3.2228 \(\int \frac{1}{(d+e x)^2 (a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=1761 \[ \text{result too large to display} \]

[Out]

(5*e*(14*c^8*d^8 - b^8*e^8 - 4*c^7*d^6*e*(14*b*d - 19*a*e) + b^6*c*e^7*(b*d + 15*a*e) + b^4*c^2*e^6*(b^2*d^2 -
16*a*b*d*e - 82*a^2*e^2) + c^6*d^4*e^2*(79*b^2*d^2 - 228*a*b*d*e + 176*a^2*e^2) - c^5*d^2*e^3*(41*b^3*d^3 - 1
97*a*b^2*d^2*e + 352*a^2*b*d*e^2 - 244*a^3*e^3) + b^2*c^3*e^5*(b^3*d^3 - 15*a*b^2*d^2*e + 95*a^2*b*d*e^2 + 187
*a^3*e^3) + c^4*e^4*(b^4*d^4 - 14*a*b^3*d^3*e + 81*a^2*b^2*d^2*e^2 - 244*a^3*b*d*e^3 - 126*a^4*e^4)))/((b^2 -
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(d + e*x)) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^4) - (8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)
*(14*c^2*d^2 - 5*b^2*e^2 - 6*c*e*(b*d - 3*a*e)) - c*(2*c*d - b*e)*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*
a*e))*x)/(12*(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)^3) + (3*a*c*e*(2*c*d - b*e)
^2*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 10*b^4*e^4 - 2*
c^3*d^2*e*(49*b*d - 78*a*e) + 5*b^2*c*e^3*(2*b*d - 15*a*e) + 3*c^2*e^2*(b^2*d^2 - 18*a*b*d*e + 42*a^2*e^2)) -
5*c*(2*c*d - b*e)*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 12*a*e) + 3*c^2*
e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)*(a + b*x
+ c*x^2)^2) - (5*(2*a*c*e*(2*c*d - b*e)^2*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*
(7*b*d - 12*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2)) - (b*c*d - b^2*e + 2*a*c*e)*(42*c^6*d^6 -
3*b^6*e^6 - 2*c^5*d^4*e*(49*b*d - 65*a*e) - 2*b^4*c*e^5*(b*d - 17*a*e) + b^2*c^2*e^4*(b^2*d^2 + 16*a*b*d*e - 1
23*a^2*e^2) + c^4*d^2*e^2*(55*b^2*d^2 - 164*a*b*d*e + 150*a^2*e^2) + 6*c^3*e^3*(b^3*d^3 - 4*a*b^2*d^2*e - 3*a^
2*b*d*e^2 + 21*a^3*e^3)) - 3*c*(2*c*d - b*e)*(14*c^6*d^6 - b^6*e^6 - 2*c^5*d^4*e*(21*b*d - 31*a*e) - 2*b^4*c*e
^5*(b*d - 7*a*e) - b^2*c^2*e^4*(3*b^2*d^2 - 26*a*b*d*e + 69*a^2*e^2) + c^4*d^2*e^2*(37*b^2*d^2 - 124*a*b*d*e +
114*a^2*e^2) - 2*c^3*e^3*(2*b^3*d^3 - 18*a*b^2*d^2*e + 57*a^2*b*d*e^2 - 65*a^3*e^3))*x))/(6*(b^2 - 4*a*c)^4*(
c*d^2 - b*d*e + a*e^2)^4*(d + e*x)*(a + b*x + c*x^2)) - (5*(28*c^10*d^10 + b^10*e^10 - 20*c^9*d^8*e*(7*b*d - 9
*a*e) - 252*a^4*c^5*e^9*(5*b*d + a*e) + 210*a^3*b^2*c^4*e^9*(4*b*d + 3*a*e) - 84*a^2*b^4*c^3*e^9*(3*b*d + 5*a*
e) + 18*a*b^6*c^2*e^9*(2*b*d + 7*a*e) - 2*b^8*c*e^9*(b*d + 9*a*e) + 18*c^8*d^6*e^2*(15*b^2*d^2 - 40*a*b*d*e +
28*a^2*e^2) - 24*c^7*d^4*e^3*(10*b^3*d^3 - 42*a*b^2*d^2*e + 63*a^2*b*d*e^2 - 35*a^3*e^3) + 84*c^6*d^2*e^4*(b^4
*d^4 - 6*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 15*a^4*e^4))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(9/2)*(c*d^2 - e*(b*d - a*e))^6) + (5*e^9*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^
2)^6 - (5*e^9*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

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Rubi [A]  time = 15.1959, antiderivative size = 1761, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {740, 822, 800, 634, 618, 206, 628} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^5),x]

[Out]

(5*e*(14*c^8*d^8 - b^8*e^8 - 4*c^7*d^6*e*(14*b*d - 19*a*e) + b^6*c*e^7*(b*d + 15*a*e) + b^4*c^2*e^6*(b^2*d^2 -
16*a*b*d*e - 82*a^2*e^2) + c^6*d^4*e^2*(79*b^2*d^2 - 228*a*b*d*e + 176*a^2*e^2) - c^5*d^2*e^3*(41*b^3*d^3 - 1
97*a*b^2*d^2*e + 352*a^2*b*d*e^2 - 244*a^3*e^3) + b^2*c^3*e^5*(b^3*d^3 - 15*a*b^2*d^2*e + 95*a^2*b*d*e^2 + 187
*a^3*e^3) + c^4*e^4*(b^4*d^4 - 14*a*b^3*d^3*e + 81*a^2*b^2*d^2*e^2 - 244*a^3*b*d*e^3 - 126*a^4*e^4)))/((b^2 -
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5*(d + e*x)) - (b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*(a + b*x + c*x^2)^4) - (8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)
*(14*c^2*d^2 - 5*b^2*e^2 - 6*c*e*(b*d - 3*a*e)) - c*(2*c*d - b*e)*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*
a*e))*x)/(12*(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)^3) + (3*a*c*e*(2*c*d - b*e)
^2*(14*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(7*b*d - 17*a*e)) - (b*c*d - b^2*e + 2*a*c*e)*(70*c^4*d^4 + 10*b^4*e^4 - 2*
c^3*d^2*e*(49*b*d - 78*a*e) + 5*b^2*c*e^3*(2*b*d - 15*a*e) + 3*c^2*e^2*(b^2*d^2 - 18*a*b*d*e + 42*a^2*e^2)) -
5*c*(2*c*d - b*e)*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*(7*b*d - 12*a*e) + 3*c^2*
e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)*(a + b*x
+ c*x^2)^2) - (5*(2*a*c*e*(2*c*d - b*e)^2*(14*c^4*d^4 + 2*b^4*e^4 + b^2*c*e^3*(5*b*d - 21*a*e) - 4*c^3*d^2*e*
(7*b*d - 12*a*e) + 3*c^2*e^2*(3*b^2*d^2 - 16*a*b*d*e + 22*a^2*e^2)) - (b*c*d - b^2*e + 2*a*c*e)*(42*c^6*d^6 -
3*b^6*e^6 - 2*c^5*d^4*e*(49*b*d - 65*a*e) - 2*b^4*c*e^5*(b*d - 17*a*e) + b^2*c^2*e^4*(b^2*d^2 + 16*a*b*d*e - 1
23*a^2*e^2) + c^4*d^2*e^2*(55*b^2*d^2 - 164*a*b*d*e + 150*a^2*e^2) + 6*c^3*e^3*(b^3*d^3 - 4*a*b^2*d^2*e - 3*a^
2*b*d*e^2 + 21*a^3*e^3)) - 3*c*(2*c*d - b*e)*(14*c^6*d^6 - b^6*e^6 - 2*c^5*d^4*e*(21*b*d - 31*a*e) - 2*b^4*c*e
^5*(b*d - 7*a*e) - b^2*c^2*e^4*(3*b^2*d^2 - 26*a*b*d*e + 69*a^2*e^2) + c^4*d^2*e^2*(37*b^2*d^2 - 124*a*b*d*e +
114*a^2*e^2) - 2*c^3*e^3*(2*b^3*d^3 - 18*a*b^2*d^2*e + 57*a^2*b*d*e^2 - 65*a^3*e^3))*x))/(6*(b^2 - 4*a*c)^4*(
c*d^2 - b*d*e + a*e^2)^4*(d + e*x)*(a + b*x + c*x^2)) - (5*(28*c^10*d^10 + b^10*e^10 - 20*c^9*d^8*e*(7*b*d - 9
*a*e) - 252*a^4*c^5*e^9*(5*b*d + a*e) + 210*a^3*b^2*c^4*e^9*(4*b*d + 3*a*e) - 84*a^2*b^4*c^3*e^9*(3*b*d + 5*a*
e) + 18*a*b^6*c^2*e^9*(2*b*d + 7*a*e) - 2*b^8*c*e^9*(b*d + 9*a*e) + 18*c^8*d^6*e^2*(15*b^2*d^2 - 40*a*b*d*e +
28*a^2*e^2) - 24*c^7*d^4*e^3*(10*b^3*d^3 - 42*a*b^2*d^2*e + 63*a^2*b*d*e^2 - 35*a^3*e^3) + 84*c^6*d^2*e^4*(b^4
*d^4 - 6*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 - 20*a^3*b*d*e^3 + 15*a^4*e^4))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c
]])/((b^2 - 4*a*c)^(9/2)*(c*d^2 - e*(b*d - a*e))^6) + (5*e^9*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^
2)^6 - (5*e^9*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

Rule 740

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

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])

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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^5} \, dx &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{\int \frac{14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)+8 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{\int \frac{2 \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )+6 c e (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{20 \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )+40 c e (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx}{24 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac{\int \frac{120 \left (14 c^8 d^8+b^8 e^8-15 a b^6 c e^8-6 c^7 d^6 e (7 b d-8 a e)-b^4 c^2 e^6 \left (b^2 d^2-2 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (37 b^2 d^2-82 a b d e+52 a^2 e^2\right )-c^5 d^2 e^3 \left (4 b^3 d^3+a b^2 d^2 e-10 a^2 b d e^2-16 a^3 e^3\right )-b^2 c^3 e^5 \left (2 b^3 d^3-17 a b^2 d^2 e+26 a^2 b d e^2+187 a^3 e^3\right )-3 c^4 e^4 \left (b^4 d^4-10 a b^3 d^3 e+35 a^2 b^2 d^2 e^2-38 a^3 b d e^3-42 a^4 e^4\right )\right )+120 c e (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{24 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{120 e^2 \left (-14 c^8 d^8+b^8 e^8+4 c^7 d^6 e (14 b d-19 a e)-b^6 c e^7 (b d+15 a e)-b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )-c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )+c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )-b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )-c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac{120 \left (b^2-4 a c\right )^4 e^{10} (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{120 \left (14 c^{10} d^{10}+b^{10} e^{10}-10 c^9 d^8 e (7 b d-9 a e)-b^8 c e^9 (2 b d+17 a e)-2 a^4 c^5 e^9 (443 b d+63 a e)+a b^6 c^2 e^9 (34 b d+111 a e)-2 a^2 b^4 c^3 e^9 (111 b d+169 a e)+a^3 b^2 c^4 e^9 (676 b d+443 a e)+9 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-12 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+42 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{24 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4}\\ &=\frac{5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac{5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}+\frac{5 \int \frac{14 c^{10} d^{10}+b^{10} e^{10}-10 c^9 d^8 e (7 b d-9 a e)-b^8 c e^9 (2 b d+17 a e)-2 a^4 c^5 e^9 (443 b d+63 a e)+a b^6 c^2 e^9 (34 b d+111 a e)-2 a^2 b^4 c^3 e^9 (111 b d+169 a e)+a^3 b^2 c^4 e^9 (676 b d+443 a e)+9 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-12 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+42 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac{5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac{5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac{\left (5 e^9 (2 c d-b e)\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 )^6}+\frac{\left (5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac{5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}+\frac{5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac{5 e^9 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^6}-\frac{\left (5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\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 )^4 \left (c d^2-b d e+a e^2\right )^6}\\ &=\frac{5 e \left (14 c^8 d^8-b^8 e^8-4 c^7 d^6 e (14 b d-19 a e)+b^6 c e^7 (b d+15 a e)+b^4 c^2 e^6 \left (b^2 d^2-16 a b d e-82 a^2 e^2\right )+c^6 d^4 e^2 \left (79 b^2 d^2-228 a b d e+176 a^2 e^2\right )-c^5 d^2 e^3 \left (41 b^3 d^3-197 a b^2 d^2 e+352 a^2 b d e^2-244 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-15 a b^2 d^2 e+95 a^2 b d e^2+187 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-14 a b^3 d^3 e+81 a^2 b^2 d^2 e^2-244 a^3 b d e^3-126 a^4 e^4\right )\right )}{\left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^5 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{4 \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 )^4}-\frac{8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (14 c^2 d^2-5 b^2 e^2-6 c e (b d-3 a e)\right )-c (2 c d-b e) \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right ) x}{12 \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 )^3}+\frac{3 a c e (2 c d-b e)^2 \left (14 c^2 d^2-5 b^2 e^2-2 c e (7 b d-17 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+10 b^4 e^4-2 c^3 d^2 e (49 b d-78 a e)+5 b^2 c e^3 (2 b d-15 a e)+3 c^2 e^2 \left (b^2 d^2-18 a b d e+42 a^2 e^2\right )\right )-5 c (2 c d-b e) \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right ) x}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b d e+a e^2\right )^3 (d+e x) \left (a+b x+c x^2\right )^2}-\frac{5 \left (2 a c e (2 c d-b e)^2 \left (14 c^4 d^4+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)-4 c^3 d^2 e (7 b d-12 a e)+3 c^2 e^2 \left (3 b^2 d^2-16 a b d e+22 a^2 e^2\right )\right )-\left (b c d-b^2 e+2 a c e\right ) \left (42 c^6 d^6-3 b^6 e^6-2 c^5 d^4 e (49 b d-65 a e)-2 b^4 c e^5 (b d-17 a e)+b^2 c^2 e^4 \left (b^2 d^2+16 a b d e-123 a^2 e^2\right )+c^4 d^2 e^2 \left (55 b^2 d^2-164 a b d e+150 a^2 e^2\right )+6 c^3 e^3 \left (b^3 d^3-4 a b^2 d^2 e-3 a^2 b d e^2+21 a^3 e^3\right )\right )-3 c (2 c d-b e) \left (14 c^6 d^6-b^6 e^6-2 c^5 d^4 e (21 b d-31 a e)-2 b^4 c e^5 (b d-7 a e)-b^2 c^2 e^4 \left (3 b^2 d^2-26 a b d e+69 a^2 e^2\right )+c^4 d^2 e^2 \left (37 b^2 d^2-124 a b d e+114 a^2 e^2\right )-2 c^3 e^3 \left (2 b^3 d^3-18 a b^2 d^2 e+57 a^2 b d e^2-65 a^3 e^3\right )\right ) x\right )}{6 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 (d+e x) \left (a+b x+c x^2\right )}-\frac{5 \left (28 c^{10} d^{10}+b^{10} e^{10}-20 c^9 d^8 e (7 b d-9 a e)-252 a^4 c^5 e^9 (5 b d+a e)+210 a^3 b^2 c^4 e^9 (4 b d+3 a e)-84 a^2 b^4 c^3 e^9 (3 b d+5 a e)+18 a b^6 c^2 e^9 (2 b d+7 a e)-2 b^8 c e^9 (b d+9 a e)+18 c^8 d^6 e^2 \left (15 b^2 d^2-40 a b d e+28 a^2 e^2\right )-24 c^7 d^4 e^3 \left (10 b^3 d^3-42 a b^2 d^2 e+63 a^2 b d e^2-35 a^3 e^3\right )+84 c^6 d^2 e^4 \left (b^4 d^4-6 a b^3 d^3 e+15 a^2 b^2 d^2 e^2-20 a^3 b d e^3+15 a^4 e^4\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} \left (c d^2-e (b d-a e)\right )^6}+\frac{5 e^9 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^6}-\frac{5 e^9 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^6}\\ \end{align*}

Mathematica [A]  time = 7.12981, size = 2147, normalized size = 1.22 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^5),x]

[Out]

-(e^9/((c*d^2 - b*d*e + a*e^2)^5*(d + e*x))) + (b*c^2*d^2 - 2*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 - 3*a*b*c*e^2
+ 2*c^3*d^2*x - 2*b*c^2*d*e*x + b^2*c*e^2*x - 2*a*c^2*e^2*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*(a +
b*x + c*x^2)^4) + (14*b*c^4*d^4 - 28*b^2*c^3*d^3*e + 9*b^3*c^2*d^2*e^2 + 48*a*b*c^3*d^2*e^2 + 13*b^4*c*d*e^3 -
112*a*b^2*c^2*d*e^3 + 128*a^2*c^3*d*e^3 - 8*b^5*e^4 + 59*a*b^3*c*e^4 - 94*a^2*b*c^2*e^4 + 28*c^5*d^4*x - 56*b
*c^4*d^3*e*x + 18*b^2*c^3*d^2*e^2*x + 96*a*c^4*d^2*e^2*x + 10*b^3*c^2*d*e^3*x - 96*a*b*c^3*d*e^3*x - 8*b^4*c*e
^4*x + 54*a*b^2*c^2*e^4*x - 60*a^2*c^3*e^4*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)
^3) + (70*b*c^6*d^6 - 210*b^2*c^5*d^5*e + 185*b^3*c^4*d^4*e^2 + 310*a*b*c^5*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 620
*a*b^2*c^4*d^3*e^3 - 15*b^5*c^2*d^2*e^4 + 180*a*b^3*c^3*d^2*e^4 + 570*a^2*b*c^4*d^2*e^4 - 28*b^6*c*d*e^5 + 346
*a*b^4*c^2*d*e^5 - 1434*a^2*b^2*c^3*d*e^5 + 1152*a^3*c^4*d*e^5 + 18*b^7*e^6 - 206*a*b^5*c*e^6 + 759*a^2*b^3*c^
2*e^6 - 822*a^3*b*c^3*e^6 + 140*c^7*d^6*x - 420*b*c^6*d^5*e*x + 370*b^2*c^5*d^4*e^2*x + 620*a*c^6*d^4*e^2*x -
40*b^3*c^4*d^3*e^3*x - 1240*a*b*c^5*d^3*e^3*x - 30*b^4*c^3*d^2*e^4*x + 360*a*b^2*c^4*d^2*e^4*x + 1140*a^2*c^5*
d^2*e^4*x - 20*b^5*c^2*d*e^5*x + 260*a*b^3*c^3*d*e^5*x - 1140*a^2*b*c^4*d*e^5*x + 18*b^6*c*e^6*x - 196*a*b^4*c
^2*e^6*x + 654*a^2*b^2*c^3*e^6*x - 492*a^3*c^4*e^6*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x
+ c*x^2)^2) + (70*b*c^8*d^8 - 280*b^2*c^7*d^7*e + 395*b^3*c^6*d^6*e^2 + 380*a*b*c^7*d^6*e^2 - 205*b^4*c^5*d^5*
e^3 - 1140*a*b^2*c^6*d^5*e^3 + 5*b^5*c^4*d^4*e^4 + 985*a*b^3*c^5*d^4*e^4 + 880*a^2*b*c^6*d^4*e^4 + 5*b^6*c^3*d
^3*e^5 - 70*a*b^4*c^4*d^3*e^5 - 1760*a^2*b^2*c^5*d^3*e^5 + 5*b^7*c^2*d^2*e^6 - 75*a*b^5*c^3*d^2*e^6 + 405*a^2*
b^3*c^4*d^2*e^6 + 1220*a^3*b*c^5*d^2*e^6 + 13*b^8*c*d*e^7 - 208*a*b^6*c^2*d*e^7 + 1243*a^2*b^4*c^3*d*e^7 - 326
8*a^3*b^2*c^4*d*e^7 + 2048*a^4*c^5*d*e^7 - 8*b^9*e^8 + 123*a*b^7*c*e^8 - 698*a^2*b^5*c^2*e^8 + 1703*a^3*b^3*c^
3*e^8 - 1398*a^4*b*c^4*e^8 + 140*c^9*d^8*x - 560*b*c^8*d^7*e*x + 790*b^2*c^7*d^6*e^2*x + 760*a*c^8*d^6*e^2*x -
410*b^3*c^6*d^5*e^3*x - 2280*a*b*c^7*d^5*e^3*x + 10*b^4*c^5*d^4*e^4*x + 1970*a*b^2*c^6*d^4*e^4*x + 1760*a^2*c
^7*d^4*e^4*x + 10*b^5*c^4*d^3*e^5*x - 140*a*b^3*c^5*d^3*e^5*x - 3520*a^2*b*c^6*d^3*e^5*x + 10*b^6*c^3*d^2*e^6*
x - 150*a*b^4*c^4*d^2*e^6*x + 810*a^2*b^2*c^5*d^2*e^6*x + 2440*a^3*c^6*d^2*e^6*x + 10*b^7*c^2*d*e^7*x - 160*a*
b^5*c^3*d*e^7*x + 950*a^2*b^3*c^4*d*e^7*x - 2440*a^3*b*c^5*d*e^7*x - 8*b^8*c*e^8*x + 118*a*b^6*c^2*e^8*x - 628
*a^2*b^4*c^3*e^8*x + 1358*a^3*b^2*c^4*e^8*x - 748*a^4*c^5*e^8*x)/(2*(-b^2 + 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^5
*(a + b*x + c*x^2)) + (5*(28*c^10*d^10 - 140*b*c^9*d^9*e + 270*b^2*c^8*d^8*e^2 + 180*a*c^9*d^8*e^2 - 240*b^3*c
^7*d^7*e^3 - 720*a*b*c^8*d^7*e^3 + 84*b^4*c^6*d^6*e^4 + 1008*a*b^2*c^7*d^6*e^4 + 504*a^2*c^8*d^6*e^4 - 504*a*b
^3*c^6*d^5*e^5 - 1512*a^2*b*c^7*d^5*e^5 + 1260*a^2*b^2*c^6*d^4*e^6 + 840*a^3*c^7*d^4*e^6 - 1680*a^3*b*c^6*d^3*
e^7 + 1260*a^4*c^6*d^2*e^8 - 2*b^9*c*d*e^9 + 36*a*b^7*c^2*d*e^9 - 252*a^2*b^5*c^3*d*e^9 + 840*a^3*b^3*c^4*d*e^
9 - 1260*a^4*b*c^5*d*e^9 + b^10*e^10 - 18*a*b^8*c*e^10 + 126*a^2*b^6*c^2*e^10 - 420*a^3*b^4*c^3*e^10 + 630*a^4
*b^2*c^4*e^10 - 252*a^5*c^5*e^10)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*c)^4*Sqrt[-b^2 + 4*a*c]*
(-(c*d^2) + b*d*e - a*e^2)^6) + (5*(2*c*d*e^9 - b*e^10)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^6 - (5*(2*c*d*e^
9 - b*e^10)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^6)

________________________________________________________________________________________

Maple [B]  time = 0.227, size = 39599, normalized size = 22.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 4.22153, size = 10855, normalized size = 6.16 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

-5*(28*c^10*d^10*e^2 - 140*b*c^9*d^9*e^3 + 270*b^2*c^8*d^8*e^4 + 180*a*c^9*d^8*e^4 - 240*b^3*c^7*d^7*e^5 - 720
*a*b*c^8*d^7*e^5 + 84*b^4*c^6*d^6*e^6 + 1008*a*b^2*c^7*d^6*e^6 + 504*a^2*c^8*d^6*e^6 - 504*a*b^3*c^6*d^5*e^7 -
1512*a^2*b*c^7*d^5*e^7 + 1260*a^2*b^2*c^6*d^4*e^8 + 840*a^3*c^7*d^4*e^8 - 1680*a^3*b*c^6*d^3*e^9 + 1260*a^4*c
^6*d^2*e^10 - 2*b^9*c*d*e^11 + 36*a*b^7*c^2*d*e^11 - 252*a^2*b^5*c^3*d*e^11 + 840*a^3*b^3*c^4*d*e^11 - 1260*a^
4*b*c^5*d*e^11 + b^10*e^12 - 18*a*b^8*c*e^12 + 126*a^2*b^6*c^2*e^12 - 420*a^3*b^4*c^3*e^12 + 630*a^4*b^2*c^4*e
^12 - 252*a^5*c^5*e^12)*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^8*c^6*d^12 - 16*a*b^6*c^7*d^12 + 96*a^2*b^4*c^8*d^12 - 256*a^3*b^2*c^9*d^12
+ 256*a^4*c^10*d^12 - 6*b^9*c^5*d^11*e + 96*a*b^7*c^6*d^11*e - 576*a^2*b^5*c^7*d^11*e + 1536*a^3*b^3*c^8*d^11*
e - 1536*a^4*b*c^9*d^11*e + 15*b^10*c^4*d^10*e^2 - 234*a*b^8*c^5*d^10*e^2 + 1344*a^2*b^6*c^6*d^10*e^2 - 3264*a
^3*b^4*c^7*d^10*e^2 + 2304*a^4*b^2*c^8*d^10*e^2 + 1536*a^5*c^9*d^10*e^2 - 20*b^11*c^3*d^9*e^3 + 290*a*b^9*c^4*
d^9*e^3 - 1440*a^2*b^7*c^5*d^9*e^3 + 2240*a^3*b^5*c^6*d^9*e^3 + 2560*a^4*b^3*c^7*d^9*e^3 - 7680*a^5*b*c^8*d^9*
e^3 + 15*b^12*c^2*d^8*e^4 - 180*a*b^10*c^3*d^8*e^4 + 495*a^2*b^8*c^4*d^8*e^4 + 1680*a^3*b^6*c^5*d^8*e^4 - 1008
0*a^4*b^4*c^6*d^8*e^4 + 11520*a^5*b^2*c^7*d^8*e^4 + 3840*a^6*c^8*d^8*e^4 - 6*b^13*c*d^7*e^5 + 36*a*b^11*c^2*d^
7*e^5 + 324*a^2*b^9*c^3*d^7*e^5 - 3264*a^3*b^7*c^4*d^7*e^5 + 8064*a^4*b^5*c^5*d^7*e^5 - 15360*a^6*b*c^7*d^7*e^
5 + b^14*d^6*e^6 + 14*a*b^12*c*d^6*e^6 - 294*a^2*b^10*c^2*d^6*e^6 + 1204*a^3*b^8*c^3*d^6*e^6 + 896*a^4*b^6*c^4
*d^6*e^6 - 13440*a^5*b^4*c^5*d^6*e^6 + 17920*a^6*b^2*c^6*d^6*e^6 + 5120*a^7*c^7*d^6*e^6 - 6*a*b^13*d^5*e^7 + 3
6*a^2*b^11*c*d^5*e^7 + 324*a^3*b^9*c^2*d^5*e^7 - 3264*a^4*b^7*c^3*d^5*e^7 + 8064*a^5*b^5*c^4*d^5*e^7 - 15360*a
^7*b*c^6*d^5*e^7 + 15*a^2*b^12*d^4*e^8 - 180*a^3*b^10*c*d^4*e^8 + 495*a^4*b^8*c^2*d^4*e^8 + 1680*a^5*b^6*c^3*d
^4*e^8 - 10080*a^6*b^4*c^4*d^4*e^8 + 11520*a^7*b^2*c^5*d^4*e^8 + 3840*a^8*c^6*d^4*e^8 - 20*a^3*b^11*d^3*e^9 +
290*a^4*b^9*c*d^3*e^9 - 1440*a^5*b^7*c^2*d^3*e^9 + 2240*a^6*b^5*c^3*d^3*e^9 + 2560*a^7*b^3*c^4*d^3*e^9 - 7680*
a^8*b*c^5*d^3*e^9 + 15*a^4*b^10*d^2*e^10 - 234*a^5*b^8*c*d^2*e^10 + 1344*a^6*b^6*c^2*d^2*e^10 - 3264*a^7*b^4*c
^3*d^2*e^10 + 2304*a^8*b^2*c^4*d^2*e^10 + 1536*a^9*c^5*d^2*e^10 - 6*a^5*b^9*d*e^11 + 96*a^6*b^7*c*d*e^11 - 576
*a^7*b^5*c^2*d*e^11 + 1536*a^8*b^3*c^3*d*e^11 - 1536*a^9*b*c^4*d*e^11 + a^6*b^8*e^12 - 16*a^7*b^6*c*e^12 + 96*
a^8*b^4*c^2*e^12 - 256*a^9*b^2*c^3*e^12 + 256*a^10*c^4*e^12)*sqrt(-b^2 + 4*a*c)) - 5/2*(2*c*d*e^9 - b*e^10)*lo
g(-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^6*d^12
- 6*b*c^5*d^11*e + 15*b^2*c^4*d^10*e^2 + 6*a*c^5*d^10*e^2 - 20*b^3*c^3*d^9*e^3 - 30*a*b*c^4*d^9*e^3 + 15*b^4*c
^2*d^8*e^4 + 60*a*b^2*c^3*d^8*e^4 + 15*a^2*c^4*d^8*e^4 - 6*b^5*c*d^7*e^5 - 60*a*b^3*c^2*d^7*e^5 - 60*a^2*b*c^3
*d^7*e^5 + b^6*d^6*e^6 + 30*a*b^4*c*d^6*e^6 + 90*a^2*b^2*c^2*d^6*e^6 + 20*a^3*c^3*d^6*e^6 - 6*a*b^5*d^5*e^7 -
60*a^2*b^3*c*d^5*e^7 - 60*a^3*b*c^2*d^5*e^7 + 15*a^2*b^4*d^4*e^8 + 60*a^3*b^2*c*d^4*e^8 + 15*a^4*c^2*d^4*e^8 -
20*a^3*b^3*d^3*e^9 - 30*a^4*b*c*d^3*e^9 + 15*a^4*b^2*d^2*e^10 + 6*a^5*c*d^2*e^10 - 6*a^5*b*d*e^11 + a^6*e^12)
- e^19/((c^5*d^10*e^10 - 5*b*c^4*d^9*e^11 + 10*b^2*c^3*d^8*e^12 + 5*a*c^4*d^8*e^12 - 10*b^3*c^2*d^7*e^13 - 20
*a*b*c^3*d^7*e^13 + 5*b^4*c*d^6*e^14 + 30*a*b^2*c^2*d^6*e^14 + 10*a^2*c^3*d^6*e^14 - b^5*d^5*e^15 - 20*a*b^3*c
*d^5*e^15 - 30*a^2*b*c^2*d^5*e^15 + 5*a*b^4*d^4*e^16 + 30*a^2*b^2*c*d^4*e^16 + 10*a^3*c^2*d^4*e^16 - 10*a^2*b^
3*d^3*e^17 - 20*a^3*b*c*d^3*e^17 + 10*a^3*b^2*d^2*e^18 + 5*a^4*c*d^2*e^18 - 5*a^4*b*d*e^19 + a^5*e^20)*(x*e +
d)) + 1/12*(840*c^13*d^9*e - 3780*b*c^12*d^8*e^2 + 6280*b^2*c^11*d^7*e^3 + 5120*a*c^12*d^7*e^3 - 4340*b^3*c^10
*d^6*e^4 - 17920*a*b*c^11*d^6*e^4 + 738*b^4*c^9*d^5*e^5 + 20136*a*b^2*c^10*d^5*e^5 + 13488*a^2*c^11*d^5*e^5 +
185*b^5*c^8*d^4*e^6 - 5540*a*b^3*c^9*d^4*e^6 - 33720*a^2*b*c^10*d^4*e^6 + 80*b^6*c^7*d^3*e^7 - 1700*a*b^4*c^8*
d^3*e^7 + 17880*a^2*b^2*c^9*d^3*e^7 + 21120*a^3*c^10*d^3*e^7 + 45*b^7*c^6*d^2*e^8 - 870*a*b^5*c^7*d^2*e^8 + 69
00*a^2*b^3*c^8*d^2*e^8 - 31680*a^3*b*c^9*d^2*e^8 - 202*b^8*c^5*d*e^9 + 3142*a*b^6*c^6*d*e^9 - 17982*a^2*b^4*c^
7*d*e^9 + 43352*a^3*b^2*c^8*d*e^9 - 27512*a^4*c^9*d*e^9 + 77*b^9*c^4*e^10 - 1184*a*b^7*c^5*e^10 + 6717*a^2*b^5
*c^6*e^10 - 16396*a^3*b^3*c^7*e^10 + 13756*a^4*b*c^8*e^10 - 4*(1470*c^13*d^10*e^2 - 7350*b*c^12*d^9*e^3 + 1431
5*b^2*c^11*d^8*e^4 + 8890*a*c^12*d^8*e^4 - 13160*b^3*c^10*d^7*e^5 - 35560*a*b*c^11*d^7*e^5 + 5186*b^4*c^9*d^6*
e^6 + 50632*a*b^2*c^10*d^6*e^6 + 23196*a^2*c^11*d^6*e^6 - 368*b^5*c^8*d^5*e^7 - 27436*a*b^3*c^9*d^5*e^7 - 6958
8*a^2*b*c^10*d^5*e^7 - 25*b^6*c^7*d^4*e^8 + 2140*a*b^4*c^8*d^4*e^8 + 60030*a^2*b^2*c^9*d^4*e^8 + 35940*a^3*c^1
0*d^4*e^8 + 10*b^7*c^6*d^3*e^9 - 40*a*b^5*c^7*d^3*e^9 - 4080*a^2*b^3*c^8*d^3*e^9 - 71880*a^3*b*c^9*d^3*e^9 - 4
22*b^8*c^5*d^2*e^10 + 6722*a*b^6*c^6*d^2*e^10 - 40272*a^2*b^4*c^7*d^2*e^10 + 111472*a^3*b^2*c^8*d^2*e^10 - 575
62*a^4*c^9*d^2*e^10 + 344*b^9*c^4*d*e^11 - 5348*a*b^7*c^5*d*e^11 + 30714*a^2*b^5*c^6*d*e^11 - 75532*a^3*b^3*c^
7*d*e^11 + 57562*a^4*b*c^8*d*e^11 - 77*b^10*c^3*e^12 + 1196*a*b^8*c^4*e^12 - 6894*a^2*b^6*c^5*e^12 + 17338*a^3
*b^4*c^6*e^12 - 15793*a^4*b^2*c^7*e^12 + 1122*a^5*c^8*e^12)*e^(-1)/(x*e + d) + 2*(8820*c^13*d^11*e^3 - 48510*b
*c^12*d^10*e^4 + 107660*b^2*c^11*d^9*e^5 + 54460*a*c^12*d^9*e^5 - 120645*b^3*c^10*d^8*e^6 - 245070*a*b*c^11*d^
8*e^6 + 68512*b^4*c^9*d^7*e^7 + 417064*a*b^2*c^10*d^7*e^7 + 146152*a^2*c^11*d^7*e^7 - 16352*b^5*c^8*d^6*e^8 -
316064*a*b^3*c^9*d^6*e^8 - 511532*a^2*b*c^10*d^6*e^8 + 744*b^6*c^7*d^5*e^9 + 89184*a*b^4*c^8*d^5*e^9 + 591456*
a^2*b^2*c^9*d^5*e^9 + 234456*a^3*c^10*d^5*e^9 + 65*b^7*c^6*d^4*e^10 - 4630*a*b^5*c^7*d^4*e^10 - 199810*a^2*b^3
*c^8*d^4*e^10 - 586140*a^3*b*c^9*d^4*e^10 - 2816*b^8*c^5*d^3*e^11 + 44796*a*b^6*c^6*d^3*e^11 - 259516*a^2*b^4*
c^7*d^3*e^11 + 958456*a^3*b^2*c^8*d^3*e^11 - 372316*a^4*c^9*d^3*e^11 + 3648*b^9*c^4*d^2*e^12 - 57216*a*b^7*c^5
*d^2*e^12 + 333318*a^2*b^5*c^6*d^2*e^12 - 851544*a^3*b^3*c^7*d^2*e^12 + 558474*a^4*b*c^8*d^2*e^12 - 1588*b^10*
c^3*d*e^13 + 24464*a*b^8*c^4*d*e^13 - 138496*a^2*b^6*c^5*d*e^13 + 331772*a^3*b^4*c^6*d*e^13 - 237772*a^4*b^2*c
^7*d*e^13 - 33172*a^5*c^8*d*e^13 + 231*b^11*c^2*e^14 - 3494*a*b^9*c^3*e^14 + 19214*a^2*b^7*c^4*e^14 - 43500*a^
3*b^5*c^5*e^14 + 25807*a^4*b^3*c^6*e^14 + 16586*a^5*b*c^7*e^14)*e^(-2)/(x*e + d)^2 - 4*(7350*c^13*d^12*e^4 - 4
4100*b*c^12*d^11*e^5 + 109375*b^2*c^11*d^10*e^6 + 47600*a*c^12*d^10*e^6 - 142625*b^3*c^10*d^9*e^7 - 238000*a*b
*c^11*d^9*e^7 + 101967*b^4*c^9*d^8*e^8 + 467889*a*b^2*c^10*d^8*e^8 + 135222*a^2*c^11*d^8*e^8 - 37218*b^5*c^8*d
^7*e^9 - 443556*a*b^3*c^9*d^7*e^9 - 540888*a^2*b*c^10*d^7*e^9 + 5487*b^6*c^7*d^6*e^10 + 194682*a*b^4*c^8*d^6*e
^10 + 773718*a^2*b^2*c^9*d^6*e^10 + 230448*a^3*c^10*d^6*e^10 - 123*b^7*c^6*d^5*e^11 - 31200*a*b^5*c^7*d^5*e^11
- 428046*a^2*b^3*c^8*d^5*e^11 - 691344*a^3*b*c^9*d^5*e^11 - 2592*b^8*c^5*d^4*e^12 + 42087*a*b^6*c^6*d^4*e^12
- 174522*a^2*b^4*c^7*d^4*e^12 + 1178802*a^3*b^2*c^8*d^4*e^12 - 314622*a^4*c^9*d^4*e^12 + 4668*b^9*c^4*d^3*e^13
- 73656*a*b^7*c^5*d^3*e^13 + 431418*a^2*b^5*c^6*d^3*e^13 - 1205364*a^3*b^3*c^7*d^3*e^13 + 629244*a^4*b*c^8*d^
3*e^13 - 3012*b^10*c^3*d^2*e^14 + 46236*a*b^8*c^4*d^2*e^14 - 259404*a^2*b^6*c^5*d^2*e^14 + 606198*a^3*b^4*c^6*
d^2*e^14 - 308373*a^4*b^2*c^7*d^2*e^14 - 130848*a^5*c^8*d^2*e^14 + 823*b^11*c^2*d*e^15 - 12082*a*b^9*c^3*d*e^1
5 + 62502*a^2*b^7*c^4*d*e^15 - 118740*a^3*b^5*c^5*d*e^15 - 6249*a^4*b^3*c^6*d*e^15 + 130848*a^5*b*c^7*d*e^15 -
77*b^12*c*e^16 + 1025*a*b^10*c^2*e^16 - 4209*a^2*b^8*c^3*e^16 + 1614*a^3*b^6*c^4*e^16 + 24843*a^4*b^4*c^5*e^1
6 - 38499*a^5*b^2*c^6*e^16 + 3858*a^6*c^7*e^16)*e^(-3)/(x*e + d)^3 + (29400*c^13*d^13*e^5 - 191100*b*c^12*d^12
*e^6 + 522200*b^2*c^11*d^11*e^7 + 204400*a*c^12*d^11*e^7 - 770000*b^3*c^10*d^10*e^8 - 1124200*a*b*c^11*d^10*e^
8 + 650450*b^4*c^9*d^9*e^9 + 2496400*a*b^2*c^10*d^9*e^9 + 628200*a^2*c^11*d^9*e^9 - 305175*b^5*c^8*d^8*e^10 -
2802300*a*b^3*c^9*d^8*e^10 - 2826900*a^2*b*c^10*d^8*e^10 + 69600*b^6*c^7*d^7*e^11 + 1606200*a*b^4*c^8*d^7*e^11
+ 4784400*a^2*b^2*c^9*d^7*e^11 + 1159200*a^3*c^10*d^7*e^11 - 5250*b^7*c^6*d^6*e^12 - 413700*a*b^5*c^7*d^6*e^1
2 - 3553200*a^2*b^3*c^8*d^6*e^12 - 4057200*a^3*b*c^9*d^6*e^12 - 11544*b^8*c^5*d^5*e^13 + 216204*a*b^6*c^6*d^5*
e^13 - 56124*a^2*b^4*c^7*d^5*e^13 + 7256064*a^3*b^2*c^8*d^5*e^13 - 1170264*a^4*c^9*d^5*e^13 + 26760*b^9*c^4*d^
4*e^14 - 423960*a*b^7*c^5*d^4*e^14 + 2427210*a^2*b^5*c^6*d^4*e^14 - 7997160*a^3*b^3*c^7*d^4*e^14 + 2925660*a^4
*b*c^8*d^4*e^14 - 22980*b^10*c^3*d^3*e^15 + 352560*a*b^8*c^4*d^3*e^15 - 1972560*a^2*b^6*c^5*d^3*e^15 + 4653960
*a^3*b^4*c^6*d^3*e^15 - 1310760*a^4*b^2*c^7*d^3*e^15 - 1291920*a^5*c^8*d^3*e^15 + 9035*b^11*c^2*d^2*e^16 - 129
830*a*b^9*c^3*d^2*e^16 + 639630*a^2*b^7*c^4*d^2*e^16 - 1012380*a^3*b^5*c^5*d^2*e^16 - 959520*a^4*b^3*c^6*d^2*e
^16 + 1937880*a^5*b*c^7*d^2*e^16 - 1550*b^12*c*d*e^17 + 19130*a*b^10*c^2*d*e^17 - 61470*a^2*b^8*c^3*d*e^17 - 9
8580*a^3*b^6*c^4*d*e^17 + 801930*a^4*b^4*c^5*d*e^17 - 899280*a^5*b^2*c^6*d*e^17 - 46440*a^6*c^7*d*e^17 + 77*b^
13*e^18 - 452*a*b^11*c*e^18 - 4593*a^2*b^9*c^2*e^18 + 48048*a^3*b^7*c^3*e^18 - 143523*a^4*b^5*c^4*e^18 + 12666
0*a^5*b^3*c^5*e^18 + 23220*a^6*b*c^6*e^18)*e^(-4)/(x*e + d)^4 - 20*(882*c^13*d^14*e^6 - 6174*b*c^12*d^13*e^7 +
18389*b^2*c^11*d^12*e^8 + 6706*a*c^12*d^12*e^8 - 30072*b^3*c^10*d^11*e^9 - 40236*a*b*c^11*d^11*e^9 + 28936*b^
4*c^9*d^10*e^10 + 99304*a*b^2*c^10*d^10*e^10 + 22690*a^2*c^11*d^10*e^10 - 16167*b^5*c^8*d^9*e^11 - 127690*a*b^
3*c^9*d^9*e^11 - 113450*a^2*b*c^10*d^9*e^11 + 4752*b^6*c^7*d^8*e^12 + 88479*a*b^4*c^8*d^8*e^12 + 220689*a^2*b^
2*c^9*d^8*e^12 + 46098*a^3*c^10*d^8*e^12 - 546*b^7*c^6*d^7*e^13 - 30372*a*b^5*c^7*d^7*e^13 - 202056*a^2*b^3*c^
8*d^7*e^13 - 184392*a^3*b*c^9*d^7*e^13 - 393*b^8*c^5*d^6*e^14 + 10110*a*b^6*c^6*d^6*e^14 + 45642*a^2*b^4*c^7*d
^6*e^14 + 349752*a^3*b^2*c^8*d^6*e^14 - 27066*a^4*c^9*d^6*e^14 + 1116*b^9*c^4*d^5*e^15 - 17730*a*b^7*c^5*d^5*e
^15 + 93780*a^2*b^5*c^6*d^5*e^15 - 403884*a^3*b^3*c^7*d^5*e^15 + 81198*a^4*b*c^8*d^5*e^15 - 1200*b^10*c^3*d^4*
e^16 + 18420*a*b^8*c^4*d^4*e^16 - 103035*a^2*b^6*c^5*d^4*e^16 + 255840*a^3*b^4*c^6*d^4*e^16 - 6825*a^4*b^2*c^7
*d^4*e^16 - 75738*a^5*c^8*d^4*e^16 + 611*b^11*c^2*d^3*e^17 - 8642*a*b^9*c^3*d^3*e^17 + 40938*a^2*b^7*c^4*d^3*e
^17 - 53664*a^3*b^5*c^5*d^3*e^17 - 121680*a^4*b^3*c^6*d^3*e^17 + 151476*a^5*b*c^7*d^3*e^17 - 147*b^12*c*d^2*e^
18 + 1695*a*b^10*c^2*d^2*e^18 - 3987*a^2*b^8*c^3*d^2*e^18 - 19674*a^3*b^6*c^4*d^2*e^18 + 99270*a^4*b^4*c^5*d^2
*e^18 - 85824*a^5*b^2*c^6*d^2*e^18 - 18522*a^6*c^7*d^2*e^18 + 13*b^13*d*e^19 - 44*a*b^11*c*d*e^19 - 1211*a^2*b
^9*c^2*d*e^19 + 9924*a^3*b^7*c^3*d*e^19 - 24897*a^4*b^5*c^4*d*e^19 + 10086*a^5*b^3*c^5*d*e^19 + 18522*a^6*b*c^
6*d*e^19 - 13*a*b^12*e^20 + 178*a^2*b^10*c*e^20 - 783*a^3*b^8*c^2*e^20 + 651*a^4*b^6*c^3*e^20 + 3417*a^5*b^4*c
^4*e^20 - 6237*a^6*b^2*c^5*e^20 + 918*a^7*c^6*e^20)*e^(-5)/(x*e + d)^5 + 30*(196*c^13*d^15*e^7 - 1470*b*c^12*d
^14*e^8 + 4732*b^2*c^11*d^13*e^9 + 1652*a*c^12*d^13*e^9 - 8463*b^3*c^10*d^12*e^10 - 10738*a*b*c^11*d^12*e^10 +
9058*b^4*c^9*d^11*e^11 + 29092*a*b^2*c^10*d^11*e^11 + 6244*a^2*c^11*d^11*e^11 - 5775*b^5*c^8*d^10*e^12 - 4188
8*a*b^3*c^9*d^10*e^12 - 34342*a^2*b*c^10*d^10*e^12 + 2016*b^6*c^7*d^9*e^13 + 33558*a*b^4*c^8*d^9*e^13 + 75208*
a^2*b^2*c^9*d^9*e^13 + 14196*a^3*c^10*d^9*e^13 - 294*b^7*c^6*d^8*e^14 - 14028*a*b^5*c^7*d^8*e^14 - 80871*a^2*b
^3*c^8*d^8*e^14 - 63882*a^3*b*c^9*d^8*e^14 - 102*b^8*c^5*d^7*e^15 + 3984*a*b^6*c^6*d^7*e^15 + 32208*a^2*b^4*c^
7*d^7*e^15 + 129768*a^3*b^2*c^8*d^7*e^15 - 2004*a^4*c^9*d^7*e^15 + 343*b^9*c^4*d^6*e^16 - 5460*a*b^7*c^5*d^6*e
^16 + 24276*a^2*b^5*c^6*d^6*e^16 - 156072*a^3*b^3*c^7*d^6*e^16 + 7014*a^4*b*c^8*d^6*e^16 - 444*b^10*c^3*d^5*e^
17 + 6822*a*b^8*c^4*d^5*e^17 - 38196*a^2*b^6*c^5*d^5*e^17 + 104232*a^3*b^4*c^6*d^5*e^17 + 25644*a^4*b^2*c^7*d^
5*e^17 - 28932*a^5*c^8*d^5*e^17 + 277*b^11*c^2*d^4*e^18 - 3874*a*b^9*c^3*d^4*e^18 + 17811*a^2*b^7*c^4*d^4*e^18
- 19458*a^3*b^5*c^5*d^4*e^18 - 81645*a^4*b^3*c^6*d^4*e^18 + 72330*a^5*b*c^7*d^4*e^18 - 84*b^12*c*d^3*e^19 + 9
08*a*b^10*c^2*d^3*e^19 - 1332*a^2*b^8*c^3*d^3*e^19 - 16644*a^3*b^6*c^4*d^3*e^19 + 69390*a^4*b^4*c^5*d^3*e^19 -
45708*a^5*b^2*c^6*d^3*e^19 - 17748*a^6*c^7*d^3*e^19 + 10*b^13*d^2*e^20 - 8*a*b^11*c*d^2*e^20 - 1274*a^2*b^9*c
^2*d^2*e^20 + 8976*a^3*b^7*c^3*d^2*e^20 - 18933*a^4*b^5*c^4*d^2*e^20 - 3768*a^5*b^3*c^5*d^2*e^20 + 26622*a^6*b
*c^6*d^2*e^20 - 20*a*b^12*d*e^21 + 248*a^2*b^10*c*d*e^21 - 804*a^3*b^8*c^2*d*e^21 - 1272*a^4*b^6*c^3*d*e^21 +
10626*a^5*b^4*c^4*d*e^21 - 12912*a^6*b^2*c^5*d*e^21 - 228*a^7*c^6*d*e^21 + 10*a^2*b^11*e^22 - 156*a^3*b^9*c*e^
22 + 903*a^4*b^7*c^2*e^22 - 2274*a^5*b^5*c^3*e^22 + 2019*a^6*b^3*c^4*e^22 + 114*a^7*b*c^5*e^22)*e^(-6)/(x*e +
d)^6 - 60*(14*c^13*d^16*e^8 - 112*b*c^12*d^15*e^9 + 387*b^2*c^11*d^14*e^10 + 132*a*c^12*d^14*e^10 - 749*b^3*c^
10*d^13*e^11 - 924*a*b*c^11*d^13*e^11 + 877*b^4*c^9*d^12*e^12 + 2721*a*b^2*c^10*d^12*e^12 + 564*a^2*c^11*d^12*
e^12 - 621*b^5*c^8*d^11*e^13 - 4314*a*b^3*c^9*d^11*e^13 - 3384*a^2*b*c^10*d^11*e^13 + 246*b^6*c^7*d^10*e^14 +
3879*a*b^4*c^8*d^10*e^14 + 8211*a^2*b^2*c^9*d^10*e^14 + 1460*a^3*c^10*d^10*e^14 - 42*b^7*c^6*d^9*e^15 - 1872*a
*b^5*c^7*d^9*e^15 - 10035*a^2*b^3*c^8*d^9*e^15 - 7300*a^3*b*c^9*d^9*e^15 - 9*b^8*c^5*d^8*e^16 + 522*a*b^6*c^6*
d^8*e^16 + 5292*a^2*b^4*c^7*d^8*e^16 + 15993*a^3*b^2*c^8*d^8*e^16 + 432*a^4*c^9*d^8*e^16 + 35*b^9*c^4*d^7*e^17
- 558*a*b^7*c^5*d^7*e^17 + 1818*a^2*b^5*c^6*d^7*e^17 - 20172*a^3*b^3*c^7*d^7*e^17 - 1728*a^4*b*c^8*d^7*e^17 -
53*b^10*c^3*d^6*e^18 + 815*a*b^8*c^4*d^6*e^18 - 4567*a^2*b^6*c^5*d^6*e^18 + 14026*a^3*b^4*c^6*d^6*e^18 + 7249
*a^4*b^2*c^7*d^6*e^18 - 3380*a^5*c^8*d^6*e^18 + 39*b^11*c^2*d^5*e^19 - 540*a*b^9*c^3*d^5*e^19 + 2415*a^2*b^7*c
^4*d^5*e^19 - 2136*a^3*b^5*c^5*d^5*e^19 - 15699*a^4*b^3*c^6*d^5*e^19 + 10140*a^5*b*c^7*d^5*e^19 - 14*b^12*c*d^
4*e^20 + 141*a*b^10*c^2*d^4*e^20 - 60*a^2*b^8*c^3*d^4*e^20 - 3705*a^3*b^6*c^4*d^4*e^20 + 13785*a^4*b^4*c^5*d^4
*e^20 - 6357*a^5*b^2*c^6*d^4*e^20 - 4212*a^6*c^7*d^4*e^20 + 2*b^13*d^3*e^21 + 4*a*b^11*c*d^3*e^21 - 326*a^2*b^
9*c^2*d^3*e^21 + 2036*a^3*b^7*c^3*d^3*e^21 - 3421*a^4*b^5*c^4*d^3*e^21 - 4186*a^5*b^3*c^5*d^3*e^21 + 8424*a^6*
b*c^6*d^3*e^21 - 6*a*b^12*d^2*e^22 + 66*a^2*b^10*c*d^2*e^22 - 114*a^3*b^8*c^2*d^2*e^22 - 1071*a^4*b^6*c^3*d^2*
e^22 + 4623*a^5*b^4*c^4*d^2*e^22 - 4071*a^6*b^2*c^5*d^2*e^22 - 1284*a^7*c^6*d^2*e^22 + 6*a^2*b^11*d*e^23 - 88*
a^3*b^9*c*d*e^23 + 453*a^4*b^7*c^2*d*e^23 - 840*a^5*b^5*c^3*d*e^23 - 141*a^6*b^3*c^4*d*e^23 + 1284*a^7*b*c^5*d
*e^23 - 2*a^3*b^10*e^24 + 32*a^4*b^8*c*e^24 - 193*a^5*b^6*c^2*e^24 + 526*a^6*b^4*c^3*e^24 - 581*a^7*b^2*c^4*e^
24 + 130*a^8*c^5*e^24)*e^(-7)/(x*e + d)^7)/((c*d^2 - b*d*e + a*e^2)^6*(b^2 - 4*a*c)^4*(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)^4)