
*d^4*e^7 - 704*a^3*b^2*c^6*d^7*e^4 - 240*a^3*b^3*c^5*d^6*e^5 + 800*a^3*b^4*c^4*d^5*e^6 - 246*a^3*b^5*c^3*d^4*e
^7 - 76*a^3*b^6*c^2*d^3*e^8 - 1440*a^4*b^2*c^5*d^5*e^6 - 240*a^4*b^3*c^4*d^4*e^7 + 480*a^4*b^4*c^3*d^3*e^8 + 2
1*a^4*b^5*c^2*d^2*e^9 - 704*a^5*b^2*c^4*d^3*e^8 - 264*a^5*b^3*c^3*d^2*e^9 - 8*a*b^3*c^7*d^10*e - 3*a*b^9*c*d^4
*e^7 + 16*a^2*b*c^8*d^10*e - 3*a^4*b^6*c*d*e^10 + 16*a*b^4*c^6*d^9*e^2 + 21*a*b^5*c^5*d^8*e^3 - 76*a*b^6*c^4*d
^7*e^4 + 62*a*b^7*c^3*d^6*e^5 - 12*a*b^8*c^2*d^5*e^6 + 2*a^2*b^8*c*d^3*e^8 + 592*a^3*b*c^7*d^8*e^3 + 2*a^3*b^7
*c*d^2*e^9 + 1696*a^4*b*c^6*d^6*e^5 + 1696*a^5*b*c^5*d^4*e^7 + 16*a^5*b^4*c^2*d*e^10 + 592*a^6*b*c^4*d^2*e^9 +
16*a^6*b^2*c^3*d*e^10)/(16*a^2*c^6*d^8 + a^4*b^4*e^8 + 16*a^6*c^2*e^8 + b^4*c^4*d^8 + b^8*d^4*e^4 - 8*a*b^2*c
^5*d^8 - 8*a^5*b^2*c*e^8 - 4*a*b^7*d^3*e^5 - 4*a^3*b^5*d*e^7 - 4*b^5*c^3*d^7*e - 4*b^7*c*d^5*e^3 + 6*a^2*b^6*d
^2*e^6 + 64*a^3*c^5*d^6*e^2 + 96*a^4*c^4*d^4*e^4 + 64*a^5*c^3*d^2*e^6 + 6*b^6*c^2*d^6*e^2 + 64*a^2*b^2*c^4*d^6
*e^2 + 32*a^2*b^3*c^3*d^5*e^3 - 74*a^2*b^4*c^2*d^4*e^4 + 144*a^3*b^2*c^3*d^4*e^4 + 32*a^3*b^3*c^2*d^3*e^5 + 64
*a^4*b^2*c^2*d^2*e^6 + 32*a*b^3*c^4*d^7*e + 4*a*b^6*c*d^4*e^4 - 64*a^2*b*c^5*d^7*e + 32*a^4*b^3*c*d*e^7 - 64*a
^5*b*c^2*d*e^7 - 44*a*b^4*c^3*d^6*e^2 + 20*a*b^5*c^2*d^5*e^3 + 20*a^2*b^5*c*d^3*e^5 - 192*a^3*b*c^4*d^5*e^3 -
44*a^3*b^4*c*d^2*e^6 - 192*a^4*b*c^3*d^3*e^5) + (x*(2*a^4*b^6*c*e^11 - 96*a^7*c^4*e^11 + 32*a^2*c^9*d^10*e + 2
*b^4*c^7*d^10*e + 2*b^10*c*d^4*e^7 - 22*a^5*b^4*c^2*e^11 + 80*a^6*b^2*c^3*e^11 + 32*a^3*c^8*d^8*e^3 - 192*a^4*
c^7*d^6*e^5 - 448*a^5*c^6*d^4*e^7 - 352*a^6*c^5*d^2*e^9 - 10*b^5*c^6*d^9*e^2 + 22*b^6*c^5*d^8*e^3 - 28*b^7*c^4
*d^7*e^4 + 22*b^8*c^3*d^6*e^5 - 10*b^9*c^2*d^5*e^6 + 336*a^2*b^2*c^7*d^8*e^3 - 384*a^2*b^3*c^6*d^7*e^4 + 180*a
^2*b^4*c^5*d^6*e^5 + 132*a^2*b^5*c^4*d^5*e^6 - 200*a^2*b^6*c^3*d^4*e^7 + 52*a^2*b^7*c^2*d^3*e^8 + 416*a^3*b^2*
c^6*d^6*e^5 - 800*a^3*b^3*c^5*d^5*e^6 + 580*a^3*b^4*c^4*d^4*e^7 + 24*a^3*b^5*c^3*d^3*e^8 - 116*a^3*b^6*c^2*d^2
*e^9 - 160*a^4*b^2*c^5*d^4*e^7 - 640*a^4*b^3*c^4*d^3*e^8 + 330*a^4*b^4*c^3*d^2*e^9 - 144*a^5*b^2*c^4*d^2*e^9 -
16*a*b^2*c^8*d^10*e - 8*a*b^9*c*d^3*e^8 - 8*a^3*b^7*c*d*e^10 + 352*a^6*b*c^4*d*e^10 + 80*a*b^3*c^7*d^9*e^2 -
174*a*b^4*c^6*d^8*e^3 + 216*a*b^5*c^5*d^7*e^4 - 156*a*b^6*c^4*d^6*e^5 + 48*a*b^7*c^3*d^5*e^6 + 10*a*b^8*c^2*d^
4*e^7 - 160*a^2*b*c^8*d^9*e^2 + 12*a^2*b^8*c*d^2*e^9 - 128*a^3*b*c^7*d^7*e^4 + 576*a^4*b*c^6*d^5*e^6 + 86*a^4*
b^5*c^2*d*e^10 + 896*a^5*b*c^5*d^3*e^8 - 304*a^5*b^3*c^3*d*e^10))/(16*a^2*c^6*d^8 + a^4*b^4*e^8 + 16*a^6*c^2*e
^8 + b^4*c^4*d^8 + b^8*d^4*e^4 - 8*a*b^2*c^5*d^8 - 8*a^5*b^2*c*e^8 - 4*a*b^7*d^3*e^5 - 4*a^3*b^5*d*e^7 - 4*b^5
*c^3*d^7*e - 4*b^7*c*d^5*e^3 + 6*a^2*b^6*d^2*e^6 + 64*a^3*c^5*d^6*e^2 + 96*a^4*c^4*d^4*e^4 + 64*a^5*c^3*d^2*e^
6 + 6*b^6*c^2*d^6*e^2 + 64*a^2*b^2*c^4*d^6*e^2 + 32*a^2*b^3*c^3*d^5*e^3 - 74*a^2*b^4*c^2*d^4*e^4 + 144*a^3*b^2
*c^3*d^4*e^4 + 32*a^3*b^3*c^2*d^3*e^5 + 64*a^4*b^2*c^2*d^2*e^6 + 32*a*b^3*c^4*d^7*e + 4*a*b^6*c*d^4*e^4 - 64*a
^2*b*c^5*d^7*e + 32*a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 - 44*a*b^4*c^3*d^6*e^2 + 20*a*b^5*c^2*d^5*e^3 + 20*a^
2*b^5*c*d^3*e^5 - 192*a^3*b*c^4*d^5*e^3 - 44*a^3*b^4*c*d^2*e^6 - 192*a^4*b*c^3*d^3*e^5)) - (x*(48*a^5*c^4*e^9*
g - 72*a^4*b*c^4*e^9*f + 16*a*c^8*d^7*e^2*f + 144*a^4*c^5*d*e^8*f - 8*a^5*b*c^3*e^9*h - 80*a^5*c^4*d*e^8*h - 4
*a^2*b^5*c^2*e^9*f + 34*a^3*b^3*c^3*e^9*f + 2*a^3*b^4*c^2*e^9*g - 20*a^4*b^2*c^3*e^9*g + 176*a^2*c^7*d^5*e^4*f
+ 304*a^3*c^6*d^3*e^6*f + 2*a^4*b^3*c^2*e^9*h - 80*a^2*c^7*d^6*e^3*g - 112*a^3*c^6*d^4*e^5*g + 16*a^4*c^5*d^2
*e^7*g - 4*b^2*c^7*d^7*e^2*f + 14*b^3*c^6*d^6*e^3*f - 10*b^4*c^5*d^5*e^4*f - 10*b^5*c^4*d^4*e^5*f + 14*b^6*c^3
*d^3*e^6*f - 4*b^7*c^2*d^2*e^7*f + 48*a^2*c^7*d^7*e^2*h + 16*a^3*c^6*d^5*e^4*h - 112*a^4*c^5*d^3*e^6*h + 2*b^3
*c^6*d^7*e^2*g - 12*b^4*c^5*d^6*e^3*g + 20*b^5*c^4*d^5*e^4*g - 12*b^6*c^3*d^4*e^5*g + 2*b^7*c^2*d^3*e^6*g + 2*
b^4*c^5*d^7*e^2*h - 2*b^5*c^4*d^6*e^3*h - 2*b^6*c^3*d^5*e^4*h + 2*b^7*c^2*d^4*e^5*h - 4*a*b^2*c^6*d^5*e^4*f +
150*a*b^3*c^5*d^4*e^5*f - 128*a*b^4*c^4*d^3*e^6*f + 14*a*b^5*c^3*d^2*e^7*f - 440*a^2*b*c^6*d^4*e^5*f - 62*a^2*
b^4*c^3*d*e^8*f - 456*a^3*b*c^5*d^2*e^7*f + 84*a^3*b^2*c^4*d*e^8*f + 68*a*b^2*c^6*d^6*e^3*g - 118*a*b^3*c^5*d^
5*e^4*g + 54*a*b^4*c^4*d^4*e^5*g + 6*a*b^5*c^3*d^3*e^6*g - 2*a*b^6*c^2*d^2*e^7*g + 152*a^2*b*c^6*d^5*e^4*g - 2
*a^2*b^5*c^2*d*e^8*g + 72*a^3*b*c^5*d^3*e^6*g + 30*a^3*b^3*c^3*d*e^8*g - 20*a*b^2*c^6*d^7*e^2*h + 30*a*b^3*c^5
*d^6*e^3*h - 4*a*b^4*c^4*d^5*e^4*h + 6*a*b^5*c^3*d^4*e^5*h - 12*a*b^6*c^2*d^3*e^6*h - 88*a^2*b*c^6*d^6*e^3*h +
72*a^3*b*c^5*d^4*e^5*h - 12*a^3*b^4*c^2*d*e^8*h + 152*a^4*b*c^4*d^2*e^7*h + 68*a^4*b^2*c^3*d*e^8*h + 212*a^2*
b^2*c^5*d^3*e^6*f + 122*a^2*b^3*c^4*d^2*e^7*f + 4*a^2*b^2*c^5*d^4*e^5*g - 74*a^2*b^3*c^4*d^3*e^6*g - 4*a^2*b^4
*c^3*d^2*e^7*g + 44*a^3*b^2*c^4*d^2*e^7*g + 44*a^2*b^2*c^5*d^5*e^4*h - 74*a^2*b^3*c^4*d^4*e^5*h + 54*a^2*b^4*c
^3*d^3*e^6*h + 20*a^2*b^5*c^2*d^2*e^7*h + 4*a^3*b^2*c^4*d^3*e^6*h - 118*a^3*b^3*c^3*d^2*e^7*h - 56*a*b*c^7*d^6
*e^3*f + 8*a*b^6*c^2*d*e^8*f - 8*a*b*c^7*d^7*e^2*g - 88*a^4*b*c^4*d*e^8*g))/(16*a^2*c^6*d^8 + a^4*b^4*e^8 + 16
*a^6*c^2*e^8 + b^4*c^4*d^8 + b^8*d^4*e^4 - 8*a*b^2*c^5*d^8 - 8*a^5*b^2*c*e^8 - 4*a*b^7*d^3*e^5 - 4*a^3*b^5*d*e
^7 - 4*b^5*c^3*d^7*e - 4*b^7*c*d^5*e^3 + 6*a^2*b^6*d^2*e^6 + 64*a^3*c^5*d^6*e^2 + 96*a^4*c^4*d^4*e^4 + 64*a^5*
c^3*d^2*e^6 + 6*b^6*c^2*d^6*e^2 + 64*a^2*b^2*c^4*d^6*e^2 + 32*a^2*b^3*c^3*d^5*e^3 - 74*a^2*b^4*c^2*d^4*e^4 + 1
44*a^3*b^2*c^3*d^4*e^4 + 32*a^3*b^3*c^2*d^3*e^5 + 64*a^4*b^2*c^2*d^2*e^6 + 32*a*b^3*c^4*d^7*e + 4*a*b^6*c*d^4*
e^4 - 64*a^2*b*c^5*d^7*e + 32*a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 - 44*a*b^4*c^3*d^6*e^2 + 20*a*b^5*c^2*d^5*e
^3 + 20*a^2*b^5*c*d^3*e^5 - 192*a^3*b*c^4*d^5*e^3 - 44*a^3*b^4*c*d^2*e^6 - 192*a^4*b*c^3*d^3*e^5)) - (32*a^2*c
^5*d^3*e^4*g^2 - 4*c^7*d^5*e^2*f^2 - a^2*b^3*c^2*e^7*g^2 - 4*b^5*c^2*e^7*f^2 - 4*b^2*c^5*d^3*e^4*f^2 - 4*b^3*c
^4*d^2*e^5*f^2 + 12*a^2*c^5*d^5*e^2*h^2 - 40*a^3*c^4*d^3*e^4*h^2 - b^2*c^5*d^5*e^2*g^2 + b^3*c^4*d^4*e^3*g^2 +
b^4*c^3*d^3*e^4*g^2 - b^5*c^2*d^2*e^5*g^2 + 24*a^3*c^4*e^7*f*g - 8*a^4*c^3*e^7*g*h + 28*a*b^3*c^3*e^7*f^2 - 4
8*a^2*b*c^4*e^7*f^2 + 4*a^3*b*c^3*e^7*g^2 - 8*a*c^6*d^3*e^4*f^2 + 60*a^2*c^5*d*e^6*f^2 + 8*b*c^6*d^4*e^3*f^2 -
32*a^3*c^4*d*e^6*g^2 + 8*b^4*c^3*d*e^6*f^2 + 12*a^4*c^3*d*e^6*h^2 + 24*a*b*c^5*d^2*e^5*f^2 - 48*a*b^2*c^4*d*e
^6*f^2 + 4*a*b*c^5*d^4*e^3*g^2 - 2*a*b^4*c^2*d*e^6*g^2 - 22*a^2*b^2*c^3*e^7*f*g - 4*a^2*b^3*c^2*e^7*f*h - 112*
a^2*c^5*d^2*e^5*f*g + 2*a^3*b^2*c^2*e^7*g*h + 80*a^2*c^5*d^3*e^4*f*h - 6*b^2*c^5*d^4*e^3*f*g + 4*b^3*c^4*d^3*e
^4*f*g - 6*b^4*c^3*d^2*e^5*f*g - 40*a^2*c^5*d^4*e^3*g*h + 80*a^3*c^4*d^2*e^5*g*h - 4*b^2*c^5*d^5*e^2*f*h + 4*b
^3*c^4*d^4*e^3*f*h + 4*b^4*c^3*d^3*e^4*f*h - 4*b^5*c^2*d^2*e^5*f*h + 2*b^3*c^4*d^5*e^2*g*h - 4*b^4*c^3*d^4*e^3
*g*h + 2*b^5*c^2*d^3*e^4*g*h - 18*a*b^2*c^4*d^3*e^4*g^2 + 12*a*b^3*c^3*d^2*e^5*g^2 - 24*a^2*b*c^4*d^2*e^5*g^2
+ 15*a^2*b^2*c^3*d*e^6*g^2 - 4*a*b^2*c^4*d^5*e^2*h^2 + 4*a*b^3*c^3*d^4*e^3*h^2 - 4*a*b^4*c^2*d^3*e^4*h^2 - 8*a
^2*b*c^4*d^4*e^3*h^2 - 8*a^3*b*c^3*d^2*e^5*h^2 - 4*a^3*b^2*c^2*d*e^6*h^2 + 4*a*b^4*c^2*e^7*f*g + 16*a^3*b*c^3*
e^7*f*h - 8*a*c^6*d^4*e^3*f*g + 8*a*c^6*d^5*e^2*f*h + 4*b*c^6*d^5*e^2*f*g - 56*a^3*c^4*d*e^6*f*h + 4*b^5*c^2*d
*e^6*f*g + 20*a^2*b^2*c^3*d^3*e^4*h^2 + 4*a^2*b^3*c^2*d^2*e^5*h^2 + 8*a*b*c^5*d^3*e^4*f*g - 40*a*b^3*c^3*d*e^6
*f*g + 100*a^2*b*c^4*d*e^6*f*g - 4*a*b*c^5*d^5*e^2*g*h - 4*a^3*b*c^3*d*e^6*g*h + 44*a*b^2*c^4*d^2*e^5*f*g - 48
*a*b^2*c^4*d^3*e^4*f*h + 32*a*b^3*c^3*d^2*e^5*f*h - 48*a^2*b*c^4*d^2*e^5*f*h + 12*a^2*b^2*c^3*d*e^6*f*h + 18*a
*b^2*c^4*d^4*e^3*g*h - 8*a*b^3*c^3*d^3*e^4*g*h + 2*a*b^4*c^2*d^2*e^5*g*h + 24*a^2*b*c^4*d^3*e^4*g*h + 2*a^2*b^
3*c^2*d*e^6*g*h - 36*a^2*b^2*c^3*d^2*e^5*g*h)/(16*a^2*c^6*d^8 + a^4*b^4*e^8 + 16*a^6*c^2*e^8 + b^4*c^4*d^8 + b
^8*d^4*e^4 - 8*a*b^2*c^5*d^8 - 8*a^5*b^2*c*e^8 - 4*a*b^7*d^3*e^5 - 4*a^3*b^5*d*e^7 - 4*b^5*c^3*d^7*e - 4*b^7*c
*d^5*e^3 + 6*a^2*b^6*d^2*e^6 + 64*a^3*c^5*d^6*e^2 + 96*a^4*c^4*d^4*e^4 + 64*a^5*c^3*d^2*e^6 + 6*b^6*c^2*d^6*e^
2 + 64*a^2*b^2*c^4*d^6*e^2 + 32*a^2*b^3*c^3*d^5*e^3 - 74*a^2*b^4*c^2*d^4*e^4 + 144*a^3*b^2*c^3*d^4*e^4 + 32*a^
3*b^3*c^2*d^3*e^5 + 64*a^4*b^2*c^2*d^2*e^6 + 32*a*b^3*c^4*d^7*e + 4*a*b^6*c*d^4*e^4 - 64*a^2*b*c^5*d^7*e + 32*
a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 - 44*a*b^4*c^3*d^6*e^2 + 20*a*b^5*c^2*d^5*e^3 + 20*a^2*b^5*c*d^3*e^5 - 19
2*a^3*b*c^4*d^5*e^3 - 44*a^3*b^4*c*d^2*e^6 - 192*a^4*b*c^3*d^3*e^5))*root(3840*a^6*b*c^5*d^5*e^7*z^3 + 3840*a^
5*b*c^6*d^7*e^5*z^3 + 1920*a^7*b*c^4*d^3*e^9*z^3 + 1920*a^4*b*c^7*d^9*e^3*z^3 - 288*a^7*b^3*c^2*d*e^11*z^3 - 2
88*a^2*b^3*c^7*d^11*e*z^3 + 210*a^4*b^7*c*d^3*e^9*z^3 + 210*a*b^7*c^4*d^9*e^3*z^3 - 174*a^5*b^6*c*d^2*e^10*z^3
- 174*a*b^6*c^5*d^10*e^2*z^3 - 120*a^3*b^8*c*d^4*e^8*z^3 - 120*a*b^8*c^3*d^8*e^4*z^3 + 12*a^2*b^9*c*d^5*e^7*z
^3 + 12*a*b^9*c^2*d^7*e^5*z^3 + 384*a^8*b*c^3*d*e^11*z^3 + 384*a^3*b*c^8*d^11*e*z^3 + 72*a^6*b^5*c*d*e^11*z^3
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z^3 - 3120*a^4*b^2*c^6*d^8*e^4*z^3 + 2160*a^4*b^4*c^4*d^6*e^6*z^3 - 1776*a^4*b^5*c^3*d^5*e^7*z^3 - 1776*a^3*b^
5*c^4*d^7*e^5*z^3 + 1740*a^5*b^4*c^3*d^4*e^8*z^3 + 1740*a^3*b^4*c^5*d^8*e^4*z^3 + 960*a^5*b^3*c^4*d^5*e^7*z^3
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d^2*e^10*z^3 + 648*a^2*b^4*c^6*d^10*e^2*z^3 - 600*a^5*b^5*c^2*d^3*e^9*z^3 - 600*a^2*b^5*c^5*d^9*e^3*z^3 + 372*
a^3*b^7*c^2*d^5*e^7*z^3 + 372*a^2*b^7*c^3*d^7*e^5*z^3 + 316*a^3*b^6*c^3*d^6*e^6*z^3 - 222*a^2*b^8*c^2*d^6*e^6*
z^3 - 160*a^6*b^3*c^3*d^3*e^9*z^3 - 160*a^3*b^3*c^6*d^9*e^3*z^3 + 15*a^4*b^6*c^2*d^4*e^8*z^3 + 15*a^2*b^6*c^4*
d^8*e^4*z^3 - 6*b^11*c*d^7*e^5*z^3 - 6*b^7*c^5*d^11*e*z^3 - 6*a^5*b^7*d*e^11*z^3 - 6*a*b^11*d^5*e^7*z^3 - 12*a
^7*b^4*c*e^12*z^3 - 12*a*b^4*c^7*d^12*z^3 - 20*b^9*c^3*d^9*e^3*z^3 + 15*b^10*c^2*d^8*e^4*z^3 + 15*b^8*c^4*d^10
*e^2*z^3 - 1280*a^6*c^6*d^6*e^6*z^3 - 960*a^7*c^5*d^4*e^8*z^3 - 960*a^5*c^7*d^8*e^4*z^3 - 384*a^8*c^4*d^2*e^10
*z^3 - 384*a^4*c^8*d^10*e^2*z^3 - 20*a^3*b^9*d^3*e^9*z^3 + 15*a^4*b^8*d^2*e^10*z^3 + 15*a^2*b^10*d^4*e^8*z^3 +
48*a^8*b^2*c^2*e^12*z^3 + 48*a^2*b^2*c^8*d^12*z^3 - 64*a^9*c^3*e^12*z^3 - 64*a^3*c^9*d^12*z^3 + b^12*d^6*e^6*
z^3 + b^6*c^6*d^12*z^3 + a^6*b^6*e^12*z^3 - 44*a^3*b^4*c*d*e^7*g*h*z - 20*a*b^6*c*d^3*e^5*g*h*z - 12*a*b^2*c^5
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*e^2*f*g*z - 804*a^3*b^2*c^3*d^3*e^5*g*h*z + 564*a^2*b^2*c^4*d^5*e^3*g*h*z + 222*a^3*b^3*c^2*d^2*e^6*g*h*z + 1
86*a^2*b^4*c^2*d^3*e^5*g*h*z - 166*a^2*b^3*c^3*d^4*e^4*g*h*z + 792*a^3*b^2*c^3*d^2*e^6*f*h*z - 744*a^2*b^2*c^4
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4*e^4*g*h*z - 48*a^2*b^5*c*d^2*e^6*g*h*z - 24*a^2*b*c^5*d^6*e^2*g*h*z + 10*a*b^3*c^4*d^6*e^2*g*h*z - 656*a^3*b
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68*a*b^3*c^4*d^5*e^3*f*h*z - 48*a^2*b*c^5*d^5*e^3*f*h*z - 24*a*b^2*c^5*d^6*e^2*f*h*z + 1320*a^3*b*c^4*d^2*e^6*
f*g*z - 732*a^3*b^2*c^3*d*e^7*f*g*z + 306*a^2*b^4*c^2*d*e^7*f*g*z - 304*a*b^4*c^3*d^3*e^5*f*g*z + 222*a*b^5*c^
2*d^2*e^6*f*g*z + 110*a*b^3*c^4*d^4*e^4*f*g*z - 84*a*b^2*c^5*d^5*e^3*f*g*z + 16*a*c^7*d^7*e*f*g*z - 8*a*b^7*d*
e^7*f*h*z + 4*a*b*c^6*d^8*g*h*z + 6*b^6*c^2*d^5*e^3*g*h*z + 6*b^5*c^3*d^6*e^2*g*h*z + 1072*a^4*c^4*d^3*e^5*g*h
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z - 960*a^4*c^4*d^2*e^6*f*h*z + 30*b^6*c^2*d^3*e^5*f*g*z + 30*b^3*c^5*d^6*e^2*f*g*z - 10*b^5*c^3*d^4*e^4*f*g*z
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92*a*b^5*c^2*d*e^7*f^2*z + 144*a*b*c^6*d^5*e^3*f^2*z + 432*a^3*b^2*c^3*d^4*e^4*h^2*z - 168*a^4*b^2*c^2*d^2*e^6
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4*d*e^7*f*g*z + 12*a^5*b*c^2*e^8*g*h*z - 2*a^4*b^3*c*e^8*g*h*z + 4*a^3*b^4*c*e^8*f*h*z - 228*a^4*b*c^3*e^8*f*g
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4*g^2*z - 102*a^3*b^3*c^2*d*e^7*g^2*z - 66*a*b^5*c^2*d^3*e^5*g^2*z + 24*a^2*b*c^5*d^5*e^3*g^2*z + 24*a*b^2*c^5
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a*b^4*c^3*d^2*e^6*f^2*z - 120*a*b^2*c^5*d^4*e^4*f^2*z + 24*a^5*c^3*e^8*f*h*z + 16*b^7*c*d*e^7*f^2*z + 16*b*c^7
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z + 240*a^5*c^3*d^2*e^6*h^2*z + 240*a^3*c^5*d^6*e^2*h^2*z - 12*b^6*c^2*d^4*e^4*g^2*z - 12*b^4*c^4*d^6*e^2*g^2*
z + 10*b^5*c^3*d^5*e^3*g^2*z + 528*a^3*c^5*d^4*e^4*g^2*z - 432*a^4*c^4*d^2*e^6*g^2*z + 20*b^4*c^4*d^4*e^4*f^2*
z - 16*b^6*c^2*d^2*e^6*f^2*z - 16*b^2*c^6*d^6*e^2*f^2*z - 16*a^2*c^6*d^6*e^2*g^2*z - 8*b^5*c^3*d^3*e^5*f^2*z -
8*b^3*c^5*d^5*e^3*f^2*z - 4*a^2*b^6*d^2*e^6*h^2*z + 912*a^3*c^5*d^2*e^6*f^2*z - 120*a^2*c^6*d^4*e^4*f^2*z - 4
5*a^4*b^2*c^2*e^8*g^2*z + 264*a^3*b^2*c^3*e^8*f^2*z - 192*a^2*b^4*c^2*e^8*f^2*z + 4*b^8*d*e^7*f*g*z - 8*a*c^7*
d^8*f*h*z + 4*b*c^7*d^8*f*g*z + 4*a*b^7*e^8*f*g*z + 6*b^7*c*d^3*e^5*g^2*z + 6*b^3*c^5*d^7*e*g^2*z - 48*a*c^7*d
^6*e^2*f^2*z + 12*a^3*b^4*c*e^8*g^2*z - b^8*d^2*e^6*g^2*z - 4*a^6*c^2*e^8*h^2*z + 48*a^5*c^3*e^8*g^2*z - 4*a^2
*c^6*d^8*h^2*z - b^2*c^6*d^8*g^2*z - 36*a^4*c^4*e^8*f^2*z - a^2*b^6*e^8*g^2*z - 4*c^8*d^8*f^2*z - 4*b^8*e^8*f^
2*z - 80*a*b*c^4*d^3*e^3*f*g*h + 24*a^2*b*c^3*d*e^5*f*g*h + 16*a*b^3*c^2*d*e^5*f*g*h - 72*a*b^2*c^3*d^2*e^4*f*
g*h - 48*a^2*b*c^3*d^3*e^3*g*h^2 + 16*a*b^3*c^2*d^3*e^3*g*h^2 - 12*a*b^2*c^3*d^3*e^3*g^2*h - 6*a^2*b^2*c^2*d*e
^5*g^2*h - 72*a^2*b^2*c^2*d*e^5*f*h^2 + 48*a*b^2*c^3*d^3*e^3*f*h^2 + 24*a^2*b*c^3*d^2*e^4*f*h^2 - 8*a*b^3*c^2*
d^2*e^4*f*h^2 - 8*b^5*c*d*e^5*f*g*h - 8*b*c^5*d^5*e*f*g*h - 8*a*b^4*c*e^6*f*g*h + 24*b^3*c^3*d^3*e^3*f*g*h + 1
6*b^4*c^2*d^2*e^4*f*g*h + 16*b^2*c^4*d^4*e^2*f*g*h + 48*a^2*c^4*d^2*e^4*f*g*h + 48*a^2*b^2*c^2*e^6*f*g*h + 40*
a^3*b*c^2*d*e^5*g*h^2 + 28*a*b*c^4*d^4*e^2*g^2*h - 8*a^2*b^3*c*d*e^5*g*h^2 - 8*a*b^4*c*d^2*e^4*g*h^2 + 96*a*b^
2*c^3*d*e^5*f^2*h + 24*a*b*c^4*d^2*e^4*f^2*h + 16*a*b*c^4*d^4*e^2*f*h^2 + 96*a*b*c^4*d^2*e^4*f*g^2 - 48*a*b^2*
c^3*d*e^5*f*g^2 + 12*a^2*b^2*c^2*d^2*e^4*g*h^2 - 56*a*c^5*d^4*e^2*f*g*h - 8*a*b*c^4*d^5*e*g*h^2 + 4*a*b^4*c*d*
e^5*g^2*h + 16*a*b^4*c*d*e^5*f*h^2 - 48*a*b*c^4*d*e^5*f^2*g - 24*a^3*c^3*e^6*f*g*h + 16*a*c^5*d^5*e*f*h^2 - 6*
b^4*c^2*d^3*e^3*g^2*h - 6*b^3*c^3*d^4*e^2*g^2*h + 4*b^4*c^2*d^4*e^2*g*h^2 + 80*a^2*c^4*d^3*e^3*g^2*h - 44*a^2*
c^4*d^4*e^2*g*h^2 + 24*a^3*c^3*d^2*e^4*g*h^2 - 16*b^3*c^3*d^2*e^4*f^2*h - 16*b^2*c^4*d^3*e^3*f^2*h - 8*b^4*c^2
*d^3*e^3*f*h^2 - 8*b^3*c^3*d^4*e^2*f*h^2 + 60*b^2*c^4*d^2*e^4*f^2*g - 48*a^2*c^4*d^3*e^3*f*h^2 - 24*b^3*c^3*d^
2*e^4*f*g^2 - 24*b^2*c^4*d^3*e^3*f*g^2 - 24*a^3*b*c^2*d^2*e^4*h^3 + 24*a^2*b*c^3*d^4*e^2*h^3 + 8*a^2*b^3*c*d^2
*e^4*h^3 - 8*a*b^3*c^2*d^4*e^2*h^3 + 18*a*b^2*c^3*d^2*e^4*g^3 + 2*b^5*c*d^2*e^4*g^2*h + 2*b^2*c^4*d^5*e*g^2*h
- 48*a^3*c^3*d*e^5*g^2*h - 8*b^4*c^2*d*e^5*f^2*h - 8*b*c^5*d^4*e^2*f^2*h - 168*a^2*c^4*d*e^5*f^2*h + 96*a*c^5*
d^3*e^3*f^2*h + 64*a^3*c^3*d*e^5*f*h^2 + 12*b^4*c^2*d*e^5*f*g^2 + 12*b*c^5*d^4*e^2*f*g^2 - 168*a*c^5*d^2*e^4*f
^2*g + 48*a^2*c^4*d*e^5*f*g^2 + 48*a*c^5*d^3*e^3*f*g^2 - 12*a^3*b*c^2*e^6*g^2*h + 2*a^2*b^3*c*e^6*g^2*h + 48*a
^2*b*c^3*e^6*f^2*h - 48*a*b^3*c^2*e^6*f^2*h - 8*a^3*b*c^2*e^6*f*h^2 - 60*a^2*b*c^3*e^6*f*g^2 + 48*a*b^2*c^3*e^
6*f^2*g + 12*a*b^3*c^2*e^6*f*g^2 + 24*a^2*b*c^3*d*e^5*g^3 - 24*a*b*c^4*d^3*e^3*g^3 - 6*a*b^3*c^2*d*e^5*g^3 - 1
2*c^6*d^4*e^2*f^2*g + 4*a^4*c^2*e^6*g*h^2 - 12*b^4*c^2*e^6*f^2*g + 36*a^2*c^4*e^6*f^2*g - 8*a^4*c^2*d*e^5*h^3
+ 8*a^2*c^4*d^5*e*h^3 - 24*b^2*c^4*d*e^5*f^3 - 24*b*c^5*d^2*e^4*f^3 + 8*c^6*d^5*e*f^2*h + 8*b^5*c*e^6*f^2*h +
144*a*c^5*d*e^5*f^3 - 72*a*b*c^4*e^6*f^3 + 10*b^3*c^3*d^3*e^3*g^3 - 3*b^4*c^2*d^2*e^4*g^3 - 3*b^2*c^4*d^4*e^2*
g^3 - 48*a^2*c^4*d^2*e^4*g^3 - 3*a^2*b^2*c^2*e^6*g^3 + 16*c^6*d^3*e^3*f^3 + 16*b^3*c^3*e^6*f^3 + 16*a^3*c^3*e^
6*g^3, z, k), k, 1, 3)