∫ f+gx______ (d+ex)2(a+bx+cx2)3 dx

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

Optimal. Leaf size=1043 \[ \frac {(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x) e^4}{\left (c d^2-b e d+a e^2\right )^4}-\frac {(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (c x^2+b x+a\right ) e^4}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac {\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e}{\left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}-\frac {\left (12 c^6 f d^6+2 c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4+10 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (-d (6 e f-5 d g) b^2-10 a e (3 e f-2 d g) b+10 a^2 e^2 g\right )-10 a b c^2 e^4 \left (-d (6 e f-5 d g) b^2-3 a e (3 e f-2 d g) b+3 a^2 e^2 g\right )-10 c^3 e^2 \left (2 b^3 g d^4-8 a b^2 e g d^3+3 a^2 b e^2 (6 e f-d g) d+6 a^3 e^3 (e f-2 d g)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (c d^2-b e d+a e^2\right )^4}-\frac {4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 (d+e x) \left (c x^2+b x+a\right )}-\frac {-e f b^2+c d f b+a e g b+2 a c e f-2 a c d g+c (2 c d f+2 a e g-b (e f+d g)) x}{2 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2} \]

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Rubi [A] time = 6.29, antiderivative size = 1043, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {822, 800, 634, 618, 206, 628} \begin {gather*} \frac {(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x) e^4}{\left (c d^2-b e d+a e^2\right )^4}-\frac {(c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (c x^2+b x+a\right ) e^4}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac {\left (6 c^4 f d^4+c^3 (4 a e (6 e f-d g)-3 b d (4 e f+d g)) d^2-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (-3 d (e f-d g) b^2-a e (21 e f-13 d g) b+7 a^2 e^2 g\right )-c^2 e \left (-b^2 (3 e f+7 d g) d^2+6 a b e (4 e f+d g) d+2 a^2 e^2 (15 e f-22 d g)\right )\right ) e}{\left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^3 (d+e x)}-\frac {\left (12 c^6 f d^6+2 c^5 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g)) d^4+10 c^4 e \left (b^2 (3 e f+2 d g) d^2-a b e (12 e f+d g) d+2 a^2 e^2 (9 e f-4 d g)\right ) d^2+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (-d (6 e f-5 d g) b^2-10 a e (3 e f-2 d g) b+10 a^2 e^2 g\right )-10 a b c^2 e^4 \left (-d (6 e f-5 d g) b^2-3 a e (3 e f-2 d g) b+3 a^2 e^2 g\right )-10 c^3 e^2 \left (2 b^3 g d^4-8 a b^2 e g d^3+3 a^2 b e^2 (6 e f-d g) d+6 a^3 e^3 (e f-2 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 e d+a e^2\right )^4}-\frac {4 a c e (2 c d-b e) (2 c d f+2 a e g-b (e f+d g))-\left (-e b^2+c d b+2 a c e\right ) \left (6 c^2 f d^2-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 f d^3+2 c^2 (2 a e (9 e f-2 d g)-3 b d (3 e f+d g)) d+b^2 e^2 (3 b e f-2 b d g-a e g)+c e \left (11 b^2 g d^2+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 (d+e x) \left (c x^2+b x+a\right )}-\frac {-e f b^2+c d f b+a e g b+2 a c e f-2 a c d 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 e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(e*(6*c^4*d^4*f - b^3*e^3*(3*b*e*f - 2*b*d*g - a*e*g) - b*c*e^2*(7*a^2*e^2*g - a*b*e*(21*e*f - 13*d*g) - 3*b^2

*d*(e*f - d*g)) + c^3*d^2*(4*a*e*(6*e*f - d*g) - 3*b*d*(4*e*f + d*g)) - c^2*e*(2*a^2*e^2*(15*e*f - 22*d*g) + 6

*a*b*d*e*(4*e*f + d*g) - b^2*d^2*(3*e*f + 7*d*g))))/((b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) - (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*(b^2 - 4*a*c)

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

b*d*(2*e*f + 3*d*g))) - c*(12*c^3*d^3*f + b^2*e^2*(3*b*e*f - 2*b*d*g - a*e*g) + 2*c^2*d*(2*a*e*(9*e*f - 2*d*g

) - 3*b*d*(3*e*f + d*g)) + c*e*(11*b^2*d^2*g + 16*a^2*e^2*g - 2*a*b*e*(9*e*f + 5*d*g)))*x)/(2*(b^2 - 4*a*c)^2*

(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*(a + b*x + c*x^2)) - ((12*c^6*d^6*f + b^5*e^5*(3*b*e*f - 2*b*d*g - a*e*g)

+ b^3*c*e^4*(10*a^2*e^2*g - b^2*d*(6*e*f - 5*d*g) - 10*a*b*e*(3*e*f - 2*d*g)) - 10*a*b*c^2*e^4*(3*a^2*e^2*g -

b^2*d*(6*e*f - 5*d*g) - 3*a*b*e*(3*e*f - 2*d*g)) - 10*c^3*e^2*(2*b^3*d^4*g - 8*a*b^2*d^3*e*g + 6*a^3*e^3*(e*f

- 2*d*g) + 3*a^2*b*d*e^2*(6*e*f - d*g)) + 2*c^5*d^4*(2*a*e*(15*e*f - 2*d*g) - 3*b*d*(6*e*f + d*g)) + 10*c^4*d^

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

*g - a*e*g))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^4*(c*d*(6*e*f - 5*d*g) - e*(3*b*e*f - 2*b*d*g - a*e*

g))*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

Rule 206

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

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

Rule 618

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 628

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

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

Rule 634

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

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

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

Rule 800

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 822

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*} \int \frac {f+g x}{(d+e x)^2 \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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))+4 c e (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x)^2 \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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}+\frac {\int \frac {-2 \left (2 c e (b d (c d-b e)+a e (4 c d-b e)) (2 c d f+2 a e g-b (e f+d g))-\frac {1}{2} \left (2 c^2 d^2-2 b^2 e^2+6 a c e^2\right ) \left (6 c^2 d^2 f-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )\right )+2 c e \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 d g)\right )\right ) x}{(d+e x)^2 \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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}+\frac {\int \left (\frac {2 e^2 \left (-6 c^4 d^4 f+b^3 e^3 (3 b e f-2 b d g-a e g)+b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )-c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))+c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac {2 \left (b^2-4 a c\right )^2 e^5 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g))}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {2 \left (6 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)-c^3 e^2 \left (10 b^3 d^4 g-40 a b^2 d^3 e g+a^2 b d e^2 (138 e f-55 d g)+30 a^3 e^3 (e f-2 d g)\right )-a b c^2 e^4 \left (23 a^2 e^2 g-9 b^2 d (6 e f-5 d g)-23 a b e (3 e f-2 d g)\right )+b^3 c e^4 \left (9 a^2 e^2 g-b^2 d (6 e f-5 d g)-9 a b e (3 e f-2 d g)\right )+c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+5 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \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 {e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}+\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}+\frac {\int \frac {6 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)-c^3 e^2 \left (10 b^3 d^4 g-40 a b^2 d^3 e g+a^2 b d e^2 (138 e f-55 d g)+30 a^3 e^3 (e f-2 d g)\right )-a b c^2 e^4 \left (23 a^2 e^2 g-9 b^2 d (6 e f-5 d g)-23 a b e (3 e f-2 d g)\right )+b^3 c e^4 \left (9 a^2 e^2 g-b^2 d (6 e f-5 d g)-9 a b e (3 e f-2 d g)\right )+c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+5 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )-c \left (b^2-4 a c\right )^2 e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e 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 )^4}\\ &=\frac {e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}+\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {\left (e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e 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 )^4}+\frac {\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 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 )^4}\\ &=\frac {e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}+\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}-\frac {\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 d g)\right )\right ) \operatorname {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 )^4}\\ &=\frac {e \left (6 c^4 d^4 f-b^3 e^3 (3 b e f-2 b d g-a e g)-b c e^2 \left (7 a^2 e^2 g-a b e (21 e f-13 d g)-3 b^2 d (e f-d g)\right )+c^3 d^2 (4 a e (6 e f-d g)-3 b d (4 e f+d g))-c^2 e \left (2 a^2 e^2 (15 e f-22 d g)+6 a b d e (4 e f+d g)-b^2 d^2 (3 e f+7 d g)\right )\right )}{\left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\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 ) (d+e x) \left (a+b x+c x^2\right )^2}-\frac {4 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-b e (3 b e f-2 b d g-a e g)+c (2 a e (5 e f-2 d g)-b d (2 e f+3 d g))\right )-c \left (12 c^3 d^3 f+b^2 e^2 (3 b e f-2 b d g-a e g)+2 c^2 d (2 a e (9 e f-2 d g)-3 b d (3 e f+d g))+c e \left (11 b^2 d^2 g+16 a^2 e^2 g-2 a b e (9 e f+5 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 (d+e x) \left (a+b x+c x^2\right )}-\frac {\left (12 c^6 d^6 f+b^5 e^5 (3 b e f-2 b d g-a e g)+b^3 c e^4 \left (10 a^2 e^2 g-b^2 d (6 e f-5 d g)-10 a b e (3 e f-2 d g)\right )-10 a b c^2 e^4 \left (3 a^2 e^2 g-b^2 d (6 e f-5 d g)-3 a b e (3 e f-2 d g)\right )-10 c^3 e^2 \left (2 b^3 d^4 g-8 a b^2 d^3 e g+6 a^3 e^3 (e f-2 d g)+3 a^2 b d e^2 (6 e f-d g)\right )+2 c^5 d^4 (2 a e (15 e f-2 d g)-3 b d (6 e f+d g))+10 c^4 d^2 e \left (2 a^2 e^2 (9 e f-4 d g)-a b d e (12 e f+d g)+b^2 d^2 (3 e f+2 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 )^4}+\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^4}-\frac {e^4 (c d (6 e f-5 d g)-e (3 b e f-2 b d g-a e g)) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^4}\\ \end {align*}

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Mathematica [A] time = 6.75, size = 1409, normalized size = 1.35 \begin {gather*} -\frac {(e f-d g) e^4}{\left (c d^2-b e d+a e^2\right )^3 (d+e x)}+\frac {\left (3 e^6 f b^6-2 d e^5 g b^6-6 c d e^5 f b^5-a e^6 g b^5+5 c d^2 e^4 g b^5-30 a c e^6 f b^4+20 a c d e^5 g b^4+60 a c^2 d e^5 f b^3+10 a^2 c e^6 g b^3-50 a c^2 d^2 e^4 g b^3-20 c^3 d^4 e^2 g b^3+90 a^2 c^2 e^6 f b^2+30 c^4 d^4 e^2 f b^2-60 a^2 c^2 d e^5 g b^2+80 a c^3 d^3 e^3 g b^2+20 c^4 d^5 e g b^2-180 a^2 c^3 d e^5 f b-120 a c^4 d^3 e^3 f b-36 c^5 d^5 e f b-6 c^5 d^6 g b-30 a^3 c^2 e^6 g b+30 a^2 c^3 d^2 e^4 g b-10 a c^4 d^4 e^2 g b+12 c^6 d^6 f-60 a^3 c^3 e^6 f+180 a^2 c^4 d^2 e^4 f+60 a c^5 d^4 e^2 f+120 a^3 c^3 d e^5 g-80 a^2 c^4 d^3 e^3 g-8 a c^5 d^5 e g\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (b^2-4 a c\right )^2 \sqrt {4 a c-b^2} \left (-c d^2+b e d-a e^2\right )^4}+\frac {\left (-3 b f e^6+a g e^6+6 c d f e^5+2 b d g e^5-5 c d^2 g e^4\right ) \log (d+e x)}{\left (c d^2-b e d+a e^2\right )^4}+\frac {\left (3 b f e^6-a g e^6-6 c d f e^5-2 b d g e^5+5 c d^2 g e^4\right ) \log \left (c x^2+b x+a\right )}{2 \left (c d^2-b e d+a e^2\right )^4}+\frac {-4 e^4 f b^5+2 d e^3 g b^5+7 c d e^3 f b^4+2 a e^4 g b^4-6 c d^2 e^2 g b^4-4 c e^4 f x b^4+2 c d e^3 g x b^4+29 a c e^4 f b^3+3 c^2 d^2 e^2 f b^3-13 a c d e^3 g b^3+7 c^2 d^3 e g b^3+6 c^2 d e^3 f x b^3+2 a c e^4 g x b^3-6 c^2 d^2 e^2 g x b^3-56 a c^2 d e^3 f b^2-12 c^3 d^3 e f b^2-3 c^3 d^4 g b^2-15 a^2 c e^4 g b^2+18 a c^2 d^2 e^2 g b^2+26 a c^2 e^4 f x b^2+6 c^3 d^2 e^2 f x b^2-10 a c^2 d e^3 g x b^2+14 c^3 d^3 e g x b^2+6 c^4 d^4 f b-46 a^2 c^2 e^4 f b+24 a c^3 d^2 e^2 f b+44 a^2 c^2 d e^3 g b-4 a c^3 d^3 e g b-48 a c^3 d e^3 f x b-24 c^4 d^3 e f x b-6 c^4 d^4 g x b-14 a^2 c^2 e^4 g x b-12 a c^3 d^2 e^2 g x b+64 a^2 c^3 d e^3 f+16 a^3 c^2 e^4 g-48 a^2 c^3 d^2 e^2 g+12 c^5 d^4 f x-28 a^2 c^3 e^4 f x+48 a c^4 d^2 e^2 f x+56 a^2 c^3 d e^3 g x-8 a c^4 d^3 e g x}{2 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )}+\frac {e^2 f b^3-2 c d e f b^2-a e^2 g b^2+c e^2 f x b^2+c^2 d^2 f b-3 a c e^2 f b+2 a c d e g b-2 c^2 d e f x b-c^2 d^2 g x b-a c e^2 g x b+4 a c^2 d e f-2 a c^2 d^2 g+2 a^2 c e^2 g+2 c^3 d^2 f x-2 a c^2 e^2 f x+4 a c^2 d e g x}{2 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

3*e^2*f - 3*a*b*c*e^2*f - 2*a*c^2*d^2*g + 2*a*b*c*d*e*g - a*b^2*e^2*g + 2*a^2*c*e^2*g + 2*c^3*d^2*f*x - 2*b*c^

2*d*e*f*x + b^2*c*e^2*f*x - 2*a*c^2*e^2*f*x - b*c^2*d^2*g*x + 4*a*c^2*d*e*g*x - a*b*c*e^2*g*x)/(2*(-b^2 + 4*a*

c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^2) + (6*b*c^4*d^4*f - 12*b^2*c^3*d^3*e*f + 3*b^3*c^2*d^2*e^2*f

+ 24*a*b*c^3*d^2*e^2*f + 7*b^4*c*d*e^3*f - 56*a*b^2*c^2*d*e^3*f + 64*a^2*c^3*d*e^3*f - 4*b^5*e^4*f + 29*a*b^3*

c*e^4*f - 46*a^2*b*c^2*e^4*f - 3*b^2*c^3*d^4*g + 7*b^3*c^2*d^3*e*g - 4*a*b*c^3*d^3*e*g - 6*b^4*c*d^2*e^2*g + 1

8*a*b^2*c^2*d^2*e^2*g - 48*a^2*c^3*d^2*e^2*g + 2*b^5*d*e^3*g - 13*a*b^3*c*d*e^3*g + 44*a^2*b*c^2*d*e^3*g + 2*a

*b^4*e^4*g - 15*a^2*b^2*c*e^4*g + 16*a^3*c^2*e^4*g + 12*c^5*d^4*f*x - 24*b*c^4*d^3*e*f*x + 6*b^2*c^3*d^2*e^2*f

*x + 48*a*c^4*d^2*e^2*f*x + 6*b^3*c^2*d*e^3*f*x - 48*a*b*c^3*d*e^3*f*x - 4*b^4*c*e^4*f*x + 26*a*b^2*c^2*e^4*f*

x - 28*a^2*c^3*e^4*f*x - 6*b*c^4*d^4*g*x + 14*b^2*c^3*d^3*e*g*x - 8*a*c^4*d^3*e*g*x - 6*b^3*c^2*d^2*e^2*g*x -

12*a*b*c^3*d^2*e^2*g*x + 2*b^4*c*d*e^3*g*x - 10*a*b^2*c^2*d*e^3*g*x + 56*a^2*c^3*d*e^3*g*x + 2*a*b^3*c*e^4*g*x

- 14*a^2*b*c^2*e^4*g*x)/(2*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)) + ((12*c^6*d^6*f - 3

6*b*c^5*d^5*e*f + 30*b^2*c^4*d^4*e^2*f + 60*a*c^5*d^4*e^2*f - 120*a*b*c^4*d^3*e^3*f + 180*a^2*c^4*d^2*e^4*f -

6*b^5*c*d*e^5*f + 60*a*b^3*c^2*d*e^5*f - 180*a^2*b*c^3*d*e^5*f + 3*b^6*e^6*f - 30*a*b^4*c*e^6*f + 90*a^2*b^2*c

^2*e^6*f - 60*a^3*c^3*e^6*f - 6*b*c^5*d^6*g + 20*b^2*c^4*d^5*e*g - 8*a*c^5*d^5*e*g - 20*b^3*c^3*d^4*e^2*g - 10

*a*b*c^4*d^4*e^2*g + 80*a*b^2*c^3*d^3*e^3*g - 80*a^2*c^4*d^3*e^3*g + 5*b^5*c*d^2*e^4*g - 50*a*b^3*c^2*d^2*e^4*

g + 30*a^2*b*c^3*d^2*e^4*g - 2*b^6*d*e^5*g + 20*a*b^4*c*d*e^5*g - 60*a^2*b^2*c^2*d*e^5*g + 120*a^3*c^3*d*e^5*g

- a*b^5*e^6*g + 10*a^2*b^3*c*e^6*g - 30*a^3*b*c^2*e^6*g)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*

c)^2*Sqrt[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^4) + ((6*c*d*e^5*f - 3*b*e^6*f - 5*c*d^2*e^4*g + 2*b*d*e^5*

g + a*e^6*g)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 + ((-6*c*d*e^5*f + 3*b*e^6*f + 5*c*d^2*e^4*g - 2*b*d*e^5*

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

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [B] time = 2.14, size = 3417, normalized size = 3.28

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*c^6*d^6*f*e^2 - 6*b*c^5*d^6*g*e^2 - 36*b*c^5*d^5*f*e^3 + 20*b^2*c^4*d^5*g*e^3 - 8*a*c^5*d^5*g*e^3 + 30*b^2

*c^4*d^4*f*e^4 + 60*a*c^5*d^4*f*e^4 - 20*b^3*c^3*d^4*g*e^4 - 10*a*b*c^4*d^4*g*e^4 - 120*a*b*c^4*d^3*f*e^5 + 80

*a*b^2*c^3*d^3*g*e^5 - 80*a^2*c^4*d^3*g*e^5 + 180*a^2*c^4*d^2*f*e^6 + 5*b^5*c*d^2*g*e^6 - 50*a*b^3*c^2*d^2*g*e

^6 + 30*a^2*b*c^3*d^2*g*e^6 - 6*b^5*c*d*f*e^7 + 60*a*b^3*c^2*d*f*e^7 - 180*a^2*b*c^3*d*f*e^7 - 2*b^6*d*g*e^7 +

20*a*b^4*c*d*g*e^7 - 60*a^2*b^2*c^2*d*g*e^7 + 120*a^3*c^3*d*g*e^7 + 3*b^6*f*e^8 - 30*a*b^4*c*f*e^8 + 90*a^2*b

^2*c^2*f*e^8 - 60*a^3*c^3*f*e^8 - a*b^5*g*e^8 + 10*a^2*b^3*c*g*e^8 - 30*a^3*b*c^2*g*e^8)*arctan((2*c*d - 2*c*d

^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*a*e^2/(x*e + d))*e^(-1)/sqrt(-b^2 + 4*a*c))*e^(-2)/((b^4*c^4*d^8 -

8*a*b^2*c^5*d^8 + 16*a^2*c^6*d^8 - 4*b^5*c^3*d^7*e + 32*a*b^3*c^4*d^7*e - 64*a^2*b*c^5*d^7*e + 6*b^6*c^2*d^6*e

^2 - 44*a*b^4*c^3*d^6*e^2 + 64*a^2*b^2*c^4*d^6*e^2 + 64*a^3*c^5*d^6*e^2 - 4*b^7*c*d^5*e^3 + 20*a*b^5*c^2*d^5*e

^3 + 32*a^2*b^3*c^3*d^5*e^3 - 192*a^3*b*c^4*d^5*e^3 + b^8*d^4*e^4 + 4*a*b^6*c*d^4*e^4 - 74*a^2*b^4*c^2*d^4*e^4

+ 144*a^3*b^2*c^3*d^4*e^4 + 96*a^4*c^4*d^4*e^4 - 4*a*b^7*d^3*e^5 + 20*a^2*b^5*c*d^3*e^5 + 32*a^3*b^3*c^2*d^3*

e^5 - 192*a^4*b*c^3*d^3*e^5 + 6*a^2*b^6*d^2*e^6 - 44*a^3*b^4*c*d^2*e^6 + 64*a^4*b^2*c^2*d^2*e^6 + 64*a^5*c^3*d

^2*e^6 - 4*a^3*b^5*d*e^7 + 32*a^4*b^3*c*d*e^7 - 64*a^5*b*c^2*d*e^7 + a^4*b^4*e^8 - 8*a^5*b^2*c*e^8 + 16*a^6*c^

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

*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^4*d^8 - 4*b*c^3*d

^7*e + 6*b^2*c^2*d^6*e^2 + 4*a*c^3*d^6*e^2 - 4*b^3*c*d^5*e^3 - 12*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d

^4*e^4 + 6*a^2*c^2*d^4*e^4 - 4*a*b^3*d^3*e^5 - 12*a^2*b*c*d^3*e^5 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 - 4*a^

3*b*d*e^7 + a^4*e^8) + (d*g*e^10/(x*e + d) - f*e^11/(x*e + d))/(c^3*d^6*e^6 - 3*b*c^2*d^5*e^7 + 3*b^2*c*d^4*e^

8 + 3*a*c^2*d^4*e^8 - b^3*d^3*e^9 - 6*a*b*c*d^3*e^9 + 3*a*b^2*d^2*e^10 + 3*a^2*c*d^2*e^10 - 3*a^2*b*d*e^11 + a

^3*e^12) + 1/2*(12*c^7*d^5*f*e - 6*b*c^6*d^5*g*e - 30*b*c^6*d^4*f*e^2 + 17*b^2*c^5*d^4*g*e^2 - 8*a*c^6*d^4*g*e

^2 + 16*b^2*c^5*d^3*f*e^3 + 56*a*c^6*d^3*f*e^3 - 12*b^3*c^4*d^3*g*e^3 - 12*a*b*c^5*d^3*g*e^3 + 6*b^3*c^4*d^2*f

*e^4 - 84*a*b*c^5*d^2*f*e^4 + 8*b^4*c^3*d^2*g*e^4 - 34*a*b^2*c^4*d^2*g*e^4 + 128*a^2*c^5*d^2*g*e^4 - 14*b^4*c^

3*d*f*e^5 + 100*a*b^2*c^4*d*f*e^5 - 116*a^2*c^5*d*f*e^5 - 2*b^5*c^2*d*g*e^5 + 18*a*b^3*c^3*d*g*e^5 - 70*a^2*b*

c^4*d*g*e^5 + 5*b^5*c^2*f*e^6 - 36*a*b^3*c^3*f*e^6 + 58*a^2*b*c^4*f*e^6 - 3*a*b^4*c^2*g*e^6 + 21*a^2*b^2*c^3*g

*e^6 - 24*a^3*c^4*g*e^6 - 2*(18*c^7*d^6*f*e^2 - 9*b*c^6*d^6*g*e^2 - 54*b*c^6*d^5*f*e^3 + 30*b^2*c^5*d^5*g*e^3

- 12*a*c^6*d^5*g*e^3 + 47*b^2*c^5*d^4*f*e^4 + 82*a*c^6*d^4*f*e^4 - 31*b^3*c^4*d^4*g*e^4 - 11*a*b*c^5*d^4*g*e^4

- 4*b^3*c^4*d^3*f*e^5 - 164*a*b*c^5*d^3*f*e^5 + 24*b^4*c^3*d^3*g*e^5 - 64*a*b^2*c^4*d^3*g*e^5 + 232*a^2*c^5*d

^3*g*e^5 - 29*b^4*c^3*d^2*f*e^6 + 244*a*b^2*c^4*d^2*f*e^6 - 242*a^2*c^5*d^2*f*e^6 - 11*b^5*c^2*d^2*g*e^6 + 67*

a*b^3*c^3*d^2*g*e^6 - 227*a^2*b*c^4*d^2*g*e^6 + 22*b^5*c^2*d*f*e^7 - 162*a*b^3*c^3*d*f*e^7 + 242*a^2*b*c^4*d*f

*e^7 + 2*b^6*c*d*g*e^7 - 24*a*b^4*c^2*d*g*e^7 + 110*a^2*b^2*c^3*d*g*e^7 - 76*a^3*c^4*d*g*e^7 - 5*b^6*c*f*e^8 +

38*a*b^4*c^2*f*e^8 - 71*a^2*b^2*c^3*f*e^8 + 14*a^3*c^4*f*e^8 + 3*a*b^5*c*g*e^8 - 22*a^2*b^3*c^2*g*e^8 + 31*a^

3*b*c^3*g*e^8)*e^(-1)/(x*e + d) + (36*c^7*d^7*f*e^3 - 18*b*c^6*d^7*g*e^3 - 126*b*c^6*d^6*f*e^4 + 69*b^2*c^5*d^

6*g*e^4 - 24*a*c^6*d^6*g*e^4 + 144*b^2*c^5*d^5*f*e^5 + 180*a*c^6*d^5*f*e^5 - 90*b^3*c^4*d^5*g*e^5 - 18*a*b*c^5

*d^5*g*e^5 - 45*b^3*c^4*d^4*f*e^6 - 450*a*b*c^5*d^4*f*e^6 + 80*b^4*c^3*d^4*g*e^6 - 145*a*b^2*c^4*d^4*g*e^6 + 5

60*a^2*c^5*d^4*g*e^6 - 70*b^4*c^3*d^3*f*e^7 + 740*a*b^2*c^4*d^3*f*e^7 - 580*a^2*c^5*d^3*f*e^7 - 50*b^5*c^2*d^3

*g*e^7 + 250*a*b^3*c^3*d^3*g*e^7 - 830*a^2*b*c^4*d^3*g*e^7 + 87*b^5*c^2*d^2*f*e^8 - 660*a*b^3*c^3*d^2*f*e^8 +

870*a^2*b*c^4*d^2*f*e^8 + 16*b^6*c*d^2*g*e^8 - 129*a*b^4*c^2*d^2*g*e^8 + 471*a^2*b^2*c^3*d^2*g*e^8 - 88*a^3*c^

4*d^2*g*e^8 - 36*b^6*c*d*f*e^9 + 258*a*b^4*c^2*d*f*e^9 - 372*a^2*b^2*c^3*d*f*e^9 - 84*a^3*c^4*d*f*e^9 - 2*b^7*

d*g*e^9 + 32*a*b^5*c*d*g*e^9 - 160*a^2*b^3*c^2*d*g*e^9 + 130*a^3*b*c^3*d*g*e^9 + 5*b^7*f*e^10 - 34*a*b^5*c*f*e

^10 + 41*a^2*b^3*c^2*f*e^10 + 42*a^3*b*c^3*f*e^10 - 3*a*b^6*g*e^10 + 19*a^2*b^4*c*g*e^10 - 11*a^3*b^2*c^2*g*e^

10 - 32*a^4*c^3*g*e^10)*e^(-2)/(x*e + d)^2 - 2*(6*c^7*d^8*f*e^4 - 3*b*c^6*d^8*g*e^4 - 24*b*c^6*d^7*f*e^5 + 13*

b^2*c^5*d^7*g*e^5 - 4*a*c^6*d^7*g*e^5 + 33*b^2*c^5*d^6*f*e^6 + 36*a*c^6*d^6*f*e^6 - 20*b^3*c^4*d^6*g*e^6 - 4*a

*b*c^5*d^6*g*e^6 - 15*b^3*c^4*d^5*f*e^7 - 108*a*b*c^5*d^5*f*e^7 + 20*b^4*c^3*d^5*g*e^7 - 25*a*b^2*c^4*d^5*g*e^

7 + 116*a^2*c^5*d^5*g*e^7 - 15*b^4*c^3*d^4*f*e^8 + 195*a*b^2*c^4*d^4*f*e^8 - 120*a^2*c^5*d^4*f*e^8 - 15*b^5*c^

2*d^4*g*e^8 + 65*a*b^3*c^3*d^4*g*e^8 - 230*a^2*b*c^4*d^4*g*e^8 + 27*b^5*c^2*d^3*f*e^9 - 210*a*b^3*c^3*d^3*f*e^

9 + 240*a^2*b*c^4*d^3*f*e^9 + 6*b^6*c*d^3*g*e^9 - 39*a*b^4*c^2*d^3*g*e^9 + 131*a^2*b^2*c^3*d^3*g*e^9 + 52*a^3*

c^4*d^3*g*e^9 - 15*b^6*c*d^2*f*e^10 + 99*a*b^4*c^2*d^2*f*e^10 - 81*a^2*b^2*c^3*d^2*f*e^10 - 132*a^3*c^4*d^2*f*

e^10 - b^7*d^2*g*e^10 + 11*a*b^5*c*d^2*g*e^10 - 46*a^2*b^3*c^2*d^2*g*e^10 - 12*a^3*b*c^3*d^2*g*e^10 + 3*b^7*d*

f*e^11 - 12*a*b^5*c*d*f*e^11 - 39*a^2*b^3*c^2*d*f*e^11 + 132*a^3*b*c^3*d*f*e^11 - a*b^6*d*g*e^11 + a^2*b^4*c*d

*g*e^11 + 41*a^3*b^2*c^2*d*g*e^11 - 68*a^4*c^3*d*g*e^11 - 3*a*b^6*f*e^12 + 24*a^2*b^4*c*f*e^12 - 51*a^3*b^2*c^

2*f*e^12 + 18*a^4*c^3*f*e^12 + 2*a^2*b^5*g*e^12 - 15*a^3*b^3*c*g*e^12 + 25*a^4*b*c^2*g*e^12)*e^(-3)/(x*e + d)^

3)/((c*d^2 - b*d*e + a*e^2)^4*(b^2 - 4*a*c)^2*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e

/(x*e + d)^2 + a*e^2/(x*e + d)^2)^2)

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maple [B] time = 0.11, size = 13679, normalized size = 13.12 \begin {gather*} \text {output too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(e*x+d)^2/(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 details)Is 4*a*c-b^2 positive or negative?

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mupad [B] time = 13.00, size = 40079, normalized size = 38.43

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(root(286720*a^9*b*c^8*d^7*e^9*z^3 + 286720*a^8*b*c^9*d^9*e^7*z^3 + 172032*a^10*b*c^7*d^5*e^11*z^3 +

172032*a^7*b*c^10*d^11*e^5*z^3 + 57344*a^11*b*c^6*d^3*e^13*z^3 + 57344*a^6*b*c^11*d^13*e^3*z^3 - 10240*a^11*b

^3*c^4*d*e^15*z^3 - 10240*a^4*b^3*c^11*d^15*e*z^3 + 5120*a^10*b^5*c^3*d*e^15*z^3 + 5120*a^3*b^5*c^10*d^15*e*z^

3 - 1280*a^9*b^7*c^2*d*e^15*z^3 - 1280*a^2*b^7*c^9*d^15*e*z^3 - 1232*a^5*b^12*c*d^4*e^12*z^3 - 1232*a*b^12*c^5

*d^12*e^4*z^3 + 1064*a^6*b^11*c*d^3*e^13*z^3 + 1064*a*b^11*c^6*d^13*e^3*z^3 + 840*a^4*b^13*c*d^5*e^11*z^3 + 84

0*a*b^13*c^4*d^11*e^5*z^3 - 552*a^7*b^10*c*d^2*e^14*z^3 - 552*a*b^10*c^7*d^14*e^2*z^3 - 280*a^3*b^14*c*d^6*e^1

0*z^3 - 280*a*b^14*c^3*d^10*e^6*z^3 - 8*a^2*b^15*c*d^7*e^9*z^3 - 8*a*b^15*c^2*d^9*e^7*z^3 + 8192*a^12*b*c^5*d*

e^15*z^3 + 8192*a^5*b*c^12*d^15*e*z^3 + 160*a^8*b^9*c*d*e^15*z^3 + 160*a*b^9*c^8*d^15*e*z^3 + 36*a*b^16*c*d^8*

e^8*z^3 - 483840*a^8*b^2*c^8*d^8*e^8*z^3 - 365568*a^7*b^5*c^6*d^7*e^9*z^3 - 365568*a^6*b^5*c^7*d^9*e^7*z^3 - 3

58400*a^9*b^2*c^7*d^6*e^10*z^3 - 358400*a^7*b^2*c^9*d^10*e^6*z^3 + 241920*a^7*b^4*c^7*d^8*e^8*z^3 + 215040*a^8

*b^4*c^6*d^6*e^10*z^3 + 215040*a^8*b^3*c^7*d^7*e^9*z^3 + 215040*a^7*b^3*c^8*d^9*e^7*z^3 + 215040*a^6*b^4*c^8*d

^10*e^6*z^3 - 193536*a^8*b^5*c^5*d^5*e^11*z^3 - 193536*a^5*b^5*c^8*d^11*e^5*z^3 - 136192*a^10*b^2*c^6*d^4*e^12

*z^3 - 136192*a^6*b^2*c^10*d^12*e^4*z^3 + 133056*a^6*b^6*c^6*d^8*e^8*z^3 + 125440*a^9*b^4*c^5*d^4*e^12*z^3 + 1

25440*a^5*b^4*c^9*d^12*e^4*z^3 - 109944*a^5*b^8*c^5*d^8*e^8*z^3 + 106752*a^6*b^7*c^5*d^7*e^9*z^3 + 106752*a^5*

b^7*c^6*d^9*e^7*z^3 + 80640*a^7*b^7*c^4*d^5*e^11*z^3 + 80640*a^4*b^7*c^7*d^11*e^5*z^3 - 77280*a^6*b^8*c^4*d^6*

e^10*z^3 - 77280*a^4*b^8*c^6*d^10*e^6*z^3 + 71680*a^9*b^3*c^6*d^5*e^11*z^3 + 71680*a^6*b^3*c^9*d^11*e^5*z^3 +

69888*a^7*b^6*c^5*d^6*e^10*z^3 + 69888*a^5*b^6*c^7*d^10*e^6*z^3 - 35840*a^9*b^5*c^4*d^3*e^13*z^3 - 35840*a^4*b

^5*c^9*d^13*e^3*z^3 + 30720*a^10*b^4*c^4*d^2*e^14*z^3 + 30720*a^4*b^4*c^10*d^14*e^2*z^3 + 26880*a^8*b^7*c^3*d^

3*e^13*z^3 + 26880*a^3*b^7*c^8*d^13*e^3*z^3 + 21510*a^4*b^10*c^4*d^8*e^8*z^3 + 18536*a^5*b^10*c^3*d^6*e^10*z^3

+ 18536*a^3*b^10*c^5*d^10*e^6*z^3 - 18480*a^7*b^8*c^3*d^4*e^12*z^3 - 18480*a^3*b^8*c^7*d^12*e^4*z^3 - 18432*a

^11*b^2*c^5*d^2*e^14*z^3 - 18432*a^5*b^2*c^11*d^14*e^2*z^3 - 16640*a^9*b^6*c^3*d^2*e^14*z^3 - 16640*a^3*b^6*c^

9*d^14*e^2*z^3 - 14336*a^10*b^3*c^5*d^3*e^13*z^3 - 14336*a^5*b^3*c^10*d^13*e^3*z^3 - 13440*a^8*b^6*c^4*d^4*e^1

2*z^3 - 13440*a^4*b^6*c^8*d^12*e^4*z^3 + 13280*a^5*b^9*c^4*d^7*e^9*z^3 + 13280*a^4*b^9*c^5*d^9*e^7*z^3 - 10840

*a^4*b^11*c^3*d^7*e^9*z^3 - 10840*a^3*b^11*c^4*d^9*e^7*z^3 + 7868*a^6*b^10*c^2*d^4*e^12*z^3 + 7868*a^2*b^10*c^

6*d^12*e^4*z^3 - 7840*a^7*b^9*c^2*d^3*e^13*z^3 - 7840*a^2*b^9*c^7*d^13*e^3*z^3 - 5600*a^6*b^9*c^3*d^5*e^11*z^3

- 5600*a^3*b^9*c^6*d^11*e^5*z^3 + 4320*a^8*b^8*c^2*d^2*e^14*z^3 + 4320*a^2*b^8*c^8*d^14*e^2*z^3 - 3528*a^5*b^

11*c^2*d^5*e^11*z^3 - 3528*a^2*b^11*c^5*d^11*e^5*z^3 + 1520*a^3*b^13*c^2*d^7*e^9*z^3 + 1520*a^2*b^13*c^3*d^9*e

^7*z^3 - 700*a^4*b^12*c^2*d^6*e^10*z^3 - 700*a^2*b^12*c^4*d^10*e^6*z^3 - 540*a^2*b^14*c^2*d^8*e^8*z^3 + 480*a^

3*b^12*c^3*d^8*e^8*z^3 - 8*b^17*c*d^9*e^7*z^3 - 8*b^11*c^7*d^15*e*z^3 - 8*a^7*b^11*d*e^15*z^3 - 8*a*b^17*d^7*e

^9*z^3 - 20*a^9*b^8*c*e^16*z^3 - 20*a*b^8*c^9*d^16*z^3 + 70*b^14*c^4*d^12*e^4*z^3 - 56*b^15*c^3*d^11*e^5*z^3 -

56*b^13*c^5*d^13*e^3*z^3 + 28*b^16*c^2*d^10*e^6*z^3 + 28*b^12*c^6*d^14*e^2*z^3 - 71680*a^9*c^9*d^8*e^8*z^3 -

57344*a^10*c^8*d^6*e^10*z^3 - 57344*a^8*c^10*d^10*e^6*z^3 - 28672*a^11*c^7*d^4*e^12*z^3 - 28672*a^7*c^11*d^12*

e^4*z^3 - 8192*a^12*c^6*d^2*e^14*z^3 - 8192*a^6*c^12*d^14*e^2*z^3 + 70*a^4*b^14*d^4*e^12*z^3 - 56*a^5*b^13*d^3

*e^13*z^3 - 56*a^3*b^15*d^5*e^11*z^3 + 28*a^6*b^12*d^2*e^14*z^3 + 28*a^2*b^16*d^6*e^10*z^3 + 1280*a^12*b^2*c^4

*e^16*z^3 - 640*a^11*b^4*c^3*e^16*z^3 + 160*a^10*b^6*c^2*e^16*z^3 + 1280*a^4*b^2*c^12*d^16*z^3 - 640*a^3*b^4*c

^11*d^16*z^3 + 160*a^2*b^6*c^10*d^16*z^3 - 1024*a^13*c^5*e^16*z^3 - 1024*a^5*c^13*d^16*z^3 + b^18*d^8*e^8*z^3

+ b^10*c^8*d^16*z^3 + a^8*b^10*e^16*z^3 + 96*a*b*c^10*d^10*e^2*f*g*z + 69900*a^4*b^2*c^6*d^3*e^9*f*g*z - 64590

*a^4*b^3*c^5*d^2*e^10*f*g*z - 40200*a^3*b^4*c^5*d^3*e^9*f*g*z + 32820*a^3*b^5*c^4*d^2*e^10*f*g*z + 10680*a^2*b

^6*c^4*d^3*e^9*f*g*z + 10500*a^3*b^3*c^6*d^4*e^8*f*g*z + 8820*a^2*b^3*c^7*d^6*e^6*f*g*z - 8460*a^2*b^7*c^3*d^2

*e^10*f*g*z - 5880*a^3*b^2*c^7*d^5*e^7*f*g*z - 5040*a^2*b^4*c^6*d^5*e^7*f*g*z - 3240*a^2*b^2*c^8*d^7*e^5*f*g*z

- 1260*a^2*b^5*c^5*d^4*e^8*f*g*z - 252*a*b^10*c*d*e^11*f*g*z + 55872*a^5*b*c^6*d^2*e^10*f*g*z - 30636*a^5*b^2

*c^5*d*e^11*f*g*z + 24180*a^4*b^4*c^4*d*e^11*f*g*z - 9720*a^3*b^6*c^3*d*e^11*f*g*z + 3690*a*b^3*c^8*d^8*e^4*f*

g*z - 3360*a^3*b*c^8*d^6*e^6*f*g*z - 3240*a*b^4*c^7*d^7*e^5*f*g*z + 2160*a^2*b^8*c^2*d*e^11*f*g*z - 2100*a^4*b

*c^7*d^4*e^8*f*g*z - 1500*a*b^2*c^9*d^9*e^3*f*g*z - 1320*a*b^8*c^3*d^3*e^9*f*g*z - 1260*a^2*b*c^9*d^8*e^4*f*g*

z + 1080*a*b^9*c^2*d^2*e^10*f*g*z + 924*a*b^6*c^5*d^5*e^7*f*g*z + 252*a*b^5*c^6*d^6*e^6*f*g*z - 150*a*b^7*c^4*

d^4*e^8*f*g*z + 48*a*c^11*d^11*e*f*g*z - 660*b^4*c^8*d^9*e^3*f*g*z + 570*b^3*c^9*d^10*e^2*f*g*z + 270*b^5*c^7*

d^8*e^4*f*g*z + 84*b^6*c^6*d^7*e^5*f*g*z + 60*b^10*c^2*d^3*e^9*f*g*z - 60*b^8*c^4*d^5*e^7*f*g*z - 42*b^7*c^5*d

^6*e^6*f*g*z + 30*b^9*c^3*d^4*e^8*f*g*z - 59280*a^5*c^7*d^3*e^9*f*g*z + 3360*a^4*c^8*d^5*e^7*f*g*z + 2400*a^3*

c^9*d^7*e^5*f*g*z + 720*a^2*c^10*d^9*e^3*f*g*z + 7410*a^5*b^3*c^4*e^12*f*g*z - 3810*a^4*b^5*c^3*e^12*f*g*z + 9

60*a^3*b^7*c^2*e^12*f*g*z + 4872*a^6*b*c^5*d*e^11*g^2*z + 90*a*b^10*c*d^2*e^10*g^2*z + 80*a^2*b^9*c*d*e^11*g^2

*z - 33048*a^5*b*c^6*d*e^11*f^2*z + 1800*a*b*c^10*d^9*e^3*f^2*z - 720*a*b^9*c^2*d*e^11*f^2*z - 31575*a^4*b^2*c

^6*d^4*e^8*g^2*z + 24700*a^4*b^3*c^5*d^3*e^9*g^2*z + 16722*a^5*b^2*c^5*d^2*e^10*g^2*z + 15700*a^3*b^4*c^5*d^4*

e^8*g^2*z - 12140*a^3*b^5*c^4*d^3*e^9*g^2*z - 11640*a^4*b^4*c^4*d^2*e^10*g^2*z - 4485*a^2*b^6*c^4*d^4*e^8*g^2*

z + 4180*a^3*b^6*c^3*d^2*e^10*g^2*z + 3120*a^2*b^7*c^3*d^3*e^9*g^2*z - 1960*a^3*b^3*c^6*d^5*e^7*g^2*z + 1820*a

^3*b^2*c^7*d^6*e^6*g^2*z + 1596*a^2*b^5*c^5*d^5*e^7*g^2*z + 1185*a^2*b^2*c^8*d^8*e^4*g^2*z - 1080*a^2*b^3*c^7*

d^7*e^5*g^2*z - 840*a^2*b^8*c^2*d^2*e^10*g^2*z - 840*a^2*b^4*c^6*d^6*e^6*g^2*z - 50760*a^4*b^2*c^6*d^2*e^10*f^

2*z + 25380*a^3*b^4*c^5*d^2*e^10*f^2*z - 12600*a^3*b^2*c^7*d^4*e^8*f^2*z - 10080*a^2*b^2*c^8*d^6*e^6*f^2*z - 6

030*a^2*b^6*c^4*d^2*e^10*f^2*z + 3150*a^2*b^4*c^6*d^4*e^8*f^2*z + 2520*a^2*b^3*c^7*d^5*e^7*f^2*z - 1260*a^2*b^

5*c^5*d^3*e^9*f^2*z - 228*b^2*c^10*d^11*e*f*g*z - 54*b^11*c*d^2*e^10*f*g*z + 12816*a^6*c^6*d*e^11*f*g*z - 5508

*a^6*b*c^5*e^12*f*g*z - 120*a^2*b^9*c*e^12*f*g*z - 24*a*b*c^10*d^11*e*g^2*z - 18360*a^5*b*c^6*d^3*e^9*g^2*z -

5340*a^5*b^3*c^4*d*e^11*g^2*z + 2580*a^4*b^5*c^3*d*e^11*g^2*z + 1680*a^4*b*c^7*d^5*e^7*g^2*z + 1380*a*b^5*c^6*

d^7*e^5*g^2*z - 1050*a*b^4*c^7*d^8*e^4*g^2*z - 686*a*b^6*c^5*d^6*e^6*g^2*z - 640*a^3*b^7*c^2*d*e^11*g^2*z + 57

0*a*b^8*c^3*d^4*e^8*g^2*z - 400*a*b^9*c^2*d^3*e^9*g^2*z - 280*a^2*b*c^9*d^9*e^3*g^2*z + 260*a*b^3*c^8*d^9*e^3*

g^2*z + 80*a^3*b*c^8*d^7*e^5*g^2*z + 50*a*b^2*c^9*d^10*e^2*g^2*z + 44460*a^4*b^3*c^5*d*e^11*f^2*z - 22860*a^3*

b^5*c^4*d*e^11*f^2*z + 15120*a^3*b*c^8*d^5*e^7*f^2*z + 12600*a^4*b*c^7*d^3*e^9*f^2*z + 7920*a^2*b*c^9*d^7*e^5*

f^2*z + 5760*a^2*b^7*c^3*d*e^11*f^2*z - 3060*a*b^2*c^9*d^8*e^4*f^2*z + 1440*a*b^3*c^8*d^7*e^5*f^2*z - 1260*a*b

^5*c^6*d^5*e^7*f^2*z + 1260*a*b^4*c^7*d^6*e^6*f^2*z + 720*a*b^8*c^3*d^2*e^10*f^2*z + 180*a*b^7*c^4*d^3*e^9*f^2

*z + 216*b*c^11*d^11*e*f^2*z + 36*b^11*c*d*e^11*f^2*z - 4*a*b^11*d*e^11*g^2*z + 180*a*b^10*c*e^12*f^2*z + 200*

b^5*c^7*d^9*e^3*g^2*z - 160*b^4*c^8*d^10*e^2*g^2*z - 85*b^6*c^6*d^8*e^4*g^2*z + 70*b^8*c^4*d^6*e^6*g^2*z - 56*

b^7*c^5*d^7*e^5*g^2*z - 25*b^10*c^2*d^4*e^8*g^2*z - 20*b^9*c^3*d^5*e^7*g^2*z + 24000*a^5*c^7*d^4*e^8*g^2*z - 1

1280*a^6*c^6*d^2*e^10*g^2*z - 1120*a^4*c^8*d^6*e^6*g^2*z + 540*b^3*c^9*d^9*e^3*f^2*z - 504*b^2*c^10*d^10*e^2*f

^2*z - 320*a^3*c^9*d^8*e^4*g^2*z - 225*b^4*c^8*d^8*e^4*f^2*z + 144*b^7*c^5*d^5*e^7*f^2*z - 126*b^6*c^6*d^6*e^6

*f^2*z - 45*b^8*c^4*d^4*e^8*f^2*z - 36*b^10*c^2*d^2*e^10*f^2*z + 36*b^5*c^7*d^7*e^5*f^2*z - 16*a^2*c^10*d^10*e

^2*g^2*z + 33048*a^5*c^7*d^2*e^10*f^2*z - 6300*a^4*c^8*d^4*e^8*f^2*z - 5040*a^3*c^9*d^6*e^6*f^2*z - 1980*a^2*c

^10*d^8*e^4*f^2*z - 1185*a^6*b^2*c^4*e^12*g^2*z + 630*a^5*b^4*c^3*e^12*g^2*z - 160*a^4*b^6*c^2*e^12*g^2*z - 11

565*a^4*b^4*c^4*e^12*f^2*z + 9612*a^5*b^2*c^5*e^12*f^2*z + 5760*a^3*b^6*c^3*e^12*f^2*z - 1440*a^2*b^8*c^2*e^12

*f^2*z + 12*b^12*d*e^11*f*g*z + 36*b*c^11*d^12*f*g*z + 6*a*b^11*e^12*f*g*z + 60*b^3*c^9*d^11*e*g^2*z + 20*b^11

*c*d^3*e^9*g^2*z - 360*a*c^11*d^10*e^2*f^2*z + 20*a^3*b^8*c*e^12*g^2*z - 4*b^12*d^2*e^10*g^2*z + 768*a^7*c^5*e

^12*g^2*z - 9*b^2*c^10*d^12*g^2*z - 900*a^6*c^6*e^12*f^2*z - a^2*b^10*e^12*g^2*z - 36*c^12*d^12*f^2*z - 9*b^12

*e^12*f^2*z + 4644*a*b^2*c^6*d^2*e^8*f^2*g + 3420*a^2*b*c^6*d^2*e^8*f*g^2 - 2436*a*b^2*c^6*d^3*e^7*f*g^2 - 214

2*a^2*b^2*c^5*d*e^9*f*g^2 - 1470*a*b^3*c^5*d^2*e^8*f*g^2 + 1020*a*b^4*c^4*d*e^9*f*g^2 + 732*a*b*c^7*d^4*e^6*f*

g^2 + 720*a*b*c^7*d^3*e^7*f^2*g - 648*a^2*b*c^6*d*e^9*f^2*g - 468*a*b^3*c^5*d*e^9*f^2*g + 981*a^2*b^2*c^5*d^2*

e^8*g^3 - 540*b^3*c^6*d^3*e^7*f^2*g + 468*b^2*c^7*d^4*e^6*f^2*g - 459*b^4*c^5*d^2*e^8*f^2*g - 438*b^2*c^7*d^5*

e^5*f*g^2 + 396*b^4*c^5*d^3*e^7*f*g^2 + 120*b^5*c^4*d^2*e^8*f*g^2 + 87*b^3*c^6*d^4*e^6*f*g^2 - 7452*a^2*c^7*d^

2*e^8*f^2*g + 2688*a^2*c^7*d^3*e^7*f*g^2 + 1512*a^2*b^2*c^5*e^10*f^2*g + 555*a^2*b^3*c^4*e^10*f*g^2 - 1184*a^2

*b*c^6*d^3*e^7*g^3 + 796*a*b^3*c^5*d^3*e^7*g^3 - 360*a*b^4*c^4*d^2*e^8*g^3 - 350*a^2*b^3*c^4*d*e^9*g^3 + 7*a*b

^2*c^6*d^4*e^6*g^3 + 216*b*c^8*d^5*e^5*f^2*g + 180*b*c^8*d^6*e^4*f*g^2 - 120*b^6*c^3*d*e^9*f*g^2 + 90*b^5*c^4*

d*e^9*f^2*g - 1332*a*c^8*d^4*e^6*f^2*g + 1008*a^3*c^6*d*e^9*f*g^2 + 240*a*c^8*d^5*e^5*f*g^2 - 1404*a^3*b*c^5*e

^10*f*g^2 - 765*a*b^4*c^4*e^10*f^2*g - 60*a*b^5*c^3*e^10*f*g^2 + 760*a^3*b*c^5*d*e^9*g^3 - 120*a*b*c^7*d^5*e^5

*g^3 + 40*a*b^5*c^3*d*e^9*g^3 - 1944*a*b*c^7*d^2*e^8*f^3 - 1728*a*b^2*c^6*d*e^9*f^3 - 180*c^9*d^6*e^4*f^2*g +

90*b^6*c^3*e^10*f^2*g + 900*a^3*c^6*e^10*f^2*g - 540*b*c^8*d^4*e^6*f^3 + 162*b^4*c^5*d*e^9*f^3 + 5400*a^2*c^7*

d*e^9*f^3 + 1296*a*c^8*d^3*e^7*f^3 - 2700*a^2*b*c^6*e^10*f^3 + 1188*a*b^3*c^5*e^10*f^3 + 138*b^3*c^6*d^5*e^5*g

^3 - 98*b^4*c^5*d^4*e^6*g^3 - 80*b^5*c^4*d^3*e^7*g^3 - 45*b^2*c^7*d^6*e^4*g^3 + 40*b^6*c^3*d^2*e^8*g^3 - 1264*

a^3*c^6*d^2*e^8*g^3 + 216*b^3*c^6*d^2*e^8*f^3 + 216*b^2*c^7*d^3*e^7*f^3 - 80*a^2*c^7*d^4*e^6*g^3 - 95*a^3*b^2*

c^4*e^10*g^3 + 10*a^2*b^4*c^3*e^10*g^3 + 216*c^9*d^5*e^5*f^3 + 256*a^4*c^5*e^10*g^3 - 135*b^5*c^4*e^10*f^3, z,

k)*(root(286720*a^9*b*c^8*d^7*e^9*z^3 + 286720*a^8*b*c^9*d^9*e^7*z^3 + 172032*a^10*b*c^7*d^5*e^11*z^3 + 17203

2*a^7*b*c^10*d^11*e^5*z^3 + 57344*a^11*b*c^6*d^3*e^13*z^3 + 57344*a^6*b*c^11*d^13*e^3*z^3 - 10240*a^11*b^3*c^4

*d*e^15*z^3 - 10240*a^4*b^3*c^11*d^15*e*z^3 + 5120*a^10*b^5*c^3*d*e^15*z^3 + 5120*a^3*b^5*c^10*d^15*e*z^3 - 12

80*a^9*b^7*c^2*d*e^15*z^3 - 1280*a^2*b^7*c^9*d^15*e*z^3 - 1232*a^5*b^12*c*d^4*e^12*z^3 - 1232*a*b^12*c^5*d^12*

e^4*z^3 + 1064*a^6*b^11*c*d^3*e^13*z^3 + 1064*a*b^11*c^6*d^13*e^3*z^3 + 840*a^4*b^13*c*d^5*e^11*z^3 + 840*a*b^

13*c^4*d^11*e^5*z^3 - 552*a^7*b^10*c*d^2*e^14*z^3 - 552*a*b^10*c^7*d^14*e^2*z^3 - 280*a^3*b^14*c*d^6*e^10*z^3

- 280*a*b^14*c^3*d^10*e^6*z^3 - 8*a^2*b^15*c*d^7*e^9*z^3 - 8*a*b^15*c^2*d^9*e^7*z^3 + 8192*a^12*b*c^5*d*e^15*z

^3 + 8192*a^5*b*c^12*d^15*e*z^3 + 160*a^8*b^9*c*d*e^15*z^3 + 160*a*b^9*c^8*d^15*e*z^3 + 36*a*b^16*c*d^8*e^8*z^

3 - 483840*a^8*b^2*c^8*d^8*e^8*z^3 - 365568*a^7*b^5*c^6*d^7*e^9*z^3 - 365568*a^6*b^5*c^7*d^9*e^7*z^3 - 358400*

a^9*b^2*c^7*d^6*e^10*z^3 - 358400*a^7*b^2*c^9*d^10*e^6*z^3 + 241920*a^7*b^4*c^7*d^8*e^8*z^3 + 215040*a^8*b^4*c

^6*d^6*e^10*z^3 + 215040*a^8*b^3*c^7*d^7*e^9*z^3 + 215040*a^7*b^3*c^8*d^9*e^7*z^3 + 215040*a^6*b^4*c^8*d^10*e^

6*z^3 - 193536*a^8*b^5*c^5*d^5*e^11*z^3 - 193536*a^5*b^5*c^8*d^11*e^5*z^3 - 136192*a^10*b^2*c^6*d^4*e^12*z^3 -

136192*a^6*b^2*c^10*d^12*e^4*z^3 + 133056*a^6*b^6*c^6*d^8*e^8*z^3 + 125440*a^9*b^4*c^5*d^4*e^12*z^3 + 125440*

a^5*b^4*c^9*d^12*e^4*z^3 - 109944*a^5*b^8*c^5*d^8*e^8*z^3 + 106752*a^6*b^7*c^5*d^7*e^9*z^3 + 106752*a^5*b^7*c^

6*d^9*e^7*z^3 + 80640*a^7*b^7*c^4*d^5*e^11*z^3 + 80640*a^4*b^7*c^7*d^11*e^5*z^3 - 77280*a^6*b^8*c^4*d^6*e^10*z

^3 - 77280*a^4*b^8*c^6*d^10*e^6*z^3 + 71680*a^9*b^3*c^6*d^5*e^11*z^3 + 71680*a^6*b^3*c^9*d^11*e^5*z^3 + 69888*

a^7*b^6*c^5*d^6*e^10*z^3 + 69888*a^5*b^6*c^7*d^10*e^6*z^3 - 35840*a^9*b^5*c^4*d^3*e^13*z^3 - 35840*a^4*b^5*c^9

*d^13*e^3*z^3 + 30720*a^10*b^4*c^4*d^2*e^14*z^3 + 30720*a^4*b^4*c^10*d^14*e^2*z^3 + 26880*a^8*b^7*c^3*d^3*e^13

*z^3 + 26880*a^3*b^7*c^8*d^13*e^3*z^3 + 21510*a^4*b^10*c^4*d^8*e^8*z^3 + 18536*a^5*b^10*c^3*d^6*e^10*z^3 + 185

36*a^3*b^10*c^5*d^10*e^6*z^3 - 18480*a^7*b^8*c^3*d^4*e^12*z^3 - 18480*a^3*b^8*c^7*d^12*e^4*z^3 - 18432*a^11*b^

2*c^5*d^2*e^14*z^3 - 18432*a^5*b^2*c^11*d^14*e^2*z^3 - 16640*a^9*b^6*c^3*d^2*e^14*z^3 - 16640*a^3*b^6*c^9*d^14

*e^2*z^3 - 14336*a^10*b^3*c^5*d^3*e^13*z^3 - 14336*a^5*b^3*c^10*d^13*e^3*z^3 - 13440*a^8*b^6*c^4*d^4*e^12*z^3

- 13440*a^4*b^6*c^8*d^12*e^4*z^3 + 13280*a^5*b^9*c^4*d^7*e^9*z^3 + 13280*a^4*b^9*c^5*d^9*e^7*z^3 - 10840*a^4*b

^11*c^3*d^7*e^9*z^3 - 10840*a^3*b^11*c^4*d^9*e^7*z^3 + 7868*a^6*b^10*c^2*d^4*e^12*z^3 + 7868*a^2*b^10*c^6*d^12

*e^4*z^3 - 7840*a^7*b^9*c^2*d^3*e^13*z^3 - 7840*a^2*b^9*c^7*d^13*e^3*z^3 - 5600*a^6*b^9*c^3*d^5*e^11*z^3 - 560

0*a^3*b^9*c^6*d^11*e^5*z^3 + 4320*a^8*b^8*c^2*d^2*e^14*z^3 + 4320*a^2*b^8*c^8*d^14*e^2*z^3 - 3528*a^5*b^11*c^2

*d^5*e^11*z^3 - 3528*a^2*b^11*c^5*d^11*e^5*z^3 + 1520*a^3*b^13*c^2*d^7*e^9*z^3 + 1520*a^2*b^13*c^3*d^9*e^7*z^3

- 700*a^4*b^12*c^2*d^6*e^10*z^3 - 700*a^2*b^12*c^4*d^10*e^6*z^3 - 540*a^2*b^14*c^2*d^8*e^8*z^3 + 480*a^3*b^12

*c^3*d^8*e^8*z^3 - 8*b^17*c*d^9*e^7*z^3 - 8*b^11*c^7*d^15*e*z^3 - 8*a^7*b^11*d*e^15*z^3 - 8*a*b^17*d^7*e^9*z^3

- 20*a^9*b^8*c*e^16*z^3 - 20*a*b^8*c^9*d^16*z^3 + 70*b^14*c^4*d^12*e^4*z^3 - 56*b^15*c^3*d^11*e^5*z^3 - 56*b^

13*c^5*d^13*e^3*z^3 + 28*b^16*c^2*d^10*e^6*z^3 + 28*b^12*c^6*d^14*e^2*z^3 - 71680*a^9*c^9*d^8*e^8*z^3 - 57344*

a^10*c^8*d^6*e^10*z^3 - 57344*a^8*c^10*d^10*e^6*z^3 - 28672*a^11*c^7*d^4*e^12*z^3 - 28672*a^7*c^11*d^12*e^4*z^

3 - 8192*a^12*c^6*d^2*e^14*z^3 - 8192*a^6*c^12*d^14*e^2*z^3 + 70*a^4*b^14*d^4*e^12*z^3 - 56*a^5*b^13*d^3*e^13*

z^3 - 56*a^3*b^15*d^5*e^11*z^3 + 28*a^6*b^12*d^2*e^14*z^3 + 28*a^2*b^16*d^6*e^10*z^3 + 1280*a^12*b^2*c^4*e^16*

z^3 - 640*a^11*b^4*c^3*e^16*z^3 + 160*a^10*b^6*c^2*e^16*z^3 + 1280*a^4*b^2*c^12*d^16*z^3 - 640*a^3*b^4*c^11*d^

16*z^3 + 160*a^2*b^6*c^10*d^16*z^3 - 1024*a^13*c^5*e^16*z^3 - 1024*a^5*c^13*d^16*z^3 + b^18*d^8*e^8*z^3 + b^10

*c^8*d^16*z^3 + a^8*b^10*e^16*z^3 + 96*a*b*c^10*d^10*e^2*f*g*z + 69900*a^4*b^2*c^6*d^3*e^9*f*g*z - 64590*a^4*b

^3*c^5*d^2*e^10*f*g*z - 40200*a^3*b^4*c^5*d^3*e^9*f*g*z + 32820*a^3*b^5*c^4*d^2*e^10*f*g*z + 10680*a^2*b^6*c^4

*d^3*e^9*f*g*z + 10500*a^3*b^3*c^6*d^4*e^8*f*g*z + 8820*a^2*b^3*c^7*d^6*e^6*f*g*z - 8460*a^2*b^7*c^3*d^2*e^10*

f*g*z - 5880*a^3*b^2*c^7*d^5*e^7*f*g*z - 5040*a^2*b^4*c^6*d^5*e^7*f*g*z - 3240*a^2*b^2*c^8*d^7*e^5*f*g*z - 126

0*a^2*b^5*c^5*d^4*e^8*f*g*z - 252*a*b^10*c*d*e^11*f*g*z + 55872*a^5*b*c^6*d^2*e^10*f*g*z - 30636*a^5*b^2*c^5*d

*e^11*f*g*z + 24180*a^4*b^4*c^4*d*e^11*f*g*z - 9720*a^3*b^6*c^3*d*e^11*f*g*z + 3690*a*b^3*c^8*d^8*e^4*f*g*z -

3360*a^3*b*c^8*d^6*e^6*f*g*z - 3240*a*b^4*c^7*d^7*e^5*f*g*z + 2160*a^2*b^8*c^2*d*e^11*f*g*z - 2100*a^4*b*c^7*d

^4*e^8*f*g*z - 1500*a*b^2*c^9*d^9*e^3*f*g*z - 1320*a*b^8*c^3*d^3*e^9*f*g*z - 1260*a^2*b*c^9*d^8*e^4*f*g*z + 10

80*a*b^9*c^2*d^2*e^10*f*g*z + 924*a*b^6*c^5*d^5*e^7*f*g*z + 252*a*b^5*c^6*d^6*e^6*f*g*z - 150*a*b^7*c^4*d^4*e^

8*f*g*z + 48*a*c^11*d^11*e*f*g*z - 660*b^4*c^8*d^9*e^3*f*g*z + 570*b^3*c^9*d^10*e^2*f*g*z + 270*b^5*c^7*d^8*e^

4*f*g*z + 84*b^6*c^6*d^7*e^5*f*g*z + 60*b^10*c^2*d^3*e^9*f*g*z - 60*b^8*c^4*d^5*e^7*f*g*z - 42*b^7*c^5*d^6*e^6

*f*g*z + 30*b^9*c^3*d^4*e^8*f*g*z - 59280*a^5*c^7*d^3*e^9*f*g*z + 3360*a^4*c^8*d^5*e^7*f*g*z + 2400*a^3*c^9*d^

7*e^5*f*g*z + 720*a^2*c^10*d^9*e^3*f*g*z + 7410*a^5*b^3*c^4*e^12*f*g*z - 3810*a^4*b^5*c^3*e^12*f*g*z + 960*a^3

*b^7*c^2*e^12*f*g*z + 4872*a^6*b*c^5*d*e^11*g^2*z + 90*a*b^10*c*d^2*e^10*g^2*z + 80*a^2*b^9*c*d*e^11*g^2*z - 3

3048*a^5*b*c^6*d*e^11*f^2*z + 1800*a*b*c^10*d^9*e^3*f^2*z - 720*a*b^9*c^2*d*e^11*f^2*z - 31575*a^4*b^2*c^6*d^4

*e^8*g^2*z + 24700*a^4*b^3*c^5*d^3*e^9*g^2*z + 16722*a^5*b^2*c^5*d^2*e^10*g^2*z + 15700*a^3*b^4*c^5*d^4*e^8*g^

2*z - 12140*a^3*b^5*c^4*d^3*e^9*g^2*z - 11640*a^4*b^4*c^4*d^2*e^10*g^2*z - 4485*a^2*b^6*c^4*d^4*e^8*g^2*z + 41

80*a^3*b^6*c^3*d^2*e^10*g^2*z + 3120*a^2*b^7*c^3*d^3*e^9*g^2*z - 1960*a^3*b^3*c^6*d^5*e^7*g^2*z + 1820*a^3*b^2

*c^7*d^6*e^6*g^2*z + 1596*a^2*b^5*c^5*d^5*e^7*g^2*z + 1185*a^2*b^2*c^8*d^8*e^4*g^2*z - 1080*a^2*b^3*c^7*d^7*e^

5*g^2*z - 840*a^2*b^8*c^2*d^2*e^10*g^2*z - 840*a^2*b^4*c^6*d^6*e^6*g^2*z - 50760*a^4*b^2*c^6*d^2*e^10*f^2*z +

25380*a^3*b^4*c^5*d^2*e^10*f^2*z - 12600*a^3*b^2*c^7*d^4*e^8*f^2*z - 10080*a^2*b^2*c^8*d^6*e^6*f^2*z - 6030*a^

2*b^6*c^4*d^2*e^10*f^2*z + 3150*a^2*b^4*c^6*d^4*e^8*f^2*z + 2520*a^2*b^3*c^7*d^5*e^7*f^2*z - 1260*a^2*b^5*c^5*

d^3*e^9*f^2*z - 228*b^2*c^10*d^11*e*f*g*z - 54*b^11*c*d^2*e^10*f*g*z + 12816*a^6*c^6*d*e^11*f*g*z - 5508*a^6*b

*c^5*e^12*f*g*z - 120*a^2*b^9*c*e^12*f*g*z - 24*a*b*c^10*d^11*e*g^2*z - 18360*a^5*b*c^6*d^3*e^9*g^2*z - 5340*a

^5*b^3*c^4*d*e^11*g^2*z + 2580*a^4*b^5*c^3*d*e^11*g^2*z + 1680*a^4*b*c^7*d^5*e^7*g^2*z + 1380*a*b^5*c^6*d^7*e^

5*g^2*z - 1050*a*b^4*c^7*d^8*e^4*g^2*z - 686*a*b^6*c^5*d^6*e^6*g^2*z - 640*a^3*b^7*c^2*d*e^11*g^2*z + 570*a*b^

8*c^3*d^4*e^8*g^2*z - 400*a*b^9*c^2*d^3*e^9*g^2*z - 280*a^2*b*c^9*d^9*e^3*g^2*z + 260*a*b^3*c^8*d^9*e^3*g^2*z

+ 80*a^3*b*c^8*d^7*e^5*g^2*z + 50*a*b^2*c^9*d^10*e^2*g^2*z + 44460*a^4*b^3*c^5*d*e^11*f^2*z - 22860*a^3*b^5*c^

4*d*e^11*f^2*z + 15120*a^3*b*c^8*d^5*e^7*f^2*z + 12600*a^4*b*c^7*d^3*e^9*f^2*z + 7920*a^2*b*c^9*d^7*e^5*f^2*z

+ 5760*a^2*b^7*c^3*d*e^11*f^2*z - 3060*a*b^2*c^9*d^8*e^4*f^2*z + 1440*a*b^3*c^8*d^7*e^5*f^2*z - 1260*a*b^5*c^6

*d^5*e^7*f^2*z + 1260*a*b^4*c^7*d^6*e^6*f^2*z + 720*a*b^8*c^3*d^2*e^10*f^2*z + 180*a*b^7*c^4*d^3*e^9*f^2*z + 2

16*b*c^11*d^11*e*f^2*z + 36*b^11*c*d*e^11*f^2*z - 4*a*b^11*d*e^11*g^2*z + 180*a*b^10*c*e^12*f^2*z + 200*b^5*c^

7*d^9*e^3*g^2*z - 160*b^4*c^8*d^10*e^2*g^2*z - 85*b^6*c^6*d^8*e^4*g^2*z + 70*b^8*c^4*d^6*e^6*g^2*z - 56*b^7*c^

5*d^7*e^5*g^2*z - 25*b^10*c^2*d^4*e^8*g^2*z - 20*b^9*c^3*d^5*e^7*g^2*z + 24000*a^5*c^7*d^4*e^8*g^2*z - 11280*a

^6*c^6*d^2*e^10*g^2*z - 1120*a^4*c^8*d^6*e^6*g^2*z + 540*b^3*c^9*d^9*e^3*f^2*z - 504*b^2*c^10*d^10*e^2*f^2*z -

320*a^3*c^9*d^8*e^4*g^2*z - 225*b^4*c^8*d^8*e^4*f^2*z + 144*b^7*c^5*d^5*e^7*f^2*z - 126*b^6*c^6*d^6*e^6*f^2*z

- 45*b^8*c^4*d^4*e^8*f^2*z - 36*b^10*c^2*d^2*e^10*f^2*z + 36*b^5*c^7*d^7*e^5*f^2*z - 16*a^2*c^10*d^10*e^2*g^2

*z + 33048*a^5*c^7*d^2*e^10*f^2*z - 6300*a^4*c^8*d^4*e^8*f^2*z - 5040*a^3*c^9*d^6*e^6*f^2*z - 1980*a^2*c^10*d^

8*e^4*f^2*z - 1185*a^6*b^2*c^4*e^12*g^2*z + 630*a^5*b^4*c^3*e^12*g^2*z - 160*a^4*b^6*c^2*e^12*g^2*z - 11565*a^

4*b^4*c^4*e^12*f^2*z + 9612*a^5*b^2*c^5*e^12*f^2*z + 5760*a^3*b^6*c^3*e^12*f^2*z - 1440*a^2*b^8*c^2*e^12*f^2*z

+ 12*b^12*d*e^11*f*g*z + 36*b*c^11*d^12*f*g*z + 6*a*b^11*e^12*f*g*z + 60*b^3*c^9*d^11*e*g^2*z + 20*b^11*c*d^3

*e^9*g^2*z - 360*a*c^11*d^10*e^2*f^2*z + 20*a^3*b^8*c*e^12*g^2*z - 4*b^12*d^2*e^10*g^2*z + 768*a^7*c^5*e^12*g^

2*z - 9*b^2*c^10*d^12*g^2*z - 900*a^6*c^6*e^12*f^2*z - a^2*b^10*e^12*g^2*z - 36*c^12*d^12*f^2*z - 9*b^12*e^12*

f^2*z + 4644*a*b^2*c^6*d^2*e^8*f^2*g + 3420*a^2*b*c^6*d^2*e^8*f*g^2 - 2436*a*b^2*c^6*d^3*e^7*f*g^2 - 2142*a^2*

b^2*c^5*d*e^9*f*g^2 - 1470*a*b^3*c^5*d^2*e^8*f*g^2 + 1020*a*b^4*c^4*d*e^9*f*g^2 + 732*a*b*c^7*d^4*e^6*f*g^2 +

720*a*b*c^7*d^3*e^7*f^2*g - 648*a^2*b*c^6*d*e^9*f^2*g - 468*a*b^3*c^5*d*e^9*f^2*g + 981*a^2*b^2*c^5*d^2*e^8*g^

3 - 540*b^3*c^6*d^3*e^7*f^2*g + 468*b^2*c^7*d^4*e^6*f^2*g - 459*b^4*c^5*d^2*e^8*f^2*g - 438*b^2*c^7*d^5*e^5*f*

g^2 + 396*b^4*c^5*d^3*e^7*f*g^2 + 120*b^5*c^4*d^2*e^8*f*g^2 + 87*b^3*c^6*d^4*e^6*f*g^2 - 7452*a^2*c^7*d^2*e^8*

f^2*g + 2688*a^2*c^7*d^3*e^7*f*g^2 + 1512*a^2*b^2*c^5*e^10*f^2*g + 555*a^2*b^3*c^4*e^10*f*g^2 - 1184*a^2*b*c^6

*d^3*e^7*g^3 + 796*a*b^3*c^5*d^3*e^7*g^3 - 360*a*b^4*c^4*d^2*e^8*g^3 - 350*a^2*b^3*c^4*d*e^9*g^3 + 7*a*b^2*c^6

*d^4*e^6*g^3 + 216*b*c^8*d^5*e^5*f^2*g + 180*b*c^8*d^6*e^4*f*g^2 - 120*b^6*c^3*d*e^9*f*g^2 + 90*b^5*c^4*d*e^9*

f^2*g - 1332*a*c^8*d^4*e^6*f^2*g + 1008*a^3*c^6*d*e^9*f*g^2 + 240*a*c^8*d^5*e^5*f*g^2 - 1404*a^3*b*c^5*e^10*f*

g^2 - 765*a*b^4*c^4*e^10*f^2*g - 60*a*b^5*c^3*e^10*f*g^2 + 760*a^3*b*c^5*d*e^9*g^3 - 120*a*b*c^7*d^5*e^5*g^3 +

40*a*b^5*c^3*d*e^9*g^3 - 1944*a*b*c^7*d^2*e^8*f^3 - 1728*a*b^2*c^6*d*e^9*f^3 - 180*c^9*d^6*e^4*f^2*g + 90*b^6

*c^3*e^10*f^2*g + 900*a^3*c^6*e^10*f^2*g - 540*b*c^8*d^4*e^6*f^3 + 162*b^4*c^5*d*e^9*f^3 + 5400*a^2*c^7*d*e^9*

f^3 + 1296*a*c^8*d^3*e^7*f^3 - 2700*a^2*b*c^6*e^10*f^3 + 1188*a*b^3*c^5*e^10*f^3 + 138*b^3*c^6*d^5*e^5*g^3 - 9

8*b^4*c^5*d^4*e^6*g^3 - 80*b^5*c^4*d^3*e^7*g^3 - 45*b^2*c^7*d^6*e^4*g^3 + 40*b^6*c^3*d^2*e^8*g^3 - 1264*a^3*c^

6*d^2*e^8*g^3 + 216*b^3*c^6*d^2*e^8*f^3 + 216*b^2*c^7*d^3*e^7*f^3 - 80*a^2*c^7*d^4*e^6*g^3 - 95*a^3*b^2*c^4*e^

10*g^3 + 10*a^2*b^4*c^3*e^10*g^3 + 216*c^9*d^5*e^5*f^3 + 256*a^4*c^5*e^10*g^3 - 135*b^5*c^4*e^10*f^3, z, k)*((

2048*a^11*c^6*d*e^14 - 256*a^11*b*c^5*e^15 - a^7*b^9*c*e^15 - b^9*c^8*d^14*e - b^16*c*d^7*e^8 + 16*a^8*b^7*c^2

*e^15 - 96*a^9*b^5*c^3*e^15 + 256*a^10*b^3*c^4*e^15 + 2048*a^5*c^12*d^13*e^2 + 12288*a^6*c^11*d^11*e^4 + 30720

*a^7*c^10*d^9*e^6 + 40960*a^8*c^9*d^7*e^8 + 30720*a^9*c^8*d^5*e^10 + 12288*a^10*c^7*d^3*e^12 + 5*b^10*c^7*d^13

*e^2 - 9*b^11*c^6*d^12*e^3 + 5*b^12*c^5*d^11*e^4 + 5*b^13*c^4*d^10*e^5 - 9*b^14*c^3*d^9*e^6 + 5*b^15*c^2*d^8*e

^7 + 352*a^2*b^6*c^9*d^13*e^2 + 16*a^2*b^7*c^8*d^12*e^3 - 1872*a^2*b^8*c^7*d^11*e^4 + 3499*a^2*b^9*c^6*d^10*e^

5 - 2549*a^2*b^10*c^5*d^9*e^6 + 438*a^2*b^11*c^4*d^8*e^7 + 334*a^2*b^12*c^3*d^7*e^8 - 113*a^2*b^13*c^2*d^6*e^9

- 512*a^3*b^4*c^10*d^13*e^2 - 2976*a^3*b^5*c^9*d^12*e^3 + 12352*a^3*b^6*c^8*d^11*e^4 - 16784*a^3*b^7*c^7*d^10

*e^5 + 6984*a^3*b^8*c^6*d^9*e^6 + 4157*a^3*b^9*c^5*d^8*e^7 - 4204*a^3*b^10*c^4*d^7*e^8 + 438*a^3*b^11*c^3*d^6*

e^9 + 284*a^3*b^12*c^2*d^5*e^10 - 768*a^4*b^2*c^11*d^13*e^2 + 11776*a^4*b^3*c^10*d^12*e^3 - 32512*a^4*b^4*c^9*

d^11*e^4 + 28704*a^4*b^5*c^8*d^10*e^5 + 13216*a^4*b^6*c^7*d^9*e^6 - 36752*a^4*b^7*c^6*d^8*e^7 + 15264*a^4*b^8*

c^5*d^7*e^8 + 4157*a^4*b^9*c^4*d^6*e^9 - 2549*a^4*b^10*c^3*d^5*e^10 - 285*a^4*b^11*c^2*d^4*e^11 + 26112*a^5*b^

2*c^10*d^11*e^4 + 14336*a^5*b^3*c^9*d^10*e^5 - 94720*a^5*b^4*c^8*d^9*e^6 + 88032*a^5*b^5*c^7*d^8*e^7 + 4480*a^

5*b^6*c^6*d^7*e^8 - 36752*a^5*b^7*c^5*d^6*e^9 + 6984*a^5*b^8*c^4*d^5*e^10 + 3499*a^5*b^9*c^3*d^4*e^11 + 70*a^5

*b^10*c^2*d^3*e^12 + 111360*a^6*b^2*c^9*d^9*e^6 - 14080*a^6*b^3*c^8*d^8*e^7 - 125440*a^6*b^4*c^7*d^7*e^8 + 880

32*a^6*b^5*c^6*d^6*e^9 + 13216*a^6*b^6*c^5*d^5*e^10 - 16784*a^6*b^7*c^4*d^4*e^11 - 1872*a^6*b^8*c^3*d^3*e^12 +

89*a^6*b^9*c^2*d^2*e^13 + 168960*a^7*b^2*c^8*d^7*e^8 - 14080*a^7*b^3*c^7*d^6*e^9 - 94720*a^7*b^4*c^6*d^5*e^10

+ 28704*a^7*b^5*c^5*d^4*e^11 + 12352*a^7*b^6*c^4*d^3*e^12 + 16*a^7*b^7*c^3*d^2*e^13 + 111360*a^8*b^2*c^7*d^5*

e^10 + 14336*a^8*b^3*c^6*d^4*e^11 - 32512*a^8*b^4*c^5*d^3*e^12 - 2976*a^8*b^5*c^4*d^2*e^13 + 26112*a^9*b^2*c^6

*d^3*e^12 + 11776*a^9*b^3*c^5*d^2*e^13 + 16*a*b^7*c^9*d^14*e + 5*a*b^15*c*d^6*e^9 - 256*a^4*b*c^12*d^14*e + 5*

a^6*b^10*c*d*e^14 - 72*a*b^8*c^8*d^13*e^2 + 89*a*b^9*c^7*d^12*e^3 + 70*a*b^10*c^6*d^11*e^4 - 285*a*b^11*c^5*d^

10*e^5 + 284*a*b^12*c^4*d^9*e^6 - 113*a*b^13*c^3*d^8*e^7 + 6*a*b^14*c^2*d^7*e^8 - 96*a^2*b^5*c^10*d^14*e - 9*a

^2*b^14*c*d^5*e^10 + 256*a^3*b^3*c^11*d^14*e + 5*a^3*b^13*c*d^4*e^11 + 5*a^4*b^12*c*d^3*e^12 - 14080*a^5*b*c^1

1*d^12*e^3 - 9*a^5*b^11*c*d^2*e^13 - 66816*a^6*b*c^10*d^10*e^5 - 131840*a^7*b*c^9*d^8*e^7 - 72*a^7*b^8*c^2*d*e

^14 - 131840*a^8*b*c^8*d^6*e^9 + 352*a^8*b^6*c^3*d*e^14 - 66816*a^9*b*c^7*d^4*e^11 - 512*a^9*b^4*c^4*d*e^14 -

14080*a^10*b*c^6*d^2*e^13 - 768*a^10*b^2*c^5*d*e^14)/(256*a^4*c^10*d^12 + a^6*b^8*e^12 + 256*a^10*c^4*e^12 + b

^8*c^6*d^12 + b^14*d^6*e^6 - 16*a*b^6*c^7*d^12 - 16*a^7*b^6*c*e^12 - 6*a*b^13*d^5*e^7 - 6*a^5*b^9*d*e^11 - 6*b

^9*c^5*d^11*e - 6*b^13*c*d^7*e^5 + 96*a^2*b^4*c^8*d^12 - 256*a^3*b^2*c^9*d^12 + 96*a^8*b^4*c^2*e^12 - 256*a^9*

b^2*c^3*e^12 + 15*a^2*b^12*d^4*e^8 - 20*a^3*b^11*d^3*e^9 + 15*a^4*b^10*d^2*e^10 + 1536*a^5*c^9*d^10*e^2 + 3840

*a^6*c^8*d^8*e^4 + 5120*a^7*c^7*d^6*e^6 + 3840*a^8*c^6*d^4*e^8 + 1536*a^9*c^5*d^2*e^10 + 15*b^10*c^4*d^10*e^2

- 20*b^11*c^3*d^9*e^3 + 15*b^12*c^2*d^8*e^4 + 1344*a^2*b^6*c^6*d^10*e^2 - 1440*a^2*b^7*c^5*d^9*e^3 + 495*a^2*b

^8*c^4*d^8*e^4 + 324*a^2*b^9*c^3*d^7*e^5 - 294*a^2*b^10*c^2*d^6*e^6 - 3264*a^3*b^4*c^7*d^10*e^2 + 2240*a^3*b^5

*c^6*d^9*e^3 + 1680*a^3*b^6*c^5*d^8*e^4 - 3264*a^3*b^7*c^4*d^7*e^5 + 1204*a^3*b^8*c^3*d^6*e^6 + 324*a^3*b^9*c^

2*d^5*e^7 + 2304*a^4*b^2*c^8*d^10*e^2 + 2560*a^4*b^3*c^7*d^9*e^3 - 10080*a^4*b^4*c^6*d^8*e^4 + 8064*a^4*b^5*c^

5*d^7*e^5 + 896*a^4*b^6*c^4*d^6*e^6 - 3264*a^4*b^7*c^3*d^5*e^7 + 495*a^4*b^8*c^2*d^4*e^8 + 11520*a^5*b^2*c^7*d

^8*e^4 - 13440*a^5*b^4*c^5*d^6*e^6 + 8064*a^5*b^5*c^4*d^5*e^7 + 1680*a^5*b^6*c^3*d^4*e^8 - 1440*a^5*b^7*c^2*d^

3*e^9 + 17920*a^6*b^2*c^6*d^6*e^6 - 10080*a^6*b^4*c^4*d^4*e^8 + 2240*a^6*b^5*c^3*d^3*e^9 + 1344*a^6*b^6*c^2*d^

2*e^10 + 11520*a^7*b^2*c^5*d^4*e^8 + 2560*a^7*b^3*c^4*d^3*e^9 - 3264*a^7*b^4*c^3*d^2*e^10 + 2304*a^8*b^2*c^4*d

^2*e^10 + 96*a*b^7*c^6*d^11*e + 14*a*b^12*c*d^6*e^6 - 1536*a^4*b*c^9*d^11*e + 96*a^6*b^7*c*d*e^11 - 1536*a^9*b

*c^4*d*e^11 - 234*a*b^8*c^5*d^10*e^2 + 290*a*b^9*c^4*d^9*e^3 - 180*a*b^10*c^3*d^8*e^4 + 36*a*b^11*c^2*d^7*e^5

- 576*a^2*b^5*c^7*d^11*e + 36*a^2*b^11*c*d^5*e^7 + 1536*a^3*b^3*c^8*d^11*e - 180*a^3*b^10*c*d^4*e^8 + 290*a^4*

b^9*c*d^3*e^9 - 7680*a^5*b*c^8*d^9*e^3 - 234*a^5*b^8*c*d^2*e^10 - 15360*a^6*b*c^7*d^7*e^5 - 15360*a^7*b*c^6*d^

5*e^7 - 576*a^7*b^5*c^2*d*e^11 - 7680*a^8*b*c^5*d^3*e^9 + 1536*a^8*b^3*c^3*d*e^11) - (x*(2*a^6*b^10*c*e^15 - 1

536*a^11*c^6*e^15 + 512*a^4*c^13*d^14*e + 2*b^8*c^9*d^14*e + 2*b^16*c*d^6*e^9 - 38*a^7*b^8*c^2*e^15 + 288*a^8*

b^6*c^3*e^15 - 1088*a^9*b^4*c^4*e^15 + 2048*a^10*b^2*c^5*e^15 + 1536*a^5*c^12*d^12*e^3 - 1536*a^6*c^11*d^10*e^

5 - 12800*a^7*c^10*d^8*e^7 - 23040*a^8*c^9*d^6*e^9 - 19968*a^9*c^8*d^4*e^11 - 8704*a^10*c^7*d^2*e^13 - 14*b^9*

c^8*d^13*e^2 + 44*b^10*c^7*d^12*e^3 - 82*b^11*c^6*d^11*e^4 + 100*b^12*c^5*d^10*e^5 - 82*b^13*c^4*d^9*e^6 + 44*

b^14*c^3*d^8*e^7 - 14*b^15*c^2*d^7*e^8 - 1344*a^2*b^5*c^10*d^13*e^2 + 4128*a^2*b^6*c^9*d^12*e^3 - 7296*a^2*b^7

*c^8*d^11*e^4 + 7962*a^2*b^8*c^7*d^10*e^5 - 4962*a^2*b^9*c^6*d^9*e^6 + 834*a^2*b^10*c^5*d^8*e^7 + 1092*a^2*b^1

1*c^4*d^7*e^8 - 714*a^2*b^12*c^3*d^6*e^9 + 78*a^2*b^13*c^2*d^5*e^10 + 3584*a^3*b^3*c^11*d^13*e^2 - 10688*a^3*b

^4*c^10*d^12*e^3 + 17536*a^3*b^5*c^9*d^11*e^4 - 15712*a^3*b^6*c^8*d^10*e^5 + 3232*a^3*b^7*c^7*d^9*e^6 + 9326*a

^3*b^8*c^6*d^8*e^7 - 10232*a^3*b^9*c^5*d^7*e^8 + 3164*a^3*b^10*c^4*d^6*e^9 + 752*a^3*b^11*c^3*d^5*e^10 - 410*a

^3*b^12*c^2*d^4*e^11 + 9728*a^4*b^2*c^11*d^12*e^3 - 11776*a^4*b^3*c^10*d^11*e^4 - 1088*a^4*b^4*c^9*d^10*e^5 +

27968*a^4*b^5*c^8*d^9*e^6 - 44576*a^4*b^6*c^7*d^8*e^7 + 27392*a^4*b^7*c^6*d^7*e^8 + 2086*a^4*b^8*c^5*d^6*e^9 -

8882*a^4*b^9*c^4*d^5*e^10 + 1520*a^4*b^10*c^3*d^4*e^11 + 670*a^4*b^11*c^2*d^3*e^12 + 27648*a^5*b^2*c^10*d^10*

e^5 - 53760*a^5*b^3*c^9*d^9*e^6 + 56640*a^5*b^4*c^8*d^8*e^7 - 5376*a^5*b^5*c^7*d^7*e^8 - 45024*a^5*b^6*c^6*d^6

*e^9 + 29472*a^5*b^7*c^5*d^5*e^10 + 2802*a^5*b^8*c^4*d^4*e^11 - 4164*a^5*b^9*c^3*d^3*e^12 - 546*a^5*b^10*c^2*d

^2*e^13 + 5120*a^6*b^2*c^9*d^8*e^7 - 66560*a^6*b^3*c^8*d^7*e^8 + 96320*a^6*b^4*c^7*d^6*e^9 - 23744*a^6*b^5*c^6

*d^5*e^10 - 32032*a^6*b^6*c^5*d^4*e^11 + 10624*a^6*b^7*c^4*d^3*e^12 + 3902*a^6*b^8*c^3*d^2*e^13 - 43520*a^7*b^

2*c^8*d^6*e^9 - 48640*a^7*b^3*c^7*d^5*e^10 + 71872*a^7*b^4*c^6*d^4*e^11 - 2944*a^7*b^5*c^5*d^3*e^12 - 13472*a^

7*b^6*c^4*d^2*e^13 - 41472*a^8*b^2*c^7*d^4*e^11 - 32256*a^8*b^3*c^6*d^3*e^12 + 21312*a^8*b^4*c^5*d^2*e^13 - 81

92*a^9*b^2*c^6*d^2*e^13 - 32*a*b^6*c^10*d^14*e - 12*a*b^15*c*d^5*e^10 - 12*a^5*b^11*c*d*e^14 + 8704*a^10*b*c^6

*d*e^14 + 224*a*b^7*c^9*d^13*e^2 - 698*a*b^8*c^8*d^12*e^3 + 1276*a*b^9*c^7*d^11*e^4 - 1498*a*b^10*c^6*d^10*e^5

+ 1132*a*b^11*c^5*d^9*e^6 - 494*a*b^12*c^4*d^8*e^7 + 68*a*b^13*c^3*d^7*e^8 + 34*a*b^14*c^2*d^6*e^9 + 192*a^2*

b^4*c^11*d^14*e + 30*a^2*b^14*c*d^4*e^11 - 512*a^3*b^2*c^12*d^14*e - 40*a^3*b^13*c*d^3*e^12 - 3584*a^4*b*c^12*

d^13*e^2 + 30*a^4*b^12*c*d^2*e^13 - 9216*a^5*b*c^11*d^11*e^4 + 7680*a^6*b*c^10*d^9*e^6 + 226*a^6*b^9*c^2*d*e^1

4 + 51200*a^7*b*c^9*d^7*e^8 - 1696*a^7*b^7*c^3*d*e^14 + 69120*a^8*b*c^8*d^5*e^10 + 6336*a^8*b^5*c^4*d*e^14 + 3

9936*a^9*b*c^7*d^3*e^12 - 11776*a^9*b^3*c^5*d*e^14))/(256*a^4*c^10*d^12 + a^6*b^8*e^12 + 256*a^10*c^4*e^12 + b