\(\int \frac {f+g x}{(d+e x) (a+b x+c x^2)^3} \, dx\) [2376]

Optimal result

Integrand size = 25, antiderivative size = 666 \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=-\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+c (2 a e (4 e f-d g)-3 b d (e f+d g))\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+c (2 a e (4 e f-d g)-3 b d (e f+d g))\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3} \]

[Out]

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

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {836, 814, 648, 632, 212, 642} \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )-6 c^2 e^2 \left (-2 a^3 e^3 g+a^2 b e^2 (d g+5 e f)-6 a b^2 d^2 e g+2 b^3 d^3 g\right )+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (d g+5 e f))+b^5 \left (-e^4\right ) (e f-d g)+12 c^5 d^5 f\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac {c x \left (3 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )+3 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac {e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[In]

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

[Out]

-1/2*(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b*(e*f + d*g))*x)/((b^2 - 4
*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^2) - (3*a*c*e*(2*c*d - b*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)
) - (b*c*d - b^2*e + 2*a*c*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*f + d*g)
) + c*(3*c*e*(b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d - b*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d
*g) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*f + d*g)))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x +
c*x^2)) - ((12*c^5*d^5*f - b^5*e^4*(e*f - d*g) + 10*a*b^3*c*e^4*(e*f - d*g) + 2*c^4*d^3*(2*a*e*(10*e*f - d*g)
- 3*b*d*(5*e*f + d*g)) - 6*c^2*e^2*(2*b^3*d^3*g - 6*a*b^2*d^2*e*g - 2*a^3*e^3*g + a^2*b*e^2*(5*e*f + d*g)) +
4*c^3*d*e*(3*a^2*e^2*(5*e*f - 2*d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)
/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*(e*f - d*g)*Log[d + e*x])/(c*d^2 -
b*d*e + a*e^2)^3 - (e^4*(e*f - d*g)*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 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rubi steps \begin{align*} \text {integral}& = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)+3 c e (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \frac {-3 c d e \left (b c d-b^2 e+2 a c e\right ) (2 c d f+2 a e g-b (e f+d g))+\left (2 c^2 d^2-b^2 e^2-c e (b d-4 a e)\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )-c e \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 \left (b^2-4 a c\right )^2 e^5 (e f-d g)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {2 \left (6 c^5 d^5 f-b^5 e^4 (e f-d g)+9 a b^3 c e^4 (e f-d g)-c^2 e^2 \left (6 b^3 d^3 g-18 a b^2 d^2 e g-6 a^3 e^3 g+a^2 b e^2 (23 e f-5 d g)\right )+c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))+2 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (e f-d g) x\right )}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {6 c^5 d^5 f-b^5 e^4 (e f-d g)+9 a b^3 c e^4 (e f-d g)-c^2 e^2 \left (6 b^3 d^3 g-18 a b^2 d^2 e g-6 a^3 e^3 g+a^2 b e^2 (23 e f-5 d g)\right )+c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))+2 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (e f-d g) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {\left (e^4 (e f-d g)\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 (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\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 )^2 \left (c d^2-b d e+a e^2\right )^3} \\ & = -\frac {b c d f-b^2 e f+2 a c e f-2 a c d g+a b e g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^2}-\frac {3 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (b c d-b^2 e+2 a c e\right ) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )+c \left (3 c e (b d-2 a e) (2 c d f+2 a e g-b (e f+d g))-(2 c d-b e) \left (6 c^2 d^2 f-2 b^2 e (e f-d g)+2 a c e (4 e f-d g)-3 b c d (e f+d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5 f-b^5 e^4 (e f-d g)+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (5 e f+d g))-6 c^2 e^2 \left (2 b^3 d^3 g-6 a b^2 d^2 e g-2 a^3 e^3 g+a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right )^3}+\frac {e^4 (e f-d g) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^4 (e f-d g) \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 = 1.84 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.00 \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\frac {1}{2} \left (\frac {-b^2 e f+b (a e g-c e f x+c d (f-g x))+2 c (-a d g+c d f x+a e (f+g x))}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))^2}+\frac {2 b^4 e^2 (e f-d g)+b^3 c e \left (5 d^2 g+2 e^2 f x+d e (f-2 g x)\right )-4 c^2 \left (-3 c^2 d^3 f x+a c d e (-7 e f+d g) x+a^2 e^2 (-4 e f+4 d g-3 e g x)\right )+2 b c \left (3 a^2 e^3 g+3 c^2 d^2 (d f-3 e f x-d g x)-a c e \left (-7 d e f+d^2 g+7 e^2 f x+5 d e g x\right )\right )+b^2 c \left (3 a e^2 (-5 e f+d g)+c d \left (-9 d e f-3 d^2 g+2 e^2 f x+10 d e g x\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 (a+x (b+c x))}-\frac {2 \left (12 c^5 d^5 f+10 a b^3 c e^4 (e f-d g)+b^5 e^4 (-e f+d g)-2 c^4 d^3 (2 a e (-10 e f+d g)+3 b d (5 e f+d g))+6 c^2 e^2 \left (-2 b^3 d^3 g+6 a b^2 d^2 e g+2 a^3 e^3 g-a^2 b e^2 (5 e f+d g)\right )+4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (5 e f+d g)+b^2 d^2 (5 e f+4 d g)\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2} \left (-c d^2+e (b d-a e)\right )^3}+\frac {2 e^4 (e f-d g) \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^3}+\frac {e^4 (-e f+d g) \log (a+x (b+c x))}{\left (c d^2+e (-b d+a e)\right )^3}\right ) \]

[In]

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

[Out]

((-(b^2*e*f) + b*(a*e*g - c*e*f*x + c*d*(f - g*x)) + 2*c*(-(a*d*g) + c*d*f*x + a*e*(f + g*x)))/((b^2 - 4*a*c)*
(-(c*d^2) + e*(b*d - a*e))*(a + x*(b + c*x))^2) + (2*b^4*e^2*(e*f - d*g) + b^3*c*e*(5*d^2*g + 2*e^2*f*x + d*e*
(f - 2*g*x)) - 4*c^2*(-3*c^2*d^3*f*x + a*c*d*e*(-7*e*f + d*g)*x + a^2*e^2*(-4*e*f + 4*d*g - 3*e*g*x)) + 2*b*c*
(3*a^2*e^3*g + 3*c^2*d^2*(d*f - 3*e*f*x - d*g*x) - a*c*e*(-7*d*e*f + d^2*g + 7*e^2*f*x + 5*d*e*g*x)) + b^2*c*(
3*a*e^2*(-5*e*f + d*g) + c*d*(-9*d*e*f - 3*d^2*g + 2*e^2*f*x + 10*d*e*g*x)))/((b^2 - 4*a*c)^2*(c*d^2 + e*(-(b*
d) + a*e))^2*(a + x*(b + c*x))) - (2*(12*c^5*d^5*f + 10*a*b^3*c*e^4*(e*f - d*g) + b^5*e^4*(-(e*f) + d*g) - 2*c
^4*d^3*(2*a*e*(-10*e*f + d*g) + 3*b*d*(5*e*f + d*g)) + 6*c^2*e^2*(-2*b^3*d^3*g + 6*a*b^2*d^2*e*g + 2*a^3*e^3*g
- a^2*b*e^2*(5*e*f + d*g)) + 4*c^3*d*e*(3*a^2*e^2*(5*e*f - 2*d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f
+ 4*d*g)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^(5/2)*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^
4*(e*f - d*g)*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^3 + (e^4*(-(e*f) + d*g)*Log[a + x*(b + c*x)])/(c*d^2 +
e*(-(b*d) + a*e))^3)/2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2086\) vs. \(2(658)=1316\).

Time = 0.58 (sec) , antiderivative size = 2087, normalized size of antiderivative = 3.13

method	result	size
default	\(\text {Expression too large to display}\)	\(2087\)
risch	\(\text {Expression too large to display}\)	\(9497\)

[In]

int((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-(d*g-e*f)*e^4/(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*(6*a^3*c*e^5*g-11*a^2*b*c*d*e^4*g
-7*a^2*b*c*e^5*f+4*a^2*c^2*d^2*e^3*g+14*a^2*c^2*d*e^4*f-a*b^3*d*e^4*g+a*b^3*e^5*f+10*a*b^2*c*d^2*e^3*g+8*a*b^2
*c*d*e^4*f-6*a*b*c^2*d^3*e^2*g-30*a*b*c^2*d^2*e^3*f-2*a*c^3*d^4*e*g+20*a*c^3*d^3*e^2*f+b^4*d^2*e^3*g-b^4*d*e^4
*f-6*b^3*c*d^3*e^2*g+8*b^2*c^2*d^4*e*g+10*b^2*c^2*d^3*e^2*f-3*b*c^3*d^5*g-15*b*c^3*d^4*e*f+6*c^4*d^5*f)/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^3+1/2*c*(18*a^3*b*c*e^5*g-16*a^3*c^2*d*e^4*g+16*a^3*c^2*e^5*f-25*a^2*b^2*c*d*e^4*g-29*a
^2*b^2*c*e^5*f+28*a^2*b*c^2*d^2*e^3*g+26*a^2*b*c^2*d*e^4*f-16*a^2*c^3*d^3*e^2*g+16*a^2*c^3*d^2*e^3*f-4*a*b^4*d
*e^4*g+4*a*b^4*e^5*f+22*a*b^3*c*d^2*e^3*g+32*a*b^3*c*d*e^4*f-10*a*b^2*c^2*d^3*e^2*g-98*a*b^2*c^2*d^2*e^3*f-6*a
*b*c^3*d^4*e*g+60*a*b*c^3*d^3*e^2*f+4*b^5*d^2*e^3*g-4*b^5*d*e^4*f-19*b^4*c*d^3*e^2*g+b^4*c*d^2*e^3*f+24*b^3*c^
2*d^4*e*g+30*b^3*c^2*d^3*e^2*f-9*b^2*c^3*d^5*g-45*b^2*c^3*d^4*e*f+18*b*c^4*d^5*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x
^2+(10*a^4*c^2*e^5*g+2*a^3*b^2*c*e^5*g-29*a^3*b*c^2*d*e^4*g-a^3*b*c^2*e^5*f+12*a^3*c^3*d^2*e^3*g+18*a^3*c^3*d*
e^4*f-6*a^2*b^3*c*e^5*f+26*a^2*b^2*c^2*d^2*e^3*g+10*a^2*b^2*c^2*d*e^4*f-26*a^2*b*c^3*d^3*e^2*g-34*a^2*b*c^3*d^
2*e^3*f+2*a^2*c^4*d^4*e*g+28*a^2*c^4*d^3*e^2*f-a*b^5*d*e^4*g+a*b^5*e^5*f+6*a*b^4*c*d*e^4*f-4*a*b^3*c^2*d^3*e^2
*g-18*a*b^3*c^2*d^2*e^3*f+10*a*b^2*c^3*d^4*e*g+26*a*b^2*c^3*d^3*e^2*f-5*a*b*c^4*d^5*g-25*a*b*c^4*d^4*e*f+10*a*
c^5*d^5*f+b^6*d^2*e^3*g-b^6*d*e^4*f-3*b^5*c*d^3*e^2*g+b^5*c*d^2*e^3*f+3*b^4*c^2*d^4*e*g+3*b^4*c^2*d^3*e^2*f-b^
3*c^3*d^5*g-5*b^3*c^3*d^4*e*f+2*b^2*c^4*d^5*f)/(16*a^2*c^2-8*a*b^2*c+b^4)*x+1/2*(10*a^4*b*c*e^5*g-24*a^4*c^2*d
*e^4*g+24*a^4*c^2*e^5*f-a^3*b^3*e^5*g-9*a^3*b^2*c*d*e^4*g-21*a^3*b^2*c*e^5*f+44*a^3*b*c^2*d^2*e^3*g-14*a^3*b*c
^2*d*e^4*f-32*a^3*c^3*d^3*e^2*g+32*a^3*c^3*d^2*e^3*f+3*a^2*b^4*e^5*f+27*a^2*b^3*c*d*e^4*f-10*a^2*b^2*c^2*d^3*e
^2*g-50*a^2*b^2*c^2*d^2*e^3*f+18*a^2*b*c^3*d^4*e*g+12*a^2*b*c^3*d^3*e^2*f-8*a^2*c^4*d^5*g+8*a^2*c^4*d^4*e*f+a*
b^5*d^2*e^3*g-4*a*b^5*d*e^4*f-3*a*b^4*c*d^3*e^2*g-a*b^4*c*d^2*e^3*f+3*a*b^3*c^2*d^4*e*g+24*a*b^3*c^2*d^3*e^2*f
-a*b^2*c^3*d^5*g-29*a*b^2*c^3*d^4*e*f+10*a*b*c^4*d^5*f+b^6*d^2*e^3*f-3*b^5*c*d^3*e^2*f+3*b^4*c^2*d^4*e*f-b^3*c
^3*d^5*f)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/(16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^3*d*e^4*g-16
*a^2*c^3*e^5*f-8*a*b^2*c^2*d*e^4*g+8*a*b^2*c^2*e^5*f+b^4*c*d*e^4*g-b^4*c*e^5*f)/c*ln(c*x^2+b*x+a)+2*(6*a^3*c^2
*e^5*g+5*a^2*b*c^2*d*e^4*g-23*a^2*b*c^2*e^5*f-12*a^2*c^3*d^2*e^3*g+30*a^2*c^3*d*e^4*f-9*a*b^3*c*d*e^4*g+9*a*b^
3*c*e^5*f+18*a*b^2*c^2*d^2*e^3*g-6*a*b*c^3*d^3*e^2*g-30*a*b*c^3*d^2*e^3*f-2*a*c^4*d^4*e*g+20*a*c^4*d^3*e^2*f+b
^5*d*e^4*g-b^5*e^5*f-6*b^3*c^2*d^3*e^2*g+8*b^2*c^3*d^4*e*g+10*b^2*c^3*d^3*e^2*f-3*b*c^4*d^5*g-15*b*c^4*d^4*e*f
+6*c^5*d^5*f-1/2*(16*a^2*c^3*d*e^4*g-16*a^2*c^3*e^5*f-8*a*b^2*c^2*d*e^4*g+8*a*b^2*c^2*e^5*f+b^4*c*d*e^4*g-b^4*
c*e^5*f)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))))

Fricas [F(-1)]

Timed out. \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((g*x+f)/(e*x+d)/(c*x^2+b*x+a)^3,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. 2398 vs. \(2 (658) = 1316\).

Time = 0.31 (sec) , antiderivative size = 2398, normalized size of antiderivative = 3.60 \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(e^5*f - d*e^4*g)*log(c*x^2 + b*x + a)/(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^6*f - d*e^5*g)*
log(abs(e*x + d))/(c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 + 3*a*c^2*d^4*e^3 - b^3*d^3*e^4 - 6*a*b*c*d^3
*e^4 + 3*a*b^2*d^2*e^5 + 3*a^2*c*d^2*e^5 - 3*a^2*b*d*e^6 + a^3*e^7) + (12*c^5*d^5*f - 30*b*c^4*d^4*e*f + 20*b^
2*c^3*d^3*e^2*f + 40*a*c^4*d^3*e^2*f - 60*a*b*c^3*d^2*e^3*f + 60*a^2*c^3*d*e^4*f - b^5*e^5*f + 10*a*b^3*c*e^5*
f - 30*a^2*b*c^2*e^5*f - 6*b*c^4*d^5*g + 16*b^2*c^3*d^4*e*g - 4*a*c^4*d^4*e*g - 12*b^3*c^2*d^3*e^2*g - 12*a*b*
c^3*d^3*e^2*g + 36*a*b^2*c^2*d^2*e^3*g - 24*a^2*c^3*d^2*e^3*g + b^5*d*e^4*g - 10*a*b^3*c*d*e^4*g - 6*a^2*b*c^2
*d*e^4*g + 12*a^3*c^2*e^5*g)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3*d^6 - 8*a*b^2*c^4*d^6 + 16*a^2*c
^5*d^6 - 3*b^5*c^2*d^5*e + 24*a*b^3*c^3*d^5*e - 48*a^2*b*c^4*d^5*e + 3*b^6*c*d^4*e^2 - 21*a*b^4*c^2*d^4*e^2 +
24*a^2*b^2*c^3*d^4*e^2 + 48*a^3*c^4*d^4*e^2 - b^7*d^3*e^3 + 2*a*b^5*c*d^3*e^3 + 32*a^2*b^3*c^2*d^3*e^3 - 96*a^
3*b*c^3*d^3*e^3 + 3*a*b^6*d^2*e^4 - 21*a^2*b^4*c*d^2*e^4 + 24*a^3*b^2*c^2*d^2*e^4 + 48*a^4*c^3*d^2*e^4 - 3*a^2
*b^5*d*e^5 + 24*a^3*b^3*c*d*e^5 - 48*a^4*b*c^2*d*e^5 + a^3*b^4*e^6 - 8*a^4*b^2*c*e^6 + 16*a^5*c^2*e^6)*sqrt(-b
^2 + 4*a*c)) - 1/2*(b^3*c^3*d^5*f - 10*a*b*c^4*d^5*f - 3*b^4*c^2*d^4*e*f + 29*a*b^2*c^3*d^4*e*f - 8*a^2*c^4*d^
4*e*f + 3*b^5*c*d^3*e^2*f - 24*a*b^3*c^2*d^3*e^2*f - 12*a^2*b*c^3*d^3*e^2*f - b^6*d^2*e^3*f + a*b^4*c*d^2*e^3*
f + 50*a^2*b^2*c^2*d^2*e^3*f - 32*a^3*c^3*d^2*e^3*f + 4*a*b^5*d*e^4*f - 27*a^2*b^3*c*d*e^4*f + 14*a^3*b*c^2*d*
e^4*f - 3*a^2*b^4*e^5*f + 21*a^3*b^2*c*e^5*f - 24*a^4*c^2*e^5*f + a*b^2*c^3*d^5*g + 8*a^2*c^4*d^5*g - 3*a*b^3*
c^2*d^4*e*g - 18*a^2*b*c^3*d^4*e*g + 3*a*b^4*c*d^3*e^2*g + 10*a^2*b^2*c^2*d^3*e^2*g + 32*a^3*c^3*d^3*e^2*g - a
*b^5*d^2*e^3*g - 44*a^3*b*c^2*d^2*e^3*g + 9*a^3*b^2*c*d*e^4*g + 24*a^4*c^2*d*e^4*g + a^3*b^3*e^5*g - 10*a^4*b*
c*e^5*g - 2*(6*c^6*d^5*f - 15*b*c^5*d^4*e*f + 10*b^2*c^4*d^3*e^2*f + 20*a*c^5*d^3*e^2*f - 30*a*b*c^4*d^2*e^3*f
- b^4*c^2*d*e^4*f + 8*a*b^2*c^3*d*e^4*f + 14*a^2*c^4*d*e^4*f + a*b^3*c^2*e^5*f - 7*a^2*b*c^3*e^5*f - 3*b*c^5*
d^5*g + 8*b^2*c^4*d^4*e*g - 2*a*c^5*d^4*e*g - 6*b^3*c^3*d^3*e^2*g - 6*a*b*c^4*d^3*e^2*g + b^4*c^2*d^2*e^3*g +
10*a*b^2*c^3*d^2*e^3*g + 4*a^2*c^4*d^2*e^3*g - a*b^3*c^2*d*e^4*g - 11*a^2*b*c^3*d*e^4*g + 6*a^3*c^3*e^5*g)*x^3
- (18*b*c^5*d^5*f - 45*b^2*c^4*d^4*e*f + 30*b^3*c^3*d^3*e^2*f + 60*a*b*c^4*d^3*e^2*f + b^4*c^2*d^2*e^3*f - 98
*a*b^2*c^3*d^2*e^3*f + 16*a^2*c^4*d^2*e^3*f - 4*b^5*c*d*e^4*f + 32*a*b^3*c^2*d*e^4*f + 26*a^2*b*c^3*d*e^4*f +
4*a*b^4*c*e^5*f - 29*a^2*b^2*c^2*e^5*f + 16*a^3*c^3*e^5*f - 9*b^2*c^4*d^5*g + 24*b^3*c^3*d^4*e*g - 6*a*b*c^4*d
^4*e*g - 19*b^4*c^2*d^3*e^2*g - 10*a*b^2*c^3*d^3*e^2*g - 16*a^2*c^4*d^3*e^2*g + 4*b^5*c*d^2*e^3*g + 22*a*b^3*c
^2*d^2*e^3*g + 28*a^2*b*c^3*d^2*e^3*g - 4*a*b^4*c*d*e^4*g - 25*a^2*b^2*c^2*d*e^4*g - 16*a^3*c^3*d*e^4*g + 18*a
^3*b*c^2*e^5*g)*x^2 - 2*(2*b^2*c^4*d^5*f + 10*a*c^5*d^5*f - 5*b^3*c^3*d^4*e*f - 25*a*b*c^4*d^4*e*f + 3*b^4*c^2
*d^3*e^2*f + 26*a*b^2*c^3*d^3*e^2*f + 28*a^2*c^4*d^3*e^2*f + b^5*c*d^2*e^3*f - 18*a*b^3*c^2*d^2*e^3*f - 34*a^2
*b*c^3*d^2*e^3*f - b^6*d*e^4*f + 6*a*b^4*c*d*e^4*f + 10*a^2*b^2*c^2*d*e^4*f + 18*a^3*c^3*d*e^4*f + a*b^5*e^5*f
- 6*a^2*b^3*c*e^5*f - a^3*b*c^2*e^5*f - b^3*c^3*d^5*g - 5*a*b*c^4*d^5*g + 3*b^4*c^2*d^4*e*g + 10*a*b^2*c^3*d^
4*e*g + 2*a^2*c^4*d^4*e*g - 3*b^5*c*d^3*e^2*g - 4*a*b^3*c^2*d^3*e^2*g - 26*a^2*b*c^3*d^3*e^2*g + b^6*d^2*e^3*g
+ 26*a^2*b^2*c^2*d^2*e^3*g + 12*a^3*c^3*d^2*e^3*g - a*b^5*d*e^4*g - 29*a^3*b*c^2*d*e^4*g + 2*a^3*b^2*c*e^5*g
+ 10*a^4*c^2*e^5*g)*x)/((c*d^2 - b*d*e + a*e^2)^3*(c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2)

Mupad [B] (verification not implemented)

Time = 18.28 (sec) , antiderivative size = 25467, normalized size of antiderivative = 38.24 \[ \int \frac {f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((24*a^3*c^2*e^3*f - 8*a^2*c^3*d^3*g - a^2*b^3*e^3*g - b^3*c^2*d^3*f + 3*a*b^4*e^3*f - b^5*d*e^2*f + 10*a*b*c^
3*d^3*f + 10*a^3*b*c*e^3*g - a*b^4*d*e^2*g + 2*b^4*c*d^2*e*f - a*b^2*c^2*d^3*g - 21*a^2*b^2*c*e^3*f + 8*a^2*c^
3*d^2*e*f - 24*a^3*c^2*d*e^2*g + 6*a*b^3*c*d*e^2*f + 2*a*b^3*c*d^2*e*g - 19*a*b^2*c^2*d^2*e*f + 10*a^2*b*c^2*d
*e^2*f + 10*a^2*b*c^2*d^2*e*g + a^2*b^2*c*d*e^2*g)/(2*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2
*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e +
16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)) + (x^2
*(16*a^2*c^3*e^3*f - 9*b^2*c^3*d^3*g + 18*b*c^4*d^3*f + 4*b^4*c*e^3*f - 4*b^4*c*d*e^2*g - 29*a*b^2*c^2*e^3*f +
18*a^2*b*c^2*e^3*g - 16*a^2*c^3*d*e^2*g - 27*b^2*c^3*d^2*e*f + 3*b^3*c^2*d*e^2*f + 15*b^3*c^2*d^2*e*g + 42*a*
b*c^3*d*e^2*f - 6*a*b*c^3*d^2*e*g - 7*a*b^2*c^2*d*e^2*g))/(2*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 +
b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d
^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3))
+ (x^3*(6*c^5*d^3*f + 6*a^2*c^3*e^3*g + b^3*c^2*e^3*f - 3*b*c^4*d^3*g - 7*a*b*c^3*e^3*f + 14*a*c^4*d*e^2*f -
2*a*c^4*d^2*e*g - 9*b*c^4*d^2*e*f + b^2*c^3*d*e^2*f + 5*b^2*c^3*d^2*e*g - b^3*c^2*d*e^2*g - 5*a*b*c^3*d*e^2*g)
)/(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e
^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^
3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3) + (x*(b^5*e^3*f + 2*b^2*c^3*d^3*f + 10*a^3*c^2*e^3*g - b^3*
c^2*d^3*g + 10*a*c^4*d^3*f - b^5*d*e^2*g - 5*a*b*c^3*d^3*g - 6*a*b^3*c*e^3*f + 2*b^4*c*d^2*e*g - a^2*b*c^2*e^3
*f + 2*a^2*b^2*c*e^3*g + 18*a^2*c^3*d*e^2*f + 2*a^2*c^3*d^2*e*g - 3*b^3*c^2*d^2*e*f - 15*a*b*c^3*d^2*e*f + 2*a
*b^3*c*d*e^2*g + 9*a*b^2*c^2*d*e^2*f + 5*a*b^2*c^2*d^2*e*g - 19*a^2*b*c^2*d*e^2*g))/(a^2*b^4*e^4 + 16*a^2*c^4*
d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*
a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3
- 32*a^3*b*c^2*d*e^3))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3) + symsum(log(root(61440*a^8*
b*c^7*d^5*e^7*z^3 + 61440*a^7*b*c^8*d^7*e^5*z^3 + 30720*a^9*b*c^6*d^3*e^9*z^3 + 30720*a^6*b*c^9*d^9*e^3*z^3 -
7680*a^9*b^3*c^4*d*e^11*z^3 - 7680*a^4*b^3*c^9*d^11*e*z^3 + 3840*a^8*b^5*c^3*d*e^11*z^3 + 3840*a^3*b^5*c^8*d^1
1*e*z^3 - 960*a^7*b^7*c^2*d*e^11*z^3 - 960*a^2*b^7*c^7*d^11*e*z^3 + 370*a^4*b^11*c*d^3*e^9*z^3 + 370*a*b^11*c^
4*d^9*e^3*z^3 - 294*a^5*b^10*c*d^2*e^10*z^3 - 294*a*b^10*c^5*d^10*e^2*z^3 - 240*a^3*b^12*c*d^4*e^8*z^3 - 240*a
*b^12*c^3*d^8*e^4*z^3 + 60*a^2*b^13*c*d^5*e^7*z^3 + 60*a*b^13*c^2*d^7*e^5*z^3 + 6144*a^10*b*c^5*d*e^11*z^3 + 6
144*a^5*b*c^10*d^11*e*z^3 + 120*a^6*b^9*c*d*e^11*z^3 + 120*a*b^9*c^6*d^11*e*z^3 + 10*a*b^14*c*d^6*e^6*z^3 + 71
680*a^6*b^4*c^6*d^6*e^6*z^3 - 66560*a^7*b^2*c^7*d^6*e^6*z^3 + 51840*a^7*b^4*c^5*d^4*e^8*z^3 + 51840*a^5*b^4*c^
7*d^8*e^4*z^3 - 42240*a^8*b^2*c^6*d^4*e^8*z^3 - 42240*a^6*b^2*c^8*d^8*e^4*z^3 - 32256*a^6*b^5*c^5*d^5*e^7*z^3
- 32256*a^5*b^5*c^6*d^7*e^5*z^3 + 21120*a^5*b^7*c^4*d^5*e^7*z^3 + 21120*a^4*b^7*c^5*d^7*e^5*z^3 - 17920*a^8*b^
3*c^5*d^3*e^9*z^3 - 17920*a^5*b^3*c^8*d^9*e^3*z^3 - 17024*a^5*b^6*c^5*d^6*e^6*z^3 - 16800*a^6*b^6*c^4*d^4*e^8*
z^3 - 16800*a^4*b^6*c^6*d^8*e^4*z^3 + 15360*a^8*b^4*c^4*d^2*e^10*z^3 - 15360*a^7*b^3*c^6*d^5*e^7*z^3 - 15360*a
^6*b^3*c^7*d^7*e^5*z^3 + 15360*a^4*b^4*c^8*d^10*e^2*z^3 - 8640*a^7*b^6*c^3*d^2*e^10*z^3 - 8640*a^3*b^6*c^7*d^1
0*e^2*z^3 + 8000*a^6*b^7*c^3*d^3*e^9*z^3 + 8000*a^3*b^7*c^6*d^9*e^3*z^3 - 7680*a^9*b^2*c^5*d^2*e^10*z^3 - 7680
*a^5*b^2*c^9*d^10*e^2*z^3 - 6400*a^7*b^5*c^4*d^3*e^9*z^3 - 6400*a^4*b^5*c^7*d^9*e^3*z^3 - 4560*a^4*b^9*c^3*d^5
*e^7*z^3 - 4560*a^3*b^9*c^4*d^7*e^5*z^3 - 3920*a^4*b^8*c^4*d^6*e^6*z^3 - 2600*a^5*b^9*c^2*d^3*e^9*z^3 - 2600*a
^2*b^9*c^5*d^9*e^3*z^3 + 2380*a^3*b^10*c^3*d^6*e^6*z^3 + 2280*a^6*b^8*c^2*d^2*e^10*z^3 + 2280*a^2*b^8*c^6*d^10
*e^2*z^3 + 1215*a^4*b^10*c^2*d^4*e^8*z^3 + 1215*a^2*b^10*c^4*d^8*e^4*z^3 - 350*a^2*b^12*c^2*d^6*e^6*z^3 - 300*
a^5*b^8*c^3*d^4*e^8*z^3 - 300*a^3*b^8*c^5*d^8*e^4*z^3 + 180*a^3*b^11*c^2*d^5*e^7*z^3 + 180*a^2*b^11*c^3*d^7*e^
5*z^3 - 6*b^15*c*d^7*e^5*z^3 - 6*b^11*c^5*d^11*e*z^3 - 6*a^5*b^11*d*e^11*z^3 - 6*a*b^15*d^5*e^7*z^3 - 20*a^7*b
^8*c*e^12*z^3 - 20*a*b^8*c^7*d^12*z^3 - 20*b^13*c^3*d^9*e^3*z^3 + 15*b^14*c^2*d^8*e^4*z^3 + 15*b^12*c^4*d^10*e
^2*z^3 - 20480*a^8*c^8*d^6*e^6*z^3 - 15360*a^9*c^7*d^4*e^8*z^3 - 15360*a^7*c^9*d^8*e^4*z^3 - 6144*a^10*c^6*d^2
*e^10*z^3 - 6144*a^6*c^10*d^10*e^2*z^3 - 20*a^3*b^13*d^3*e^9*z^3 + 15*a^4*b^12*d^2*e^10*z^3 + 15*a^2*b^14*d^4*
e^8*z^3 + 1280*a^10*b^2*c^4*e^12*z^3 - 640*a^9*b^4*c^3*e^12*z^3 + 160*a^8*b^6*c^2*e^12*z^3 + 1280*a^4*b^2*c^10
*d^12*z^3 - 640*a^3*b^4*c^9*d^12*z^3 + 160*a^2*b^6*c^8*d^12*z^3 - 1024*a^11*c^5*e^12*z^3 - 1024*a^5*c^11*d^12*
z^3 + b^16*d^6*e^6*z^3 + b^10*c^6*d^12*z^3 + a^6*b^10*e^12*z^3 + 132*a*b*c^8*d^8*e^2*f*g*z + 1960*a^2*b^3*c^5*
d^4*e^6*f*g*z - 1560*a^3*b^2*c^5*d^3*e^7*f*g*z - 1500*a^2*b^2*c^6*d^5*e^5*f*g*z + 960*a^3*b^3*c^4*d^2*e^8*f*g*
z - 420*a^2*b^4*c^4*d^3*e^7*f*g*z - 222*a^2*b^5*c^3*d^2*e^8*f*g*z - 40*a*b^8*c*d*e^9*f*g*z + 1830*a^4*b^2*c^4*
d*e^9*f*g*z + 1440*a*b^3*c^6*d^6*e^4*f*g*z - 1080*a^3*b^4*c^3*d*e^9*f*g*z - 856*a*b^2*c^7*d^7*e^3*f*g*z - 840*
a*b^4*c^5*d^5*e^5*f*g*z + 302*a^2*b^6*c^2*d*e^9*f*g*z + 180*a^4*b*c^5*d^2*e^8*f*g*z - 120*a^3*b*c^6*d^4*e^6*f*
g*z + 84*a*b^6*c^3*d^3*e^7*f*g*z - 24*a^2*b*c^7*d^6*e^4*f*g*z + 18*a*b^7*c^2*d^2*e^8*f*g*z - 2*a*b^5*c^4*d^4*e
^6*f*g*z + 24*a*c^9*d^9*e*f*g*z + 372*b^3*c^7*d^8*e^2*f*g*z - 340*b^4*c^6*d^7*e^3*f*g*z + 114*b^5*c^5*d^6*e^4*
f*g*z + 12*b^6*c^4*d^5*e^5*f*g*z - 6*b^8*c^2*d^3*e^7*f*g*z - 2*b^7*c^3*d^4*e^6*f*g*z + 528*a^3*c^7*d^5*e^5*f*g
*z + 480*a^4*c^6*d^3*e^7*f*g*z + 224*a^2*c^8*d^7*e^3*f*g*z - 60*a^4*b^3*c^3*e^10*f*g*z + 6*a^3*b^5*c^2*e^10*f*
g*z + 36*a^5*b*c^4*d*e^9*g^2*z + 20*a*b^8*c*d^2*e^8*g^2*z + 960*a*b*c^8*d^7*e^3*f^2*z + 900*a^4*b*c^5*d*e^9*f^
2*z - 1185*a^4*b^2*c^4*d^2*e^8*g^2*z + 450*a^3*b^4*c^3*d^2*e^8*g^2*z - 420*a^2*b^4*c^4*d^4*e^6*g^2*z + 300*a^3
*b^2*c^5*d^4*e^6*g^2*z + 210*a^2*b^2*c^6*d^6*e^4*g^2*z + 192*a^2*b^5*c^3*d^3*e^7*g^2*z - 142*a^2*b^6*c^2*d^2*e
^8*g^2*z + 100*a^2*b^3*c^5*d^5*e^5*g^2*z + 60*a^3*b^3*c^4*d^3*e^7*g^2*z - 1950*a^2*b^2*c^6*d^4*e^6*f^2*z - 900
*a^3*b^2*c^5*d^2*e^8*f^2*z + 300*a^2*b^4*c^4*d^2*e^8*f^2*z + 100*a^2*b^3*c^5*d^3*e^7*f^2*z - 186*b^2*c^8*d^9*e
*f*g*z - 1896*a^5*c^5*d*e^9*f*g*z + 180*a^5*b*c^4*e^10*f*g*z - 12*a*b*c^8*d^9*e*g^2*z - 390*a*b^4*c^5*d^6*e^4*
g^2*z + 298*a*b^5*c^4*d^5*e^5*g^2*z + 180*a*b^3*c^6*d^7*e^3*g^2*z - 120*a^3*b*c^6*d^5*e^5*g^2*z - 96*a^2*b*c^7
*d^7*e^3*g^2*z + 60*a^4*b^3*c^3*d*e^9*g^2*z - 54*a*b^6*c^3*d^4*e^6*g^2*z - 18*a*b^7*c^2*d^3*e^7*g^2*z - 6*a^3*
b^5*c^2*d*e^9*g^2*z - 4*a*b^2*c^7*d^8*e^2*g^2*z + 2400*a^3*b*c^6*d^3*e^7*f^2*z + 2280*a^2*b*c^7*d^5*e^5*f^2*z
- 1300*a*b^2*c^7*d^6*e^4*f^2*z + 540*a*b^3*c^6*d^5*e^5*f^2*z - 300*a^3*b^3*c^4*d*e^9*f^2*z + 150*a*b^4*c^5*d^4
*e^6*f^2*z - 80*a*b^5*c^4*d^3*e^7*f^2*z + 30*a^2*b^5*c^3*d*e^9*f^2*z - 30*a*b^6*c^3*d^2*e^8*f^2*z + 180*b*c^9*
d^9*e*f^2*z + 20*a*b^8*c*e^10*f^2*z - 100*b^4*c^6*d^8*e^2*g^2*z + 96*b^5*c^5*d^7*e^3*g^2*z - 33*b^6*c^4*d^6*e^
4*g^2*z - 8*b^7*c^3*d^5*e^5*g^2*z + 6*b^8*c^2*d^4*e^6*g^2*z + 912*a^5*c^5*d^2*e^8*g^2*z - 345*b^2*c^8*d^8*e^2*
f^2*z + 300*b^3*c^7*d^7*e^3*f^2*z - 120*a^4*c^6*d^4*e^6*g^2*z - 100*b^4*c^6*d^6*e^4*f^2*z - 48*a^3*c^7*d^6*e^4
*g^2*z - 15*b^6*c^4*d^4*e^6*f^2*z + 10*b^7*c^3*d^3*e^7*f^2*z + 6*b^5*c^5*d^5*e^5*f^2*z - 4*a^2*c^8*d^8*e^2*g^2
*z - 1200*a^3*c^7*d^4*e^6*f^2*z - 900*a^4*c^6*d^2*e^8*f^2*z - 760*a^2*c^8*d^6*e^4*f^2*z - 1185*a^4*b^2*c^4*e^1
0*f^2*z + 630*a^3*b^4*c^3*e^10*f^2*z - 160*a^2*b^6*c^2*e^10*f^2*z + 2*b^10*d*e^9*f*g*z + 36*b*c^9*d^10*f*g*z +
48*b^3*c^7*d^9*e*g^2*z - 240*a*c^9*d^8*e^2*f^2*z - b^10*d^2*e^8*g^2*z - 36*a^6*c^4*e^10*g^2*z - 9*b^2*c^8*d^1
0*g^2*z + 768*a^5*c^5*e^10*f^2*z - 36*c^10*d^10*f^2*z - b^10*e^10*f^2*z - 177*a*b^2*c^4*d^2*e^7*f*g^2 + 285*a*
b^2*c^4*d*e^8*f^2*g + 252*a^2*b*c^4*d*e^8*f*g^2 - 120*a*b^3*c^3*d*e^8*f*g^2 + 108*a*b*c^5*d^3*e^6*f*g^2 + 36*a
*b*c^5*d^2*e^7*f^2*g - 132*a*b*c^5*d*e^8*f^3 - 69*b^2*c^5*d^4*e^5*f*g^2 + 57*b^2*c^5*d^3*e^6*f^2*g - 45*b^3*c^
4*d^2*e^7*f^2*g + 30*b^4*c^3*d^2*e^7*f*g^2 + 9*b^3*c^4*d^3*e^6*f*g^2 + 156*a^2*c^5*d^2*e^7*f*g^2 - 72*a^2*b*c^
4*d^2*e^7*g^3 + 60*a*b^3*c^3*d^2*e^7*g^3 - 13*a*b^2*c^4*d^3*e^6*g^3 + 36*b*c^6*d^5*e^4*f*g^2 + 36*b*c^6*d^4*e^
5*f^2*g - 30*b^4*c^3*d*e^8*f^2*g + 12*b^5*c^2*d*e^8*f*g^2 - 408*a^2*c^5*d*e^8*f^2*g - 156*a*c^6*d^3*e^6*f^2*g
+ 24*a*c^6*d^4*e^5*f*g^2 - 180*a^2*b*c^4*e^9*f^2*g + 60*a*b^3*c^3*e^9*f^2*g - 12*a*b*c^5*d^4*e^5*g^3 - 36*c^7*
d^5*e^4*f^2*g - 6*b^5*c^2*e^9*f^2*g + 36*a^3*c^4*e^9*f*g^2 - 72*b*c^6*d^3*e^6*f^3 - 36*a^3*c^4*d*e^8*g^3 + 15*
b^3*c^4*d*e^8*f^3 + 132*a*c^6*d^2*e^7*f^3 - 95*a*b^2*c^4*e^9*f^3 + 21*b^3*c^4*d^4*e^5*g^3 - 10*b^4*c^3*d^3*e^6
*g^3 - 9*b^2*c^5*d^5*e^4*g^3 - 6*b^5*c^2*d^2*e^7*g^3 + 21*b^2*c^5*d^2*e^7*f^3 - 4*a^2*c^5*d^3*e^6*g^3 + 36*c^7
*d^4*e^5*f^3 + 10*b^4*c^3*e^9*f^3 + 256*a^2*c^5*e^9*f^3, z, k)*((a^2*b^9*c*e^10*f - 96*a^7*c^5*e^10*g + 368*a^
6*b*c^5*e^10*f + 96*a^2*c^10*d^9*e*f + 32*a^6*c^6*d*e^9*f + 6*b^4*c^8*d^9*e*f + b^11*c*d^2*e^8*f - 3*b^5*c^7*d
^9*e*g - b^11*c*d^3*e^7*g - 17*a^3*b^7*c^2*e^10*f + 111*a^4*b^5*c^3*e^10*f - 328*a^5*b^3*c^4*e^10*f - 6*a^5*b^
4*c^3*e^10*g + 48*a^6*b^2*c^4*e^10*g + 320*a^3*c^9*d^7*e^3*f + 384*a^4*c^8*d^5*e^5*f + 192*a^5*c^7*d^3*e^7*f -
32*a^3*c^9*d^8*e^2*g - 192*a^4*c^8*d^6*e^4*g - 384*a^5*c^7*d^4*e^6*g - 320*a^6*c^6*d^2*e^8*g - 21*b^5*c^7*d^8
*e^2*f + 25*b^6*c^6*d^7*e^3*f - 10*b^7*c^5*d^6*e^4*f - b^10*c^2*d^3*e^7*f + 11*b^6*c^6*d^8*e^2*g - 14*b^7*c^5*
d^7*e^3*g + 6*b^8*c^4*d^6*e^4*g + b^10*c^2*d^4*e^6*g + 168*a*b^3*c^8*d^8*e^2*f - 180*a*b^4*c^7*d^7*e^3*f + 36*
a*b^5*c^6*d^6*e^4*f + 15*a*b^6*c^5*d^5*e^5*f + 11*a*b^7*c^4*d^4*e^6*f + 17*a*b^8*c^3*d^3*e^7*f - 17*a*b^9*c^2*
d^2*e^8*f - 336*a^2*b*c^9*d^8*e^2*f + 35*a^2*b^8*c^2*d*e^9*f - 704*a^3*b*c^8*d^6*e^4*f - 239*a^3*b^6*c^3*d*e^9
*f - 32*a^4*b*c^7*d^4*e^6*f + 746*a^4*b^4*c^4*d*e^9*f + 704*a^5*b*c^6*d^2*e^8*f - 896*a^5*b^2*c^5*d*e^9*f - 90
*a*b^4*c^7*d^8*e^2*g + 108*a*b^5*c^6*d^7*e^3*g - 27*a*b^6*c^5*d^6*e^4*g - 11*a*b^7*c^4*d^5*e^5*g - 23*a*b^8*c^
3*d^4*e^6*g + 17*a*b^9*c^2*d^3*e^7*g - 64*a^3*b*c^8*d^7*e^3*g + 17*a^3*b^7*c^2*d*e^9*g + 32*a^4*b*c^7*d^5*e^5*
g - 87*a^4*b^5*c^3*d*e^9*g + 64*a^5*b*c^6*d^3*e^7*g + 136*a^5*b^3*c^4*d*e^9*g - 2*a*b^10*c*d*e^9*f + 240*a^2*b
^2*c^8*d^7*e^3*f + 192*a^2*b^3*c^7*d^6*e^4*f - 96*a^2*b^4*c^6*d^5*e^5*f - 90*a^2*b^5*c^5*d^4*e^6*f - 153*a^2*b
^6*c^4*d^3*e^7*f + 112*a^2*b^7*c^3*d^2*e^8*f + 48*a^3*b^2*c^7*d^5*e^5*f + 192*a^3*b^3*c^6*d^4*e^6*f + 676*a^3*
b^4*c^5*d^3*e^7*f - 292*a^3*b^5*c^4*d^2*e^8*f - 1136*a^4*b^2*c^6*d^3*e^7*f + 32*a^4*b^3*c^5*d^2*e^8*f + 192*a^
2*b^2*c^8*d^8*e^2*g - 192*a^2*b^3*c^7*d^7*e^3*g - 84*a^2*b^4*c^6*d^6*e^4*g + 90*a^2*b^5*c^5*d^5*e^5*g + 165*a^
2*b^6*c^4*d^4*e^6*g - 88*a^2*b^7*c^3*d^3*e^7*g - 35*a^2*b^8*c^2*d^2*e^8*g + 432*a^3*b^2*c^7*d^6*e^4*g - 192*a^
3*b^3*c^6*d^5*e^5*g - 496*a^3*b^4*c^5*d^4*e^6*g + 148*a^3*b^5*c^4*d^3*e^7*g + 203*a^3*b^6*c^3*d^2*e^8*g + 656*
a^4*b^2*c^6*d^4*e^6*g - 32*a^4*b^3*c^5*d^3*e^7*g - 476*a^4*b^4*c^4*d^2*e^8*g + 464*a^5*b^2*c^5*d^2*e^8*g - 48*
a*b^2*c^9*d^9*e*f + 24*a*b^3*c^8*d^9*e*g + 2*a*b^10*c*d^2*e^8*g - 48*a^2*b*c^9*d^9*e*g - a^2*b^9*c*d*e^9*g + 1
6*a^6*b*c^5*d*e^9*g)/(a^4*b^8*e^8 + 256*a^4*c^8*d^8 + 256*a^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*
c^5*d^8 - 16*a^5*b^6*c*e^8 - 4*a*b^11*d^3*e^5 - 4*a^3*b^9*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*
b^4*c^6*d^8 - 256*a^3*b^2*c^7*d^8 + 96*a^6*b^4*c^2*e^8 - 256*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c
^7*d^6*e^2 + 1536*a^6*c^6*d^4*e^4 + 1024*a^7*c^5*d^2*e^6 + 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*
a^2*b^7*c^3*d^5*e^3 - 90*a^2*b^8*c^2*d^4*e^4 - 1152*a^3*b^4*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^
6*c^3*d^4*e^4 - 192*a^3*b^7*c^2*d^3*e^5 + 512*a^4*b^2*c^6*d^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^
4*d^4*e^4 - 128*a^4*b^5*c^3*d^3*e^5 + 512*a^4*b^6*c^2*d^2*e^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^
3*e^5 - 1152*a^5*b^4*c^3*d^2*e^6 + 512*a^6*b^2*c^4*d^2*e^6 + 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^
4*b*c^7*d^7*e + 64*a^4*b^7*c*d*e^7 - 1024*a^7*b*c^4*d*e^7 - 92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*
a^2*b^5*c^5*d^7*e + 52*a^2*b^9*c*d^3*e^5 + 1024*a^3*b^3*c^6*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*
e^3 - 384*a^5*b^5*c^2*d*e^7 - 3072*a^6*b*c^5*d^3*e^5 + 1024*a^6*b^3*c^3*d*e^7) - root(61440*a^8*b*c^7*d^5*e^7*
z^3 + 61440*a^7*b*c^8*d^7*e^5*z^3 + 30720*a^9*b*c^6*d^3*e^9*z^3 + 30720*a^6*b*c^9*d^9*e^3*z^3 - 7680*a^9*b^3*c
^4*d*e^11*z^3 - 7680*a^4*b^3*c^9*d^11*e*z^3 + 3840*a^8*b^5*c^3*d*e^11*z^3 + 3840*a^3*b^5*c^8*d^11*e*z^3 - 960*
a^7*b^7*c^2*d*e^11*z^3 - 960*a^2*b^7*c^7*d^11*e*z^3 + 370*a^4*b^11*c*d^3*e^9*z^3 + 370*a*b^11*c^4*d^9*e^3*z^3
- 294*a^5*b^10*c*d^2*e^10*z^3 - 294*a*b^10*c^5*d^10*e^2*z^3 - 240*a^3*b^12*c*d^4*e^8*z^3 - 240*a*b^12*c^3*d^8*
e^4*z^3 + 60*a^2*b^13*c*d^5*e^7*z^3 + 60*a*b^13*c^2*d^7*e^5*z^3 + 6144*a^10*b*c^5*d*e^11*z^3 + 6144*a^5*b*c^10
*d^11*e*z^3 + 120*a^6*b^9*c*d*e^11*z^3 + 120*a*b^9*c^6*d^11*e*z^3 + 10*a*b^14*c*d^6*e^6*z^3 + 71680*a^6*b^4*c^
6*d^6*e^6*z^3 - 66560*a^7*b^2*c^7*d^6*e^6*z^3 + 51840*a^7*b^4*c^5*d^4*e^8*z^3 + 51840*a^5*b^4*c^7*d^8*e^4*z^3
- 42240*a^8*b^2*c^6*d^4*e^8*z^3 - 42240*a^6*b^2*c^8*d^8*e^4*z^3 - 32256*a^6*b^5*c^5*d^5*e^7*z^3 - 32256*a^5*b^
5*c^6*d^7*e^5*z^3 + 21120*a^5*b^7*c^4*d^5*e^7*z^3 + 21120*a^4*b^7*c^5*d^7*e^5*z^3 - 17920*a^8*b^3*c^5*d^3*e^9*
z^3 - 17920*a^5*b^3*c^8*d^9*e^3*z^3 - 17024*a^5*b^6*c^5*d^6*e^6*z^3 - 16800*a^6*b^6*c^4*d^4*e^8*z^3 - 16800*a^
4*b^6*c^6*d^8*e^4*z^3 + 15360*a^8*b^4*c^4*d^2*e^10*z^3 - 15360*a^7*b^3*c^6*d^5*e^7*z^3 - 15360*a^6*b^3*c^7*d^7
*e^5*z^3 + 15360*a^4*b^4*c^8*d^10*e^2*z^3 - 8640*a^7*b^6*c^3*d^2*e^10*z^3 - 8640*a^3*b^6*c^7*d^10*e^2*z^3 + 80
00*a^6*b^7*c^3*d^3*e^9*z^3 + 8000*a^3*b^7*c^6*d^9*e^3*z^3 - 7680*a^9*b^2*c^5*d^2*e^10*z^3 - 7680*a^5*b^2*c^9*d
^10*e^2*z^3 - 6400*a^7*b^5*c^4*d^3*e^9*z^3 - 6400*a^4*b^5*c^7*d^9*e^3*z^3 - 4560*a^4*b^9*c^3*d^5*e^7*z^3 - 456
0*a^3*b^9*c^4*d^7*e^5*z^3 - 3920*a^4*b^8*c^4*d^6*e^6*z^3 - 2600*a^5*b^9*c^2*d^3*e^9*z^3 - 2600*a^2*b^9*c^5*d^9
*e^3*z^3 + 2380*a^3*b^10*c^3*d^6*e^6*z^3 + 2280*a^6*b^8*c^2*d^2*e^10*z^3 + 2280*a^2*b^8*c^6*d^10*e^2*z^3 + 121
5*a^4*b^10*c^2*d^4*e^8*z^3 + 1215*a^2*b^10*c^4*d^8*e^4*z^3 - 350*a^2*b^12*c^2*d^6*e^6*z^3 - 300*a^5*b^8*c^3*d^
4*e^8*z^3 - 300*a^3*b^8*c^5*d^8*e^4*z^3 + 180*a^3*b^11*c^2*d^5*e^7*z^3 + 180*a^2*b^11*c^3*d^7*e^5*z^3 - 6*b^15
*c*d^7*e^5*z^3 - 6*b^11*c^5*d^11*e*z^3 - 6*a^5*b^11*d*e^11*z^3 - 6*a*b^15*d^5*e^7*z^3 - 20*a^7*b^8*c*e^12*z^3
- 20*a*b^8*c^7*d^12*z^3 - 20*b^13*c^3*d^9*e^3*z^3 + 15*b^14*c^2*d^8*e^4*z^3 + 15*b^12*c^4*d^10*e^2*z^3 - 20480
*a^8*c^8*d^6*e^6*z^3 - 15360*a^9*c^7*d^4*e^8*z^3 - 15360*a^7*c^9*d^8*e^4*z^3 - 6144*a^10*c^6*d^2*e^10*z^3 - 61
44*a^6*c^10*d^10*e^2*z^3 - 20*a^3*b^13*d^3*e^9*z^3 + 15*a^4*b^12*d^2*e^10*z^3 + 15*a^2*b^14*d^4*e^8*z^3 + 1280
*a^10*b^2*c^4*e^12*z^3 - 640*a^9*b^4*c^3*e^12*z^3 + 160*a^8*b^6*c^2*e^12*z^3 + 1280*a^4*b^2*c^10*d^12*z^3 - 64
0*a^3*b^4*c^9*d^12*z^3 + 160*a^2*b^6*c^8*d^12*z^3 - 1024*a^11*c^5*e^12*z^3 - 1024*a^5*c^11*d^12*z^3 + b^16*d^6
*e^6*z^3 + b^10*c^6*d^12*z^3 + a^6*b^10*e^12*z^3 + 132*a*b*c^8*d^8*e^2*f*g*z + 1960*a^2*b^3*c^5*d^4*e^6*f*g*z
- 1560*a^3*b^2*c^5*d^3*e^7*f*g*z - 1500*a^2*b^2*c^6*d^5*e^5*f*g*z + 960*a^3*b^3*c^4*d^2*e^8*f*g*z - 420*a^2*b^
4*c^4*d^3*e^7*f*g*z - 222*a^2*b^5*c^3*d^2*e^8*f*g*z - 40*a*b^8*c*d*e^9*f*g*z + 1830*a^4*b^2*c^4*d*e^9*f*g*z +
1440*a*b^3*c^6*d^6*e^4*f*g*z - 1080*a^3*b^4*c^3*d*e^9*f*g*z - 856*a*b^2*c^7*d^7*e^3*f*g*z - 840*a*b^4*c^5*d^5*
e^5*f*g*z + 302*a^2*b^6*c^2*d*e^9*f*g*z + 180*a^4*b*c^5*d^2*e^8*f*g*z - 120*a^3*b*c^6*d^4*e^6*f*g*z + 84*a*b^6
*c^3*d^3*e^7*f*g*z - 24*a^2*b*c^7*d^6*e^4*f*g*z + 18*a*b^7*c^2*d^2*e^8*f*g*z - 2*a*b^5*c^4*d^4*e^6*f*g*z + 24*
a*c^9*d^9*e*f*g*z + 372*b^3*c^7*d^8*e^2*f*g*z - 340*b^4*c^6*d^7*e^3*f*g*z + 114*b^5*c^5*d^6*e^4*f*g*z + 12*b^6
*c^4*d^5*e^5*f*g*z - 6*b^8*c^2*d^3*e^7*f*g*z - 2*b^7*c^3*d^4*e^6*f*g*z + 528*a^3*c^7*d^5*e^5*f*g*z + 480*a^4*c
^6*d^3*e^7*f*g*z + 224*a^2*c^8*d^7*e^3*f*g*z - 60*a^4*b^3*c^3*e^10*f*g*z + 6*a^3*b^5*c^2*e^10*f*g*z + 36*a^5*b
*c^4*d*e^9*g^2*z + 20*a*b^8*c*d^2*e^8*g^2*z + 960*a*b*c^8*d^7*e^3*f^2*z + 900*a^4*b*c^5*d*e^9*f^2*z - 1185*a^4
*b^2*c^4*d^2*e^8*g^2*z + 450*a^3*b^4*c^3*d^2*e^8*g^2*z - 420*a^2*b^4*c^4*d^4*e^6*g^2*z + 300*a^3*b^2*c^5*d^4*e
^6*g^2*z + 210*a^2*b^2*c^6*d^6*e^4*g^2*z + 192*a^2*b^5*c^3*d^3*e^7*g^2*z - 142*a^2*b^6*c^2*d^2*e^8*g^2*z + 100
*a^2*b^3*c^5*d^5*e^5*g^2*z + 60*a^3*b^3*c^4*d^3*e^7*g^2*z - 1950*a^2*b^2*c^6*d^4*e^6*f^2*z - 900*a^3*b^2*c^5*d
^2*e^8*f^2*z + 300*a^2*b^4*c^4*d^2*e^8*f^2*z + 100*a^2*b^3*c^5*d^3*e^7*f^2*z - 186*b^2*c^8*d^9*e*f*g*z - 1896*
a^5*c^5*d*e^9*f*g*z + 180*a^5*b*c^4*e^10*f*g*z - 12*a*b*c^8*d^9*e*g^2*z - 390*a*b^4*c^5*d^6*e^4*g^2*z + 298*a*
b^5*c^4*d^5*e^5*g^2*z + 180*a*b^3*c^6*d^7*e^3*g^2*z - 120*a^3*b*c^6*d^5*e^5*g^2*z - 96*a^2*b*c^7*d^7*e^3*g^2*z
+ 60*a^4*b^3*c^3*d*e^9*g^2*z - 54*a*b^6*c^3*d^4*e^6*g^2*z - 18*a*b^7*c^2*d^3*e^7*g^2*z - 6*a^3*b^5*c^2*d*e^9*
g^2*z - 4*a*b^2*c^7*d^8*e^2*g^2*z + 2400*a^3*b*c^6*d^3*e^7*f^2*z + 2280*a^2*b*c^7*d^5*e^5*f^2*z - 1300*a*b^2*c
^7*d^6*e^4*f^2*z + 540*a*b^3*c^6*d^5*e^5*f^2*z - 300*a^3*b^3*c^4*d*e^9*f^2*z + 150*a*b^4*c^5*d^4*e^6*f^2*z - 8
0*a*b^5*c^4*d^3*e^7*f^2*z + 30*a^2*b^5*c^3*d*e^9*f^2*z - 30*a*b^6*c^3*d^2*e^8*f^2*z + 180*b*c^9*d^9*e*f^2*z +
20*a*b^8*c*e^10*f^2*z - 100*b^4*c^6*d^8*e^2*g^2*z + 96*b^5*c^5*d^7*e^3*g^2*z - 33*b^6*c^4*d^6*e^4*g^2*z - 8*b^
7*c^3*d^5*e^5*g^2*z + 6*b^8*c^2*d^4*e^6*g^2*z + 912*a^5*c^5*d^2*e^8*g^2*z - 345*b^2*c^8*d^8*e^2*f^2*z + 300*b^
3*c^7*d^7*e^3*f^2*z - 120*a^4*c^6*d^4*e^6*g^2*z - 100*b^4*c^6*d^6*e^4*f^2*z - 48*a^3*c^7*d^6*e^4*g^2*z - 15*b^
6*c^4*d^4*e^6*f^2*z + 10*b^7*c^3*d^3*e^7*f^2*z + 6*b^5*c^5*d^5*e^5*f^2*z - 4*a^2*c^8*d^8*e^2*g^2*z - 1200*a^3*
c^7*d^4*e^6*f^2*z - 900*a^4*c^6*d^2*e^8*f^2*z - 760*a^2*c^8*d^6*e^4*f^2*z - 1185*a^4*b^2*c^4*e^10*f^2*z + 630*
a^3*b^4*c^3*e^10*f^2*z - 160*a^2*b^6*c^2*e^10*f^2*z + 2*b^10*d*e^9*f*g*z + 36*b*c^9*d^10*f*g*z + 48*b^3*c^7*d^
9*e*g^2*z - 240*a*c^9*d^8*e^2*f^2*z - b^10*d^2*e^8*g^2*z - 36*a^6*c^4*e^10*g^2*z - 9*b^2*c^8*d^10*g^2*z + 768*
a^5*c^5*e^10*f^2*z - 36*c^10*d^10*f^2*z - b^10*e^10*f^2*z - 177*a*b^2*c^4*d^2*e^7*f*g^2 + 285*a*b^2*c^4*d*e^8*
f^2*g + 252*a^2*b*c^4*d*e^8*f*g^2 - 120*a*b^3*c^3*d*e^8*f*g^2 + 108*a*b*c^5*d^3*e^6*f*g^2 + 36*a*b*c^5*d^2*e^7
*f^2*g - 132*a*b*c^5*d*e^8*f^3 - 69*b^2*c^5*d^4*e^5*f*g^2 + 57*b^2*c^5*d^3*e^6*f^2*g - 45*b^3*c^4*d^2*e^7*f^2*
g + 30*b^4*c^3*d^2*e^7*f*g^2 + 9*b^3*c^4*d^3*e^6*f*g^2 + 156*a^2*c^5*d^2*e^7*f*g^2 - 72*a^2*b*c^4*d^2*e^7*g^3
+ 60*a*b^3*c^3*d^2*e^7*g^3 - 13*a*b^2*c^4*d^3*e^6*g^3 + 36*b*c^6*d^5*e^4*f*g^2 + 36*b*c^6*d^4*e^5*f^2*g - 30*b
^4*c^3*d*e^8*f^2*g + 12*b^5*c^2*d*e^8*f*g^2 - 408*a^2*c^5*d*e^8*f^2*g - 156*a*c^6*d^3*e^6*f^2*g + 24*a*c^6*d^4
*e^5*f*g^2 - 180*a^2*b*c^4*e^9*f^2*g + 60*a*b^3*c^3*e^9*f^2*g - 12*a*b*c^5*d^4*e^5*g^3 - 36*c^7*d^5*e^4*f^2*g
- 6*b^5*c^2*e^9*f^2*g + 36*a^3*c^4*e^9*f*g^2 - 72*b*c^6*d^3*e^6*f^3 - 36*a^3*c^4*d*e^8*g^3 + 15*b^3*c^4*d*e^8*
f^3 + 132*a*c^6*d^2*e^7*f^3 - 95*a*b^2*c^4*e^9*f^3 + 21*b^3*c^4*d^4*e^5*g^3 - 10*b^4*c^3*d^3*e^6*g^3 - 9*b^2*c
^5*d^5*e^4*g^3 - 6*b^5*c^2*d^2*e^7*g^3 + 21*b^2*c^5*d^2*e^7*f^3 - 4*a^2*c^5*d^3*e^6*g^3 + 36*c^7*d^4*e^5*f^3 +
10*b^4*c^3*e^9*f^3 + 256*a^2*c^5*e^9*f^3, z, k)*((a^5*b^9*c*e^11 + 256*a^9*b*c^5*e^11 - 2048*a^9*c^6*d*e^10 +
b^9*c^6*d^10*e + b^14*c*d^5*e^6 - 16*a^6*b^7*c^2*e^11 + 96*a^7*b^5*c^3*e^11 - 256*a^8*b^3*c^4*e^11 - 2048*a^5
*c^10*d^9*e^2 - 8192*a^6*c^9*d^7*e^4 - 12288*a^7*c^8*d^5*e^6 - 8192*a^8*c^7*d^3*e^8 - 3*b^10*c^5*d^9*e^2 + 2*b
^11*c^4*d^8*e^3 + 2*b^12*c^3*d^7*e^4 - 3*b^13*c^2*d^6*e^5 - 160*a^2*b^6*c^7*d^9*e^2 - 400*a^2*b^7*c^6*d^8*e^3
+ 1120*a^2*b^8*c^5*d^7*e^4 - 790*a^2*b^9*c^4*d^6*e^5 + 46*a^2*b^10*c^3*d^5*e^6 + 86*a^2*b^11*c^2*d^4*e^7 + 304
0*a^3*b^5*c^7*d^8*e^3 - 5760*a^3*b^6*c^6*d^7*e^4 + 2720*a^3*b^7*c^5*d^6*e^5 + 1136*a^3*b^8*c^4*d^5*e^6 - 790*a
^3*b^9*c^3*d^4*e^7 - 92*a^3*b^10*c^2*d^3*e^8 + 1280*a^4*b^2*c^9*d^9*e^2 - 8960*a^4*b^3*c^8*d^8*e^3 + 12800*a^4
*b^4*c^7*d^7*e^4 - 320*a^4*b^5*c^6*d^6*e^5 - 8896*a^4*b^6*c^5*d^5*e^6 + 2720*a^4*b^7*c^4*d^4*e^7 + 1120*a^4*b^
8*c^3*d^3*e^8 + 5*a^4*b^9*c^2*d^2*e^9 - 7168*a^5*b^2*c^8*d^7*e^4 - 17408*a^5*b^3*c^7*d^6*e^5 + 23552*a^5*b^4*c
^6*d^5*e^6 - 320*a^5*b^5*c^5*d^4*e^7 - 5760*a^5*b^6*c^4*d^3*e^8 - 400*a^5*b^7*c^3*d^2*e^9 - 16896*a^6*b^2*c^7*
d^5*e^6 - 17408*a^6*b^3*c^6*d^4*e^7 + 12800*a^6*b^4*c^5*d^3*e^8 + 3040*a^6*b^5*c^4*d^2*e^9 - 7168*a^7*b^2*c^6*
d^3*e^8 - 8960*a^7*b^3*c^5*d^2*e^9 - 16*a*b^7*c^7*d^10*e - 3*a*b^13*c*d^4*e^7 + 256*a^4*b*c^10*d^10*e - 3*a^4*
b^10*c*d*e^10 + 40*a*b^8*c^6*d^9*e^2 + 5*a*b^9*c^5*d^8*e^3 - 92*a*b^10*c^4*d^7*e^4 + 86*a*b^11*c^3*d^6*e^5 - 2
0*a*b^12*c^2*d^5*e^6 + 96*a^2*b^5*c^8*d^10*e + 2*a^2*b^12*c*d^3*e^8 - 256*a^3*b^3*c^9*d^10*e + 2*a^3*b^11*c*d^
2*e^9 + 9472*a^5*b*c^9*d^8*e^3 + 40*a^5*b^8*c^2*d*e^10 + 27136*a^6*b*c^8*d^6*e^5 - 160*a^6*b^6*c^3*d*e^10 + 27
136*a^7*b*c^7*d^4*e^7 + 9472*a^8*b*c^6*d^2*e^9 + 1280*a^8*b^2*c^5*d*e^10)/(a^4*b^8*e^8 + 256*a^4*c^8*d^8 + 256
*a^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 16*a^5*b^6*c*e^8 - 4*a*b^11*d^3*e^5 - 4*a^3*b^9
*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8 - 256*a^3*b^2*c^7*d^8 + 96*a^6*b^4*c^2*e^8 -
256*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c^7*d^6*e^2 + 1536*a^6*c^6*d^4*e^4 + 1024*a^7*c^5*d^2*e^6
+ 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3*d^5*e^3 - 90*a^2*b^8*c^2*d^4*e^4 - 1152*a^3*b
^4*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^6*c^3*d^4*e^4 - 192*a^3*b^7*c^2*d^3*e^5 + 512*a^4*b^2*c^6
*d^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^4*d^4*e^4 - 128*a^4*b^5*c^3*d^3*e^5 + 512*a^4*b^6*c^2*d^2
*e^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^3*e^5 - 1152*a^5*b^4*c^3*d^2*e^6 + 512*a^6*b^2*c^4*d^2*e^
6 + 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^4*b*c^7*d^7*e + 64*a^4*b^7*c*d*e^7 - 1024*a^7*b*c^4*d*e^7
- 92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*a^2*b^5*c^5*d^7*e + 52*a^2*b^9*c*d^3*e^5 + 1024*a^3*b^3*c
^6*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*e^3 - 384*a^5*b^5*c^2*d*e^7 - 3072*a^6*b*c^5*d^3*e^5 + 10
24*a^6*b^3*c^3*d*e^7) - (x*(1536*a^9*c^6*e^11 - 2*a^4*b^10*c*e^11 - 512*a^4*c^11*d^10*e - 2*b^8*c^7*d^10*e - 2
*b^14*c*d^4*e^7 + 38*a^5*b^8*c^2*e^11 - 288*a^6*b^6*c^3*e^11 + 1088*a^7*b^4*c^4*e^11 - 2048*a^8*b^2*c^5*e^11 -
512*a^5*c^10*d^8*e^3 + 3072*a^6*c^9*d^6*e^5 + 7168*a^7*c^8*d^4*e^7 + 5632*a^8*c^7*d^2*e^9 + 10*b^9*c^6*d^9*e^
2 - 22*b^10*c^5*d^8*e^3 + 28*b^11*c^4*d^7*e^4 - 22*b^12*c^3*d^6*e^5 + 10*b^13*c^2*d^5*e^6 + 960*a^2*b^5*c^8*d^
9*e^2 - 2080*a^2*b^6*c^7*d^8*e^3 + 2560*a^2*b^7*c^6*d^7*e^4 - 1780*a^2*b^8*c^5*d^6*e^5 + 412*a^2*b^9*c^4*d^5*e
^6 + 248*a^2*b^10*c^3*d^4*e^7 - 116*a^2*b^11*c^2*d^3*e^8 - 2560*a^3*b^3*c^9*d^9*e^2 + 5440*a^3*b^4*c^8*d^8*e^3
- 6400*a^3*b^5*c^7*d^7*e^4 + 3520*a^3*b^6*c^6*d^6*e^5 + 1088*a^3*b^7*c^5*d^5*e^6 - 2340*a^3*b^8*c^4*d^4*e^7 +
520*a^3*b^9*c^3*d^3*e^8 + 212*a^3*b^10*c^2*d^2*e^9 - 5120*a^4*b^2*c^9*d^8*e^3 + 5120*a^4*b^3*c^8*d^7*e^4 + 64
0*a^4*b^4*c^7*d^6*e^5 - 9088*a^4*b^5*c^6*d^5*e^6 + 8000*a^4*b^6*c^5*d^4*e^7 - 1450*a^4*b^8*c^3*d^2*e^9 - 8192*
a^5*b^2*c^8*d^6*e^5 + 17408*a^5*b^3*c^7*d^5*e^6 - 10112*a^5*b^4*c^6*d^4*e^7 - 6400*a^5*b^5*c^5*d^3*e^8 + 4640*
a^5*b^6*c^4*d^2*e^9 - 1024*a^6*b^2*c^7*d^4*e^7 + 17408*a^6*b^3*c^6*d^3*e^8 - 6080*a^6*b^4*c^5*d^2*e^9 - 512*a^
7*b^2*c^6*d^2*e^9 + 32*a*b^6*c^8*d^10*e + 8*a*b^13*c*d^3*e^8 + 8*a^3*b^11*c*d*e^10 - 5632*a^8*b*c^6*d*e^10 - 1
60*a*b^7*c^7*d^9*e^2 + 350*a*b^8*c^6*d^8*e^3 - 440*a*b^9*c^5*d^7*e^4 + 332*a*b^10*c^4*d^6*e^5 - 128*a*b^11*c^3
*d^5*e^6 + 6*a*b^12*c^2*d^4*e^7 - 192*a^2*b^4*c^9*d^10*e - 12*a^2*b^12*c*d^2*e^9 + 512*a^3*b^2*c^10*d^10*e + 2
560*a^4*b*c^10*d^9*e^2 - 150*a^4*b^9*c^2*d*e^10 + 2048*a^5*b*c^9*d^7*e^4 + 1120*a^5*b^7*c^3*d*e^10 - 9216*a^6*
b*c^8*d^5*e^6 - 4160*a^6*b^5*c^4*d*e^10 - 14336*a^7*b*c^7*d^3*e^8 + 7680*a^7*b^3*c^5*d*e^10))/(a^4*b^8*e^8 + 2
56*a^4*c^8*d^8 + 256*a^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 16*a^5*b^6*c*e^8 - 4*a*b^11
*d^3*e^5 - 4*a^3*b^9*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8 - 256*a^3*b^2*c^7*d^8 + 9
6*a^6*b^4*c^2*e^8 - 256*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c^7*d^6*e^2 + 1536*a^6*c^6*d^4*e^4 + 1
024*a^7*c^5*d^2*e^6 + 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3*d^5*e^3 - 90*a^2*b^8*c^2*
d^4*e^4 - 1152*a^3*b^4*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^6*c^3*d^4*e^4 - 192*a^3*b^7*c^2*d^3*e
^5 + 512*a^4*b^2*c^6*d^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^4*d^4*e^4 - 128*a^4*b^5*c^3*d^3*e^5 +
512*a^4*b^6*c^2*d^2*e^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^3*e^5 - 1152*a^5*b^4*c^3*d^2*e^6 + 51
2*a^6*b^2*c^4*d^2*e^6 + 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^4*b*c^7*d^7*e + 64*a^4*b^7*c*d*e^7 -
1024*a^7*b*c^4*d*e^7 - 92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*a^2*b^5*c^5*d^7*e + 52*a^2*b^9*c*d^3*
e^5 + 1024*a^3*b^3*c^6*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*e^3 - 384*a^5*b^5*c^2*d*e^7 - 3072*a^
6*b*c^5*d^3*e^5 + 1024*a^6*b^3*c^3*d*e^7)) + (x*(768*a^6*c^6*e^10*f - 96*a^6*b*c^5*e^10*g - 576*a^6*c^6*d*e^9*
g + 2*a^2*b^8*c^2*e^10*f - 33*a^3*b^6*c^3*e^10*f + 216*a^4*b^4*c^4*e^10*f - 656*a^5*b^2*c^5*e^10*f - 6*a^4*b^5
*c^3*e^10*g + 48*a^5*b^3*c^4*e^10*g + 192*a^2*c^10*d^8*e^2*f + 832*a^3*c^9*d^6*e^4*f + 1856*a^4*c^8*d^4*e^6*f
+ 1984*a^5*c^7*d^2*e^8*f - 64*a^3*c^9*d^7*e^3*g - 704*a^4*c^8*d^5*e^5*g - 1216*a^5*c^7*d^3*e^7*g + 12*b^4*c^8*
d^8*e^2*f - 48*b^5*c^7*d^7*e^3*f + 71*b^6*c^6*d^6*e^4*f - 45*b^7*c^5*d^5*e^5*f + 11*b^8*c^4*d^4*e^6*f - 3*b^9*
c^3*d^3*e^7*f + 2*b^10*c^2*d^2*e^8*f - 6*b^5*c^7*d^8*e^2*g + 25*b^6*c^6*d^7*e^3*g - 39*b^7*c^5*d^6*e^4*g + 25*
b^8*c^4*d^5*e^5*g - 3*b^9*c^3*d^4*e^6*g - 2*b^10*c^2*d^3*e^7*g - 96*a*b^2*c^9*d^8*e^2*f + 384*a*b^3*c^8*d^7*e^
3*f - 516*a*b^4*c^7*d^6*e^4*f + 204*a*b^5*c^6*d^5*e^5*f + 49*a*b^6*c^5*d^4*e^6*f + 10*a*b^7*c^4*d^3*e^7*f - 31
*a*b^8*c^3*d^2*e^8*f - 768*a^2*b*c^9*d^7*e^3*f + 67*a^2*b^7*c^3*d*e^9*f - 2496*a^3*b*c^8*d^5*e^5*f - 468*a^3*b
^5*c^4*d*e^9*f - 3712*a^4*b*c^7*d^3*e^7*f + 1552*a^4*b^3*c^5*d*e^9*f + 48*a*b^3*c^8*d^8*e^2*g - 204*a*b^4*c^7*
d^7*e^3*g + 300*a*b^5*c^6*d^6*e^4*g - 121*a*b^6*c^5*d^5*e^5*g - 82*a*b^7*c^4*d^4*e^6*g + 55*a*b^8*c^3*d^3*e^7*
g + 4*a*b^9*c^2*d^2*e^8*g - 96*a^2*b*c^9*d^8*e^2*g - 2*a^2*b^8*c^2*d*e^9*g - 192*a^3*b*c^8*d^6*e^4*g + 57*a^3*
b^6*c^3*d*e^9*g + 832*a^4*b*c^7*d^4*e^6*g - 396*a^4*b^4*c^4*d*e^9*g + 832*a^5*b*c^6*d^2*e^8*g + 944*a^5*b^2*c^
5*d*e^9*g + 720*a^2*b^2*c^8*d^6*e^4*f + 528*a^2*b^3*c^7*d^5*e^5*f - 804*a^2*b^4*c^6*d^4*e^6*f - 168*a^2*b^5*c^
5*d^3*e^7*f + 233*a^2*b^6*c^4*d^2*e^8*f + 1264*a^3*b^2*c^7*d^4*e^6*f + 1632*a^3*b^3*c^6*d^3*e^7*f - 764*a^3*b^
4*c^5*d^2*e^8*f + 304*a^4*b^2*c^6*d^2*e^8*f + 432*a^2*b^2*c^8*d^7*e^3*g - 528*a^2*b^3*c^7*d^6*e^4*g - 276*a^2*
b^4*c^6*d^5*e^5*g + 852*a^2*b^5*c^5*d^4*e^6*g - 281*a^2*b^6*c^4*d^3*e^7*g - 103*a^2*b^7*c^3*d^2*e^8*g + 1616*a
^3*b^2*c^7*d^5*e^5*g - 2112*a^3*b^3*c^6*d^4*e^6*g + 44*a^3*b^4*c^5*d^3*e^7*g + 684*a^3*b^5*c^4*d^2*e^8*g + 161
6*a^4*b^2*c^6*d^3*e^7*g - 1552*a^4*b^3*c^5*d^2*e^8*g - 4*a*b^9*c^2*d*e^9*f - 1984*a^5*b*c^6*d*e^9*f))/(a^4*b^8
*e^8 + 256*a^4*c^8*d^8 + 256*a^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 16*a^5*b^6*c*e^8 -
4*a*b^11*d^3*e^5 - 4*a^3*b^9*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8 - 256*a^3*b^2*c^7
*d^8 + 96*a^6*b^4*c^2*e^8 - 256*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c^7*d^6*e^2 + 1536*a^6*c^6*d^4
*e^4 + 1024*a^7*c^5*d^2*e^6 + 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3*d^5*e^3 - 90*a^2*
b^8*c^2*d^4*e^4 - 1152*a^3*b^4*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^6*c^3*d^4*e^4 - 192*a^3*b^7*c
^2*d^3*e^5 + 512*a^4*b^2*c^6*d^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^4*d^4*e^4 - 128*a^4*b^5*c^3*d
^3*e^5 + 512*a^4*b^6*c^2*d^2*e^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^3*e^5 - 1152*a^5*b^4*c^3*d^2*
e^6 + 512*a^6*b^2*c^4*d^2*e^6 + 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^4*b*c^7*d^7*e + 64*a^4*b^7*c*
d*e^7 - 1024*a^7*b*c^4*d*e^7 - 92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*a^2*b^5*c^5*d^7*e + 52*a^2*b^
9*c*d^3*e^5 + 1024*a^3*b^3*c^6*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*e^3 - 384*a^5*b^5*c^2*d*e^7 -
3072*a^6*b*c^5*d^3*e^5 + 1024*a^6*b^3*c^3*d*e^7)) - (6*b^3*c^6*d^4*e^5*f^2 - 36*c^9*d^7*e^2*f^2 - 72*a^2*b^3*
c^4*e^9*f^2 - 292*a^2*c^7*d^3*e^6*f^2 - 4*a^2*c^7*d^5*e^4*g^2 - 8*a^3*c^6*d^3*e^6*g^2 - 93*b^2*c^7*d^5*e^4*f^2
- b^7*c^2*e^9*f^2 + 11*b^4*c^5*d^3*e^6*f^2 + 7*b^5*c^4*d^2*e^7*f^2 - 9*b^2*c^7*d^7*e^2*g^2 + 30*b^3*c^6*d^6*e
^3*g^2 - 31*b^4*c^5*d^5*e^4*g^2 + 7*b^5*c^4*d^4*e^5*g^2 + 4*b^6*c^3*d^3*e^6*g^2 - b^7*c^2*d^2*e^7*g^2 - 96*a^4
*c^5*e^9*f*g + 15*a*b^5*c^3*e^9*f^2 + 112*a^3*b*c^5*e^9*f^2 - 168*a*c^8*d^5*e^4*f^2 - 224*a^3*c^6*d*e^8*f^2 +
108*b*c^8*d^6*e^3*f^2 + 60*a^4*c^5*d*e^8*g^2 - 2*b^6*c^3*d*e^8*f^2 + 336*a*b*c^7*d^4*e^5*f^2 + 8*a*b^4*c^4*d*e
^8*f^2 - 12*a*b*c^7*d^6*e^3*g^2 - 6*a^2*b^4*c^3*e^9*f*g + 48*a^3*b^2*c^4*e^9*f*g + 80*a^2*c^7*d^4*e^5*f*g + 88
*a^3*c^6*d^2*e^7*f*g - 114*b^2*c^7*d^6*e^3*f*g + 108*b^3*c^6*d^5*e^4*f*g - 16*b^4*c^5*d^4*e^5*f*g - 14*b^5*c^4
*d^3*e^6*f*g - 2*b^6*c^3*d^2*e^7*f*g - 106*a*b^2*c^6*d^3*e^6*f^2 - 86*a*b^3*c^5*d^2*e^7*f^2 + 340*a^2*b*c^6*d^
2*e^7*f^2 + 47*a^2*b^2*c^5*d*e^8*f^2 - 10*a*b^2*c^6*d^5*e^4*g^2 + 70*a*b^3*c^5*d^4*e^5*g^2 - 52*a*b^4*c^4*d^3*
e^6*g^2 + 3*a*b^5*c^3*d^2*e^7*g^2 - 32*a^2*b*c^6*d^4*e^5*g^2 + 6*a^2*b^4*c^3*d*e^8*g^2 - 20*a^3*b*c^5*d^2*e^7*
g^2 - 48*a^3*b^2*c^4*d*e^8*g^2 + 24*a*c^8*d^6*e^3*f*g + 36*b*c^8*d^7*e^2*f*g + 2*b^7*c^2*d*e^8*f*g + 11*a^2*b^
2*c^5*d^3*e^6*g^2 + 36*a^2*b^3*c^4*d^2*e^7*g^2 + 108*a*b*c^7*d^5*e^4*f*g - 18*a*b^5*c^3*d*e^8*f*g + 52*a^3*b*c
^5*d*e^8*f*g - 316*a*b^2*c^6*d^4*e^5*f*g + 160*a*b^3*c^5*d^3*e^6*f*g + 44*a*b^4*c^4*d^2*e^7*f*g + 124*a^2*b*c^
6*d^3*e^6*f*g + 36*a^2*b^3*c^4*d*e^8*f*g - 274*a^2*b^2*c^5*d^2*e^7*f*g)/(a^4*b^8*e^8 + 256*a^4*c^8*d^8 + 256*a
^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 16*a^5*b^6*c*e^8 - 4*a*b^11*d^3*e^5 - 4*a^3*b^9*d
*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8 - 256*a^3*b^2*c^7*d^8 + 96*a^6*b^4*c^2*e^8 - 25
6*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c^7*d^6*e^2 + 1536*a^6*c^6*d^4*e^4 + 1024*a^7*c^5*d^2*e^6 +
6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3*d^5*e^3 - 90*a^2*b^8*c^2*d^4*e^4 - 1152*a^3*b^4
*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^6*c^3*d^4*e^4 - 192*a^3*b^7*c^2*d^3*e^5 + 512*a^4*b^2*c^6*d
^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^4*d^4*e^4 - 128*a^4*b^5*c^3*d^3*e^5 + 512*a^4*b^6*c^2*d^2*e
^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^3*e^5 - 1152*a^5*b^4*c^3*d^2*e^6 + 512*a^6*b^2*c^4*d^2*e^6
+ 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^4*b*c^7*d^7*e + 64*a^4*b^7*c*d*e^7 - 1024*a^7*b*c^4*d*e^7 -
92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*a^2*b^5*c^5*d^7*e + 52*a^2*b^9*c*d^3*e^5 + 1024*a^3*b^3*c^6
*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*e^3 - 384*a^5*b^5*c^2*d*e^7 - 3072*a^6*b*c^5*d^3*e^5 + 1024
*a^6*b^3*c^3*d*e^7) + (x*(36*a^4*c^5*e^9*g^2 + b^6*c^3*e^9*f^2 + 36*c^9*d^6*e^3*f^2 + 49*a^2*b^2*c^5*e^9*f^2 +
196*a^2*c^7*d^2*e^7*f^2 + 4*a^2*c^7*d^4*e^5*g^2 - 24*a^3*c^6*d^2*e^7*g^2 + 93*b^2*c^7*d^4*e^5*f^2 - 6*b^3*c^6
*d^3*e^6*f^2 - 17*b^4*c^5*d^2*e^7*f^2 + 9*b^2*c^7*d^6*e^3*g^2 - 30*b^3*c^6*d^5*e^4*g^2 + 31*b^4*c^5*d^4*e^5*g^
2 - 10*b^5*c^4*d^3*e^6*g^2 + b^6*c^3*d^2*e^7*g^2 - 14*a*b^4*c^4*e^9*f^2 + 168*a*c^8*d^4*e^5*f^2 - 108*b*c^8*d^
5*e^4*f^2 + 2*b^5*c^4*d*e^8*f^2 - 336*a*b*c^7*d^3*e^6*f^2 + 14*a*b^3*c^5*d*e^8*f^2 - 196*a^2*b*c^6*d*e^8*f^2 +
12*a*b*c^7*d^5*e^4*g^2 - 60*a^3*b*c^5*d*e^8*g^2 + 12*a^2*b^3*c^4*e^9*f*g + 16*a^2*c^7*d^3*e^6*f*g + 114*b^2*c
^7*d^5*e^4*f*g - 108*b^3*c^6*d^4*e^5*f*g + 22*b^4*c^5*d^3*e^6*f*g + 8*b^5*c^4*d^2*e^7*f*g + 154*a*b^2*c^6*d^2*
e^7*f^2 + 10*a*b^2*c^6*d^4*e^5*g^2 - 46*a*b^3*c^5*d^3*e^6*g^2 + 10*a*b^4*c^4*d^2*e^7*g^2 - 16*a^2*b*c^6*d^3*e^
6*g^2 - 12*a^2*b^3*c^4*d*e^8*g^2 - 84*a^3*b*c^5*e^9*f*g - 24*a*c^8*d^5*e^4*f*g + 168*a^3*c^6*d*e^8*f*g - 36*b*
c^8*d^6*e^3*f*g - 2*b^6*c^3*d*e^8*f*g + 85*a^2*b^2*c^5*d^2*e^7*g^2 - 108*a*b*c^7*d^4*e^5*f*g + 4*a*b^4*c^4*d*e
^8*f*g + 268*a*b^2*c^6*d^3*e^6*f*g - 112*a*b^3*c^5*d^2*e^7*f*g - 220*a^2*b*c^6*d^2*e^7*f*g + 82*a^2*b^2*c^5*d*
e^8*f*g))/(a^4*b^8*e^8 + 256*a^4*c^8*d^8 + 256*a^8*c^4*e^8 + b^8*c^4*d^8 + b^12*d^4*e^4 - 16*a*b^6*c^5*d^8 - 1
6*a^5*b^6*c*e^8 - 4*a*b^11*d^3*e^5 - 4*a^3*b^9*d*e^7 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 96*a^2*b^4*c^6*d^8
- 256*a^3*b^2*c^7*d^8 + 96*a^6*b^4*c^2*e^8 - 256*a^7*b^2*c^3*e^8 + 6*a^2*b^10*d^2*e^6 + 1024*a^5*c^7*d^6*e^2
+ 1536*a^6*c^6*d^4*e^4 + 1024*a^7*c^5*d^2*e^6 + 6*b^10*c^2*d^6*e^2 + 512*a^2*b^6*c^4*d^6*e^2 - 192*a^2*b^7*c^3
*d^5*e^3 - 90*a^2*b^8*c^2*d^4*e^4 - 1152*a^3*b^4*c^5*d^6*e^2 - 128*a^3*b^5*c^4*d^5*e^3 + 800*a^3*b^6*c^3*d^4*e
^4 - 192*a^3*b^7*c^2*d^3*e^5 + 512*a^4*b^2*c^6*d^6*e^2 + 2048*a^4*b^3*c^5*d^5*e^3 - 2240*a^4*b^4*c^4*d^4*e^4 -
128*a^4*b^5*c^3*d^3*e^5 + 512*a^4*b^6*c^2*d^2*e^6 + 1536*a^5*b^2*c^5*d^4*e^4 + 2048*a^5*b^3*c^4*d^3*e^5 - 115
2*a^5*b^4*c^3*d^2*e^6 + 512*a^6*b^2*c^4*d^2*e^6 + 64*a*b^7*c^4*d^7*e - 4*a*b^10*c*d^4*e^4 - 1024*a^4*b*c^7*d^7
*e + 64*a^4*b^7*c*d*e^7 - 1024*a^7*b*c^4*d*e^7 - 92*a*b^8*c^3*d^6*e^2 + 52*a*b^9*c^2*d^5*e^3 - 384*a^2*b^5*c^5
*d^7*e + 52*a^2*b^9*c*d^3*e^5 + 1024*a^3*b^3*c^6*d^7*e - 92*a^3*b^8*c*d^2*e^6 - 3072*a^5*b*c^6*d^5*e^3 - 384*a
^5*b^5*c^2*d*e^7 - 3072*a^6*b*c^5*d^3*e^5 + 1024*a^6*b^3*c^3*d*e^7))*root(61440*a^8*b*c^7*d^5*e^7*z^3 + 61440*
a^7*b*c^8*d^7*e^5*z^3 + 30720*a^9*b*c^6*d^3*e^9*z^3 + 30720*a^6*b*c^9*d^9*e^3*z^3 - 7680*a^9*b^3*c^4*d*e^11*z^
3 - 7680*a^4*b^3*c^9*d^11*e*z^3 + 3840*a^8*b^5*c^3*d*e^11*z^3 + 3840*a^3*b^5*c^8*d^11*e*z^3 - 960*a^7*b^7*c^2*
d*e^11*z^3 - 960*a^2*b^7*c^7*d^11*e*z^3 + 370*a^4*b^11*c*d^3*e^9*z^3 + 370*a*b^11*c^4*d^9*e^3*z^3 - 294*a^5*b^
10*c*d^2*e^10*z^3 - 294*a*b^10*c^5*d^10*e^2*z^3 - 240*a^3*b^12*c*d^4*e^8*z^3 - 240*a*b^12*c^3*d^8*e^4*z^3 + 60
*a^2*b^13*c*d^5*e^7*z^3 + 60*a*b^13*c^2*d^7*e^5*z^3 + 6144*a^10*b*c^5*d*e^11*z^3 + 6144*a^5*b*c^10*d^11*e*z^3
+ 120*a^6*b^9*c*d*e^11*z^3 + 120*a*b^9*c^6*d^11*e*z^3 + 10*a*b^14*c*d^6*e^6*z^3 + 71680*a^6*b^4*c^6*d^6*e^6*z^
3 - 66560*a^7*b^2*c^7*d^6*e^6*z^3 + 51840*a^7*b^4*c^5*d^4*e^8*z^3 + 51840*a^5*b^4*c^7*d^8*e^4*z^3 - 42240*a^8*
b^2*c^6*d^4*e^8*z^3 - 42240*a^6*b^2*c^8*d^8*e^4*z^3 - 32256*a^6*b^5*c^5*d^5*e^7*z^3 - 32256*a^5*b^5*c^6*d^7*e^
5*z^3 + 21120*a^5*b^7*c^4*d^5*e^7*z^3 + 21120*a^4*b^7*c^5*d^7*e^5*z^3 - 17920*a^8*b^3*c^5*d^3*e^9*z^3 - 17920*
a^5*b^3*c^8*d^9*e^3*z^3 - 17024*a^5*b^6*c^5*d^6*e^6*z^3 - 16800*a^6*b^6*c^4*d^4*e^8*z^3 - 16800*a^4*b^6*c^6*d^
8*e^4*z^3 + 15360*a^8*b^4*c^4*d^2*e^10*z^3 - 15360*a^7*b^3*c^6*d^5*e^7*z^3 - 15360*a^6*b^3*c^7*d^7*e^5*z^3 + 1
5360*a^4*b^4*c^8*d^10*e^2*z^3 - 8640*a^7*b^6*c^3*d^2*e^10*z^3 - 8640*a^3*b^6*c^7*d^10*e^2*z^3 + 8000*a^6*b^7*c
^3*d^3*e^9*z^3 + 8000*a^3*b^7*c^6*d^9*e^3*z^3 - 7680*a^9*b^2*c^5*d^2*e^10*z^3 - 7680*a^5*b^2*c^9*d^10*e^2*z^3
- 6400*a^7*b^5*c^4*d^3*e^9*z^3 - 6400*a^4*b^5*c^7*d^9*e^3*z^3 - 4560*a^4*b^9*c^3*d^5*e^7*z^3 - 4560*a^3*b^9*c^
4*d^7*e^5*z^3 - 3920*a^4*b^8*c^4*d^6*e^6*z^3 - 2600*a^5*b^9*c^2*d^3*e^9*z^3 - 2600*a^2*b^9*c^5*d^9*e^3*z^3 + 2
380*a^3*b^10*c^3*d^6*e^6*z^3 + 2280*a^6*b^8*c^2*d^2*e^10*z^3 + 2280*a^2*b^8*c^6*d^10*e^2*z^3 + 1215*a^4*b^10*c
^2*d^4*e^8*z^3 + 1215*a^2*b^10*c^4*d^8*e^4*z^3 - 350*a^2*b^12*c^2*d^6*e^6*z^3 - 300*a^5*b^8*c^3*d^4*e^8*z^3 -
300*a^3*b^8*c^5*d^8*e^4*z^3 + 180*a^3*b^11*c^2*d^5*e^7*z^3 + 180*a^2*b^11*c^3*d^7*e^5*z^3 - 6*b^15*c*d^7*e^5*z
^3 - 6*b^11*c^5*d^11*e*z^3 - 6*a^5*b^11*d*e^11*z^3 - 6*a*b^15*d^5*e^7*z^3 - 20*a^7*b^8*c*e^12*z^3 - 20*a*b^8*c
^7*d^12*z^3 - 20*b^13*c^3*d^9*e^3*z^3 + 15*b^14*c^2*d^8*e^4*z^3 + 15*b^12*c^4*d^10*e^2*z^3 - 20480*a^8*c^8*d^6
*e^6*z^3 - 15360*a^9*c^7*d^4*e^8*z^3 - 15360*a^7*c^9*d^8*e^4*z^3 - 6144*a^10*c^6*d^2*e^10*z^3 - 6144*a^6*c^10*
d^10*e^2*z^3 - 20*a^3*b^13*d^3*e^9*z^3 + 15*a^4*b^12*d^2*e^10*z^3 + 15*a^2*b^14*d^4*e^8*z^3 + 1280*a^10*b^2*c^
4*e^12*z^3 - 640*a^9*b^4*c^3*e^12*z^3 + 160*a^8*b^6*c^2*e^12*z^3 + 1280*a^4*b^2*c^10*d^12*z^3 - 640*a^3*b^4*c^
9*d^12*z^3 + 160*a^2*b^6*c^8*d^12*z^3 - 1024*a^11*c^5*e^12*z^3 - 1024*a^5*c^11*d^12*z^3 + b^16*d^6*e^6*z^3 + b
^10*c^6*d^12*z^3 + a^6*b^10*e^12*z^3 + 132*a*b*c^8*d^8*e^2*f*g*z + 1960*a^2*b^3*c^5*d^4*e^6*f*g*z - 1560*a^3*b
^2*c^5*d^3*e^7*f*g*z - 1500*a^2*b^2*c^6*d^5*e^5*f*g*z + 960*a^3*b^3*c^4*d^2*e^8*f*g*z - 420*a^2*b^4*c^4*d^3*e^
7*f*g*z - 222*a^2*b^5*c^3*d^2*e^8*f*g*z - 40*a*b^8*c*d*e^9*f*g*z + 1830*a^4*b^2*c^4*d*e^9*f*g*z + 1440*a*b^3*c
^6*d^6*e^4*f*g*z - 1080*a^3*b^4*c^3*d*e^9*f*g*z - 856*a*b^2*c^7*d^7*e^3*f*g*z - 840*a*b^4*c^5*d^5*e^5*f*g*z +
302*a^2*b^6*c^2*d*e^9*f*g*z + 180*a^4*b*c^5*d^2*e^8*f*g*z - 120*a^3*b*c^6*d^4*e^6*f*g*z + 84*a*b^6*c^3*d^3*e^7
*f*g*z - 24*a^2*b*c^7*d^6*e^4*f*g*z + 18*a*b^7*c^2*d^2*e^8*f*g*z - 2*a*b^5*c^4*d^4*e^6*f*g*z + 24*a*c^9*d^9*e*
f*g*z + 372*b^3*c^7*d^8*e^2*f*g*z - 340*b^4*c^6*d^7*e^3*f*g*z + 114*b^5*c^5*d^6*e^4*f*g*z + 12*b^6*c^4*d^5*e^5
*f*g*z - 6*b^8*c^2*d^3*e^7*f*g*z - 2*b^7*c^3*d^4*e^6*f*g*z + 528*a^3*c^7*d^5*e^5*f*g*z + 480*a^4*c^6*d^3*e^7*f
*g*z + 224*a^2*c^8*d^7*e^3*f*g*z - 60*a^4*b^3*c^3*e^10*f*g*z + 6*a^3*b^5*c^2*e^10*f*g*z + 36*a^5*b*c^4*d*e^9*g
^2*z + 20*a*b^8*c*d^2*e^8*g^2*z + 960*a*b*c^8*d^7*e^3*f^2*z + 900*a^4*b*c^5*d*e^9*f^2*z - 1185*a^4*b^2*c^4*d^2
*e^8*g^2*z + 450*a^3*b^4*c^3*d^2*e^8*g^2*z - 420*a^2*b^4*c^4*d^4*e^6*g^2*z + 300*a^3*b^2*c^5*d^4*e^6*g^2*z + 2
10*a^2*b^2*c^6*d^6*e^4*g^2*z + 192*a^2*b^5*c^3*d^3*e^7*g^2*z - 142*a^2*b^6*c^2*d^2*e^8*g^2*z + 100*a^2*b^3*c^5
*d^5*e^5*g^2*z + 60*a^3*b^3*c^4*d^3*e^7*g^2*z - 1950*a^2*b^2*c^6*d^4*e^6*f^2*z - 900*a^3*b^2*c^5*d^2*e^8*f^2*z
+ 300*a^2*b^4*c^4*d^2*e^8*f^2*z + 100*a^2*b^3*c^5*d^3*e^7*f^2*z - 186*b^2*c^8*d^9*e*f*g*z - 1896*a^5*c^5*d*e^
9*f*g*z + 180*a^5*b*c^4*e^10*f*g*z - 12*a*b*c^8*d^9*e*g^2*z - 390*a*b^4*c^5*d^6*e^4*g^2*z + 298*a*b^5*c^4*d^5*
e^5*g^2*z + 180*a*b^3*c^6*d^7*e^3*g^2*z - 120*a^3*b*c^6*d^5*e^5*g^2*z - 96*a^2*b*c^7*d^7*e^3*g^2*z + 60*a^4*b^
3*c^3*d*e^9*g^2*z - 54*a*b^6*c^3*d^4*e^6*g^2*z - 18*a*b^7*c^2*d^3*e^7*g^2*z - 6*a^3*b^5*c^2*d*e^9*g^2*z - 4*a*
b^2*c^7*d^8*e^2*g^2*z + 2400*a^3*b*c^6*d^3*e^7*f^2*z + 2280*a^2*b*c^7*d^5*e^5*f^2*z - 1300*a*b^2*c^7*d^6*e^4*f
^2*z + 540*a*b^3*c^6*d^5*e^5*f^2*z - 300*a^3*b^3*c^4*d*e^9*f^2*z + 150*a*b^4*c^5*d^4*e^6*f^2*z - 80*a*b^5*c^4*
d^3*e^7*f^2*z + 30*a^2*b^5*c^3*d*e^9*f^2*z - 30*a*b^6*c^3*d^2*e^8*f^2*z + 180*b*c^9*d^9*e*f^2*z + 20*a*b^8*c*e
^10*f^2*z - 100*b^4*c^6*d^8*e^2*g^2*z + 96*b^5*c^5*d^7*e^3*g^2*z - 33*b^6*c^4*d^6*e^4*g^2*z - 8*b^7*c^3*d^5*e^
5*g^2*z + 6*b^8*c^2*d^4*e^6*g^2*z + 912*a^5*c^5*d^2*e^8*g^2*z - 345*b^2*c^8*d^8*e^2*f^2*z + 300*b^3*c^7*d^7*e^
3*f^2*z - 120*a^4*c^6*d^4*e^6*g^2*z - 100*b^4*c^6*d^6*e^4*f^2*z - 48*a^3*c^7*d^6*e^4*g^2*z - 15*b^6*c^4*d^4*e^
6*f^2*z + 10*b^7*c^3*d^3*e^7*f^2*z + 6*b^5*c^5*d^5*e^5*f^2*z - 4*a^2*c^8*d^8*e^2*g^2*z - 1200*a^3*c^7*d^4*e^6*
f^2*z - 900*a^4*c^6*d^2*e^8*f^2*z - 760*a^2*c^8*d^6*e^4*f^2*z - 1185*a^4*b^2*c^4*e^10*f^2*z + 630*a^3*b^4*c^3*
e^10*f^2*z - 160*a^2*b^6*c^2*e^10*f^2*z + 2*b^10*d*e^9*f*g*z + 36*b*c^9*d^10*f*g*z + 48*b^3*c^7*d^9*e*g^2*z -
240*a*c^9*d^8*e^2*f^2*z - b^10*d^2*e^8*g^2*z - 36*a^6*c^4*e^10*g^2*z - 9*b^2*c^8*d^10*g^2*z + 768*a^5*c^5*e^10
*f^2*z - 36*c^10*d^10*f^2*z - b^10*e^10*f^2*z - 177*a*b^2*c^4*d^2*e^7*f*g^2 + 285*a*b^2*c^4*d*e^8*f^2*g + 252*
a^2*b*c^4*d*e^8*f*g^2 - 120*a*b^3*c^3*d*e^8*f*g^2 + 108*a*b*c^5*d^3*e^6*f*g^2 + 36*a*b*c^5*d^2*e^7*f^2*g - 132
*a*b*c^5*d*e^8*f^3 - 69*b^2*c^5*d^4*e^5*f*g^2 + 57*b^2*c^5*d^3*e^6*f^2*g - 45*b^3*c^4*d^2*e^7*f^2*g + 30*b^4*c
^3*d^2*e^7*f*g^2 + 9*b^3*c^4*d^3*e^6*f*g^2 + 156*a^2*c^5*d^2*e^7*f*g^2 - 72*a^2*b*c^4*d^2*e^7*g^3 + 60*a*b^3*c
^3*d^2*e^7*g^3 - 13*a*b^2*c^4*d^3*e^6*g^3 + 36*b*c^6*d^5*e^4*f*g^2 + 36*b*c^6*d^4*e^5*f^2*g - 30*b^4*c^3*d*e^8
*f^2*g + 12*b^5*c^2*d*e^8*f*g^2 - 408*a^2*c^5*d*e^8*f^2*g - 156*a*c^6*d^3*e^6*f^2*g + 24*a*c^6*d^4*e^5*f*g^2 -
180*a^2*b*c^4*e^9*f^2*g + 60*a*b^3*c^3*e^9*f^2*g - 12*a*b*c^5*d^4*e^5*g^3 - 36*c^7*d^5*e^4*f^2*g - 6*b^5*c^2*
e^9*f^2*g + 36*a^3*c^4*e^9*f*g^2 - 72*b*c^6*d^3*e^6*f^3 - 36*a^3*c^4*d*e^8*g^3 + 15*b^3*c^4*d*e^8*f^3 + 132*a*
c^6*d^2*e^7*f^3 - 95*a*b^2*c^4*e^9*f^3 + 21*b^3*c^4*d^4*e^5*g^3 - 10*b^4*c^3*d^3*e^6*g^3 - 9*b^2*c^5*d^5*e^4*g
^3 - 6*b^5*c^2*d^2*e^7*g^3 + 21*b^2*c^5*d^2*e^7*f^3 - 4*a^2*c^5*d^3*e^6*g^3 + 36*c^7*d^4*e^5*f^3 + 10*b^4*c^3*
e^9*f^3 + 256*a^2*c^5*e^9*f^3, z, k), k, 1, 3)