3.2227 \(\int \frac{1}{(d+e x) (a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=1324 \[ \text{result too large to display} \]

[Out]

-(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)*(a + b*x + c*x^2)^4) -
(7*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(14*c^2*d^2 - 4*b^2*e^2 - c*e*(7*b*d - 16*a*e)) - 2*c*(2
*c*d - b*e)*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e))*x)/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a
+ b*x + c*x^2)^3) + (5*a*c*e*(2*c*d - b*e)^2*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e)) - (b*c*d - b^2*e
+ 2*a*c*e)*(70*c^4*d^4 + 6*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 24*a*e) - 15*c^3*d^2*e*(7*b*d - 10*a*e) + 3*c^2*e^2*
(5*b^2*d^2 - 25*a*b*d*e + 32*a^2*e^2)) - 2*c*(2*c*d - b*e)*(35*c^4*d^4 + 3*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 17*a
*e) - 10*c^3*d^2*e*(7*b*d - 11*a*e) + c^2*e^2*(25*b^2*d^2 - 110*a*b*d*e + 123*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3
*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)^2) + (b^7*c*d*e^6 + 2*b^8*e^7 + 256*a^4*c^4*e^7 + b^6*c*e^5*(c*d^
2 - 31*a*e^2) + b^5*c^2*d*e^4*(c*d^2 - 14*a*e^2) - b^4*c^2*e^3*(125*c^2*d^4 + 13*a*c*d^2*e^2 - 178*a^2*e^4) +
b^3*c^3*d*e^2*(295*c^2*d^4 + 492*a*c*d^2*e^2 + 69*a^2*e^4) + 2*b*c^4*d*(35*c^3*d^6 + 145*a*c^2*d^4*e^2 + 233*a
^2*c*d^2*e^4 + 187*a^3*e^6) - b^2*c^3*e*(245*c^3*d^6 + 725*a*c^2*d^4*e^2 + 699*a^2*c*d^2*e^4 + 443*a^3*e^6) +
2*c*(2*c*d - b*e)*(35*c^6*d^6 - b^6*e^6 - 5*c^5*d^4*e*(21*b*d - 29*a*e) - 3*b^4*c*e^5*(b*d - 5*a*e) - b^2*c^2*
e^4*(7*b^2*d^2 - 44*a*b*d*e + 82*a^2*e^2) + c^4*d^2*e^2*(95*b^2*d^2 - 290*a*b*d*e + 233*a^2*e^2) - c^3*e^3*(15
*b^3*d^3 - 101*a*b^2*d^2*e + 233*a^2*b*d*e^2 - 187*a^3*e^3))*x)/(2*(b^2 - 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(
a + b*x + c*x^2)) - ((140*c^9*d^9 - b^9*e^9 + 18*a*b^7*c*e^9 - 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3*e^9 - 630
*a^4*b*c^4*e^9 - 90*c^8*d^7*e*(7*b*d - 8*a*e) + 72*c^7*d^5*e^2*(15*b^2*d^2 - 35*a*b*d*e + 21*a^2*e^2) - 84*c^6
*d^3*e^3*(10*b^3*d^3 - 36*a*b^2*d^2*e + 45*a^2*b*d*e^2 - 20*a^3*e^3) + 252*c^5*d*e^4*(b^4*d^4 - 5*a*b^3*d^3*e
+ 10*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 + 5*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))^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*x^2])/(2*(
c*d^2 - b*d*e + a*e^2)^5)

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Rubi [A]  time = 13.9008, antiderivative size = 1324, 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} \[ \frac{\log (d+e x) e^9}{\left (c d^2-b e d+a e^2\right )^5}-\frac{\log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )^5}-\frac{\left (140 c^9 d^9-90 c^8 e (7 b d-8 a e) d^7+72 c^7 e^2 \left (15 b^2 d^2-35 a b e d+21 a^2 e^2\right ) d^5-84 c^6 e^3 \left (10 b^3 d^3-36 a b^2 e d^2+45 a^2 b e^2 d-20 a^3 e^3\right ) d^3+252 c^5 e^4 \left (b^4 d^4-5 a b^3 e d^3+10 a^2 b^2 e^2 d^2-10 a^3 b e^3 d+5 a^4 e^4\right ) d-b^9 e^9-630 a^4 b c^4 e^9+420 a^3 b^3 c^3 e^9-126 a^2 b^5 c^2 e^9+18 a b^7 c e^9\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 )^5}+\frac{2 e^7 b^8+c d e^6 b^7+c e^5 \left (c d^2-31 a e^2\right ) b^6+c^2 d e^4 \left (c d^2-14 a e^2\right ) b^5-c^2 e^3 \left (125 c^2 d^4+13 a c e^2 d^2-178 a^2 e^4\right ) b^4+c^3 d e^2 \left (295 c^2 d^4+492 a c e^2 d^2+69 a^2 e^4\right ) b^3-c^3 e \left (245 c^3 d^6+725 a c^2 e^2 d^4+699 a^2 c e^4 d^2+443 a^3 e^6\right ) b^2+2 c^4 d \left (35 c^3 d^6+145 a c^2 e^2 d^4+233 a^2 c e^4 d^2+187 a^3 e^6\right ) b+256 a^4 c^4 e^7+2 c (2 c d-b e) \left (35 c^6 d^6-5 c^5 e (21 b d-29 a e) d^4+c^4 e^2 \left (95 b^2 d^2-290 a b e d+233 a^2 e^2\right ) d^2-b^6 e^6-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b e d+82 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 e d^2+233 a^2 b e^2 d-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b e d+a e^2\right )^4 \left (c x^2+b x+a\right )}+\frac{5 a c e \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) (2 c d-b e)^2-2 c \left (35 c^4 d^4-10 c^3 e (7 b d-11 a e) d^2+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b e d+123 a^2 e^2\right )\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (70 c^4 d^4-15 c^3 e (7 b d-10 a e) d^2+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b e d+32 a^2 e^2\right )\right )}{12 \left (b^2-4 a c\right )^3 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )^2}-\frac{7 a c e (2 c d-b e)^2-2 c \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)\right )}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^3}-\frac{-e b^2+c d b+2 a c e+c (2 c d-b e) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)*(a + b*x + c*x^2)^4) -
(7*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(14*c^2*d^2 - 4*b^2*e^2 - c*e*(7*b*d - 16*a*e)) - 2*c*(2
*c*d - b*e)*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e))*x)/(12*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a
+ b*x + c*x^2)^3) + (5*a*c*e*(2*c*d - b*e)^2*(7*c^2*d^2 - 2*b^2*e^2 - c*e*(7*b*d - 15*a*e)) - (b*c*d - b^2*e
+ 2*a*c*e)*(70*c^4*d^4 + 6*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 24*a*e) - 15*c^3*d^2*e*(7*b*d - 10*a*e) + 3*c^2*e^2*
(5*b^2*d^2 - 25*a*b*d*e + 32*a^2*e^2)) - 2*c*(2*c*d - b*e)*(35*c^4*d^4 + 3*b^4*e^4 + 2*b^2*c*e^3*(5*b*d - 17*a
*e) - 10*c^3*d^2*e*(7*b*d - 11*a*e) + c^2*e^2*(25*b^2*d^2 - 110*a*b*d*e + 123*a^2*e^2))*x)/(12*(b^2 - 4*a*c)^3
*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)^2) + (b^7*c*d*e^6 + 2*b^8*e^7 + 256*a^4*c^4*e^7 + b^6*c*e^5*(c*d^
2 - 31*a*e^2) + b^5*c^2*d*e^4*(c*d^2 - 14*a*e^2) - b^4*c^2*e^3*(125*c^2*d^4 + 13*a*c*d^2*e^2 - 178*a^2*e^4) +
b^3*c^3*d*e^2*(295*c^2*d^4 + 492*a*c*d^2*e^2 + 69*a^2*e^4) + 2*b*c^4*d*(35*c^3*d^6 + 145*a*c^2*d^4*e^2 + 233*a
^2*c*d^2*e^4 + 187*a^3*e^6) - b^2*c^3*e*(245*c^3*d^6 + 725*a*c^2*d^4*e^2 + 699*a^2*c*d^2*e^4 + 443*a^3*e^6) +
2*c*(2*c*d - b*e)*(35*c^6*d^6 - b^6*e^6 - 5*c^5*d^4*e*(21*b*d - 29*a*e) - 3*b^4*c*e^5*(b*d - 5*a*e) - b^2*c^2*
e^4*(7*b^2*d^2 - 44*a*b*d*e + 82*a^2*e^2) + c^4*d^2*e^2*(95*b^2*d^2 - 290*a*b*d*e + 233*a^2*e^2) - c^3*e^3*(15
*b^3*d^3 - 101*a*b^2*d^2*e + 233*a^2*b*d*e^2 - 187*a^3*e^3))*x)/(2*(b^2 - 4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(
a + b*x + c*x^2)) - ((140*c^9*d^9 - b^9*e^9 + 18*a*b^7*c*e^9 - 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3*e^9 - 630
*a^4*b*c^4*e^9 - 90*c^8*d^7*e*(7*b*d - 8*a*e) + 72*c^7*d^5*e^2*(15*b^2*d^2 - 35*a*b*d*e + 21*a^2*e^2) - 84*c^6
*d^3*e^3*(10*b^3*d^3 - 36*a*b^2*d^2*e + 45*a^2*b*d*e^2 - 20*a^3*e^3) + 252*c^5*d*e^4*(b^4*d^4 - 5*a*b^3*d^3*e
+ 10*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 + 5*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))^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x + c*x^2])/(2*(
c*d^2 - b*d*e + a*e^2)^5)

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) \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 ) \left (a+b x+c x^2\right )^4}-\frac{\int \frac{14 c^2 d^2-4 b^2 e^2-c e (7 b d-16 a e)+7 c e (2 c d-b e) x}{(d+e x) \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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{\int \frac{2 \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )+10 c e (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right ) x}{(d+e x) \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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}-\frac{\int \frac{12 \left (70 c^6 d^6-2 b^6 e^6-5 c^5 d^4 e (35 b d-44 a e)-3 b^4 c e^5 (b d-8 a e)+6 c^4 d^2 e^2 \left (20 b^2 d^2-55 a b d e+41 a^2 e^2\right )-2 b^2 c^2 e^4 \left (2 b^2 d^2-17 a b d e+48 a^2 e^2\right )-c^3 e^3 \left (5 b^3 d^3-42 a b^2 d^2 e+123 a^2 b d e^2-128 a^3 e^3\right )\right )+12 c e (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 a^2 e^2\right )\right ) x}{(d+e x) \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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}+\frac{\int \frac{24 \left (70 c^8 d^8+b^8 e^8-5 c^7 d^6 e (49 b d-58 a e)+b^6 c e^7 (b d-16 a e)+b^4 c^2 e^6 \left (b^2 d^2-15 a b d e+96 a^2 e^2\right )+c^6 d^4 e^2 \left (295 b^2 d^2-725 a b d e+466 a^2 e^2\right )-c^5 d^2 e^3 \left (125 b^3 d^3-492 a b^2 d^2 e+699 a^2 b d e^2-374 a^3 e^3\right )+b^2 c^3 e^5 \left (b^3 d^3-14 a b^2 d^2 e+82 a^2 b d e^2-256 a^3 e^3\right )+c^4 e^4 \left (b^4 d^4-13 a b^3 d^3 e+69 a^2 b^2 d^2 e^2-187 a^3 b d e^3+256 a^4 e^4\right )\right )+24 c e (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{(d+e x) \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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}+\frac{\int \left (\frac{24 \left (b^2-4 a c\right )^4 e^{10}}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{24 \left (70 c^9 d^9-b^9 e^9+17 a b^7 c e^9-111 a^2 b^5 c^2 e^9+338 a^3 b^3 c^3 e^9-443 a^4 b c^4 e^9-45 c^8 d^7 e (7 b d-8 a e)+36 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-42 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+126 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 x\right )}{\left (c d^2-b d e+a e^2\right ) \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{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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}+\frac{e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}+\frac{\int \frac{70 c^9 d^9-b^9 e^9+17 a b^7 c e^9-111 a^2 b^5 c^2 e^9+338 a^3 b^3 c^3 e^9-443 a^4 b c^4 e^9-45 c^8 d^7 e (7 b d-8 a e)+36 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-42 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+126 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\right )-c \left (b^2-4 a c\right )^4 e^9 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 )^5}\\ &=-\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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}+\frac{e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}-\frac{e^9 \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 )^5}+\frac{\left (140 c^9 d^9-b^9 e^9+18 a b^7 c e^9-126 a^2 b^5 c^2 e^9+420 a^3 b^3 c^3 e^9-630 a^4 b c^4 e^9-90 c^8 d^7 e (7 b d-8 a e)+72 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-84 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+252 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\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 )^5}\\ &=-\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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}+\frac{e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}-\frac{e^9 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^5}-\frac{\left (140 c^9 d^9-b^9 e^9+18 a b^7 c e^9-126 a^2 b^5 c^2 e^9+420 a^3 b^3 c^3 e^9-630 a^4 b c^4 e^9-90 c^8 d^7 e (7 b d-8 a e)+72 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-84 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+252 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 a^4 e^4\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 )^5}\\ &=-\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 ) \left (a+b x+c x^2\right )^4}-\frac{7 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^2-4 b^2 e^2-c e (7 b d-16 a e)\right )-2 c (2 c d-b e) \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 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 \left (a+b x+c x^2\right )^3}+\frac{5 a c e (2 c d-b e)^2 \left (7 c^2 d^2-2 b^2 e^2-c e (7 b d-15 a e)\right )-\left (b c d-b^2 e+2 a c e\right ) \left (70 c^4 d^4+6 b^4 e^4+2 b^2 c e^3 (5 b d-24 a e)-15 c^3 d^2 e (7 b d-10 a e)+3 c^2 e^2 \left (5 b^2 d^2-25 a b d e+32 a^2 e^2\right )\right )-2 c (2 c d-b e) \left (35 c^4 d^4+3 b^4 e^4+2 b^2 c e^3 (5 b d-17 a e)-10 c^3 d^2 e (7 b d-11 a e)+c^2 e^2 \left (25 b^2 d^2-110 a b d e+123 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 \left (a+b x+c x^2\right )^2}+\frac{b^7 c d e^6+2 b^8 e^7+256 a^4 c^4 e^7+b^6 c e^5 \left (c d^2-31 a e^2\right )+b^5 c^2 d e^4 \left (c d^2-14 a e^2\right )-b^4 c^2 e^3 \left (125 c^2 d^4+13 a c d^2 e^2-178 a^2 e^4\right )+b^3 c^3 d e^2 \left (295 c^2 d^4+492 a c d^2 e^2+69 a^2 e^4\right )+2 b c^4 d \left (35 c^3 d^6+145 a c^2 d^4 e^2+233 a^2 c d^2 e^4+187 a^3 e^6\right )-b^2 c^3 e \left (245 c^3 d^6+725 a c^2 d^4 e^2+699 a^2 c d^2 e^4+443 a^3 e^6\right )+2 c (2 c d-b e) \left (35 c^6 d^6-b^6 e^6-5 c^5 d^4 e (21 b d-29 a e)-3 b^4 c e^5 (b d-5 a e)-b^2 c^2 e^4 \left (7 b^2 d^2-44 a b d e+82 a^2 e^2\right )+c^4 d^2 e^2 \left (95 b^2 d^2-290 a b d e+233 a^2 e^2\right )-c^3 e^3 \left (15 b^3 d^3-101 a b^2 d^2 e+233 a^2 b d e^2-187 a^3 e^3\right )\right ) x}{2 \left (b^2-4 a c\right )^4 \left (c d^2-b d e+a e^2\right )^4 \left (a+b x+c x^2\right )}-\frac{\left (140 c^9 d^9-b^9 e^9+18 a b^7 c e^9-126 a^2 b^5 c^2 e^9+420 a^3 b^3 c^3 e^9-630 a^4 b c^4 e^9-90 c^8 d^7 e (7 b d-8 a e)+72 c^7 d^5 e^2 \left (15 b^2 d^2-35 a b d e+21 a^2 e^2\right )-84 c^6 d^3 e^3 \left (10 b^3 d^3-36 a b^2 d^2 e+45 a^2 b d e^2-20 a^3 e^3\right )+252 c^5 d e^4 \left (b^4 d^4-5 a b^3 d^3 e+10 a^2 b^2 d^2 e^2-10 a^3 b d e^3+5 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 )^5}+\frac{e^9 \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^5}-\frac{e^9 \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^5}\\ \end{align*}

Mathematica [A]  time = 6.58957, size = 1601, normalized size = 1.21 \[ \frac{\log (d+e x) e^9}{\left (c d^2-b e d+a e^2\right )^5}-\frac{\log \left (c x^2+b x+a\right ) e^9}{2 \left (c d^2-b e d+a e^2\right )^5}+\frac{\left (-140 c^9 d^9+630 b c^8 e d^8-720 a c^8 e^2 d^7-1080 b^2 c^7 e^2 d^7+2520 a b c^7 e^3 d^6+840 b^3 c^6 e^3 d^6-1512 a^2 c^7 e^4 d^5-3024 a b^2 c^6 e^4 d^5-252 b^4 c^5 e^4 d^5+3780 a^2 b c^6 e^5 d^4+1260 a b^3 c^5 e^5 d^4-1680 a^3 c^6 e^6 d^3-2520 a^2 b^2 c^5 e^6 d^3+2520 a^3 b c^5 e^7 d^2-1260 a^4 c^5 e^8 d+b^9 e^9+630 a^4 b c^4 e^9-420 a^3 b^3 c^3 e^9+126 a^2 b^5 c^2 e^9-18 a b^7 c e^9\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (b^2-4 a c\right )^4 \sqrt{4 a c-b^2} \left (-c d^2+b e d-a e^2\right )^5}+\frac{2 e^7 b^8+c d e^6 b^7+2 c e^7 x b^7-31 a c e^7 b^6+c^2 d^2 e^5 b^6+2 c^2 d e^6 x b^6-14 a c^2 d e^6 b^5+c^3 d^3 e^4 b^5-30 a c^2 e^7 x b^5+2 c^3 d^2 e^5 x b^5+178 a^2 c^2 e^7 b^4-13 a c^3 d^2 e^5 b^4-125 c^4 d^4 e^3 b^4-28 a c^3 d e^6 x b^4+2 c^4 d^3 e^4 x b^4+69 a^2 c^3 d e^6 b^3+492 a c^4 d^3 e^4 b^3+295 c^5 d^5 e^2 b^3+164 a^2 c^3 e^7 x b^3-26 a c^4 d^2 e^5 x b^3-250 c^5 d^4 e^3 x b^3-443 a^3 c^3 e^7 b^2-699 a^2 c^4 d^2 e^5 b^2-725 a c^5 d^4 e^3 b^2-245 c^6 d^6 e b^2+138 a^2 c^4 d e^6 x b^2+984 a c^5 d^3 e^4 x b^2+590 c^6 d^5 e^2 x b^2+70 c^7 d^7 b+374 a^3 c^4 d e^6 b+466 a^2 c^5 d^3 e^4 b+290 a c^6 d^5 e^2 b-374 a^3 c^4 e^7 x b-1398 a^2 c^5 d^2 e^5 x b-1450 a c^6 d^4 e^3 x b-490 c^7 d^6 e x b+256 a^4 c^4 e^7+140 c^8 d^7 x+748 a^3 c^5 d e^6 x+932 a^2 c^6 d^3 e^4 x+580 a c^7 d^5 e^2 x}{2 \left (4 a c-b^2\right )^4 \left (c d^2-b e d+a e^2\right )^4 \left (c x^2+b x+a\right )}+\frac{-6 e^5 b^6-4 c d e^4 b^5-6 c e^5 x b^5+70 a c e^5 b^4-5 c^2 d^2 e^3 b^4-8 c^2 d e^4 x b^4+42 a c^2 d e^4 b^3+120 c^3 d^3 e^2 b^3+68 a c^2 e^5 x b^3-10 c^3 d^2 e^3 x b^3-267 a^2 c^2 e^5 b^2-330 a c^3 d^2 e^3 b^2-175 c^4 d^4 e b^2+84 a c^3 d e^4 x b^2+240 c^4 d^3 e^2 x b^2+70 c^5 d^5 b+246 a^2 c^3 d e^4 b+220 a c^4 d^3 e^2 b-246 a^2 c^3 e^5 x b-660 a c^4 d^2 e^3 x b-350 c^5 d^4 e x b+192 a^3 c^3 e^5+140 c^6 d^5 x+492 a^2 c^4 d e^4 x+440 a c^5 d^3 e^2 x}{12 \left (4 a c-b^2\right )^3 \left (c d^2-b e d+a e^2\right )^3 \left (c x^2+b x+a\right )^2}+\frac{4 e^3 b^4+3 c d e^2 b^3+4 c e^3 x b^3-31 a c e^3 b^2-21 c^2 d^2 e b^2+6 c^2 d e^2 x b^2+14 c^3 d^3 b+30 a c^2 d e^2 b-30 a c^2 e^3 x b-42 c^3 d^2 e x b+32 a^2 c^2 e^3+28 c^4 d^3 x+60 a c^3 d e^2 x}{12 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )^3}+\frac{-e b^2+c d b-c e x b+2 a c e+2 c^2 d x}{4 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*d - b^2*e + 2*a*c*e + 2*c^2*d*x - b*c*e*x)/(4*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^4)
+ (14*b*c^3*d^3 - 21*b^2*c^2*d^2*e + 3*b^3*c*d*e^2 + 30*a*b*c^2*d*e^2 + 4*b^4*e^3 - 31*a*b^2*c*e^3 + 32*a^2*c
^2*e^3 + 28*c^4*d^3*x - 42*b*c^3*d^2*e*x + 6*b^2*c^2*d*e^2*x + 60*a*c^3*d*e^2*x + 4*b^3*c*e^3*x - 30*a*b*c^2*e
^3*x)/(12*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2)^3) + (70*b*c^5*d^5 - 175*b^2*c^4*d^4*e
+ 120*b^3*c^3*d^3*e^2 + 220*a*b*c^4*d^3*e^2 - 5*b^4*c^2*d^2*e^3 - 330*a*b^2*c^3*d^2*e^3 - 4*b^5*c*d*e^4 + 42*a
*b^3*c^2*d*e^4 + 246*a^2*b*c^3*d*e^4 - 6*b^6*e^5 + 70*a*b^4*c*e^5 - 267*a^2*b^2*c^2*e^5 + 192*a^3*c^3*e^5 + 14
0*c^6*d^5*x - 350*b*c^5*d^4*e*x + 240*b^2*c^4*d^3*e^2*x + 440*a*c^5*d^3*e^2*x - 10*b^3*c^3*d^2*e^3*x - 660*a*b
*c^4*d^2*e^3*x - 8*b^4*c^2*d*e^4*x + 84*a*b^2*c^3*d*e^4*x + 492*a^2*c^4*d*e^4*x - 6*b^5*c*e^5*x + 68*a*b^3*c^2
*e^5*x - 246*a^2*b*c^3*e^5*x)/(12*(-b^2 + 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)^2) + (70*b*c^7*
d^7 - 245*b^2*c^6*d^6*e + 295*b^3*c^5*d^5*e^2 + 290*a*b*c^6*d^5*e^2 - 125*b^4*c^4*d^4*e^3 - 725*a*b^2*c^5*d^4*
e^3 + b^5*c^3*d^3*e^4 + 492*a*b^3*c^4*d^3*e^4 + 466*a^2*b*c^5*d^3*e^4 + b^6*c^2*d^2*e^5 - 13*a*b^4*c^3*d^2*e^5
- 699*a^2*b^2*c^4*d^2*e^5 + b^7*c*d*e^6 - 14*a*b^5*c^2*d*e^6 + 69*a^2*b^3*c^3*d*e^6 + 374*a^3*b*c^4*d*e^6 + 2
*b^8*e^7 - 31*a*b^6*c*e^7 + 178*a^2*b^4*c^2*e^7 - 443*a^3*b^2*c^3*e^7 + 256*a^4*c^4*e^7 + 140*c^8*d^7*x - 490*
b*c^7*d^6*e*x + 590*b^2*c^6*d^5*e^2*x + 580*a*c^7*d^5*e^2*x - 250*b^3*c^5*d^4*e^3*x - 1450*a*b*c^6*d^4*e^3*x +
2*b^4*c^4*d^3*e^4*x + 984*a*b^2*c^5*d^3*e^4*x + 932*a^2*c^6*d^3*e^4*x + 2*b^5*c^3*d^2*e^5*x - 26*a*b^3*c^4*d^
2*e^5*x - 1398*a^2*b*c^5*d^2*e^5*x + 2*b^6*c^2*d*e^6*x - 28*a*b^4*c^3*d*e^6*x + 138*a^2*b^2*c^4*d*e^6*x + 748*
a^3*c^5*d*e^6*x + 2*b^7*c*e^7*x - 30*a*b^5*c^2*e^7*x + 164*a^2*b^3*c^3*e^7*x - 374*a^3*b*c^4*e^7*x)/(2*(-b^2 +
4*a*c)^4*(c*d^2 - b*d*e + a*e^2)^4*(a + b*x + c*x^2)) + ((-140*c^9*d^9 + 630*b*c^8*d^8*e - 1080*b^2*c^7*d^7*e
^2 - 720*a*c^8*d^7*e^2 + 840*b^3*c^6*d^6*e^3 + 2520*a*b*c^7*d^6*e^3 - 252*b^4*c^5*d^5*e^4 - 3024*a*b^2*c^6*d^5
*e^4 - 1512*a^2*c^7*d^5*e^4 + 1260*a*b^3*c^5*d^4*e^5 + 3780*a^2*b*c^6*d^4*e^5 - 2520*a^2*b^2*c^5*d^3*e^6 - 168
0*a^3*c^6*d^3*e^6 + 2520*a^3*b*c^5*d^2*e^7 - 1260*a^4*c^5*d*e^8 + b^9*e^9 - 18*a*b^7*c*e^9 + 126*a^2*b^5*c^2*e
^9 - 420*a^3*b^3*c^3*e^9 + 630*a^4*b*c^4*e^9)*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)^5) + (e^9*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^5 - (e^9*Log[a + b*x +
c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^5)

________________________________________________________________________________________

Maple [B]  time = 0.213, size = 32834, normalized size = 24.8 \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)/(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)/(c*x^2+b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.39818, size = 8597, normalized size = 6.49 \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)/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

-1/2*e^9*log(c*x^2 + b*x + a)/(c^5*d^10 - 5*b*c^4*d^9*e + 10*b^2*c^3*d^8*e^2 + 5*a*c^4*d^8*e^2 - 10*b^3*c^2*d^
7*e^3 - 20*a*b*c^3*d^7*e^3 + 5*b^4*c*d^6*e^4 + 30*a*b^2*c^2*d^6*e^4 + 10*a^2*c^3*d^6*e^4 - b^5*d^5*e^5 - 20*a*
b^3*c*d^5*e^5 - 30*a^2*b*c^2*d^5*e^5 + 5*a*b^4*d^4*e^6 + 30*a^2*b^2*c*d^4*e^6 + 10*a^3*c^2*d^4*e^6 - 10*a^2*b^
3*d^3*e^7 - 20*a^3*b*c*d^3*e^7 + 10*a^3*b^2*d^2*e^8 + 5*a^4*c*d^2*e^8 - 5*a^4*b*d*e^9 + a^5*e^10) + e^10*log(a
bs(x*e + d))/(c^5*d^10*e - 5*b*c^4*d^9*e^2 + 10*b^2*c^3*d^8*e^3 + 5*a*c^4*d^8*e^3 - 10*b^3*c^2*d^7*e^4 - 20*a*
b*c^3*d^7*e^4 + 5*b^4*c*d^6*e^5 + 30*a*b^2*c^2*d^6*e^5 + 10*a^2*c^3*d^6*e^5 - b^5*d^5*e^6 - 20*a*b^3*c*d^5*e^6
- 30*a^2*b*c^2*d^5*e^6 + 5*a*b^4*d^4*e^7 + 30*a^2*b^2*c*d^4*e^7 + 10*a^3*c^2*d^4*e^7 - 10*a^2*b^3*d^3*e^8 - 2
0*a^3*b*c*d^3*e^8 + 10*a^3*b^2*d^2*e^9 + 5*a^4*c*d^2*e^9 - 5*a^4*b*d*e^10 + a^5*e^11) + (140*c^9*d^9 - 630*b*c
^8*d^8*e + 1080*b^2*c^7*d^7*e^2 + 720*a*c^8*d^7*e^2 - 840*b^3*c^6*d^6*e^3 - 2520*a*b*c^7*d^6*e^3 + 252*b^4*c^5
*d^5*e^4 + 3024*a*b^2*c^6*d^5*e^4 + 1512*a^2*c^7*d^5*e^4 - 1260*a*b^3*c^5*d^4*e^5 - 3780*a^2*b*c^6*d^4*e^5 + 2
520*a^2*b^2*c^5*d^3*e^6 + 1680*a^3*c^6*d^3*e^6 - 2520*a^3*b*c^5*d^2*e^7 + 1260*a^4*c^5*d*e^8 - b^9*e^9 + 18*a*
b^7*c*e^9 - 126*a^2*b^5*c^2*e^9 + 420*a^3*b^3*c^3*e^9 - 630*a^4*b*c^4*e^9)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*
c))/((b^8*c^5*d^10 - 16*a*b^6*c^6*d^10 + 96*a^2*b^4*c^7*d^10 - 256*a^3*b^2*c^8*d^10 + 256*a^4*c^9*d^10 - 5*b^9
*c^4*d^9*e + 80*a*b^7*c^5*d^9*e - 480*a^2*b^5*c^6*d^9*e + 1280*a^3*b^3*c^7*d^9*e - 1280*a^4*b*c^8*d^9*e + 10*b
^10*c^3*d^8*e^2 - 155*a*b^8*c^4*d^8*e^2 + 880*a^2*b^6*c^5*d^8*e^2 - 2080*a^3*b^4*c^6*d^8*e^2 + 1280*a^4*b^2*c^
7*d^8*e^2 + 1280*a^5*c^8*d^8*e^2 - 10*b^11*c^2*d^7*e^3 + 140*a*b^9*c^3*d^7*e^3 - 640*a^2*b^7*c^4*d^7*e^3 + 640
*a^3*b^5*c^5*d^7*e^3 + 2560*a^4*b^3*c^6*d^7*e^3 - 5120*a^5*b*c^7*d^7*e^3 + 5*b^12*c*d^6*e^4 - 50*a*b^10*c^2*d^
6*e^4 + 10*a^2*b^8*c^3*d^6*e^4 + 1440*a^3*b^6*c^4*d^6*e^4 - 5440*a^4*b^4*c^5*d^6*e^4 + 5120*a^5*b^2*c^6*d^6*e^
4 + 2560*a^6*c^7*d^6*e^4 - b^13*d^5*e^5 - 4*a*b^11*c*d^5*e^5 + 194*a^2*b^9*c^2*d^5*e^5 - 1184*a^3*b^7*c^3*d^5*
e^5 + 1984*a^4*b^5*c^4*d^5*e^5 + 2560*a^5*b^3*c^5*d^5*e^5 - 7680*a^6*b*c^6*d^5*e^5 + 5*a*b^12*d^4*e^6 - 50*a^2
*b^10*c*d^4*e^6 + 10*a^3*b^8*c^2*d^4*e^6 + 1440*a^4*b^6*c^3*d^4*e^6 - 5440*a^5*b^4*c^4*d^4*e^6 + 5120*a^6*b^2*
c^5*d^4*e^6 + 2560*a^7*c^6*d^4*e^6 - 10*a^2*b^11*d^3*e^7 + 140*a^3*b^9*c*d^3*e^7 - 640*a^4*b^7*c^2*d^3*e^7 + 6
40*a^5*b^5*c^3*d^3*e^7 + 2560*a^6*b^3*c^4*d^3*e^7 - 5120*a^7*b*c^5*d^3*e^7 + 10*a^3*b^10*d^2*e^8 - 155*a^4*b^8
*c*d^2*e^8 + 880*a^5*b^6*c^2*d^2*e^8 - 2080*a^6*b^4*c^3*d^2*e^8 + 1280*a^7*b^2*c^4*d^2*e^8 + 1280*a^8*c^5*d^2*
e^8 - 5*a^4*b^9*d*e^9 + 80*a^5*b^7*c*d*e^9 - 480*a^6*b^5*c^2*d*e^9 + 1280*a^7*b^3*c^3*d*e^9 - 1280*a^8*b*c^4*d
*e^9 + a^5*b^8*e^10 - 16*a^6*b^6*c*e^10 + 96*a^7*b^4*c^2*e^10 - 256*a^8*b^2*c^3*e^10 + 256*a^9*c^4*e^10)*sqrt(
-b^2 + 4*a*c)) - 1/12*(3*b^7*c^5*d^9 - 50*a*b^5*c^6*d^9 + 326*a^2*b^3*c^7*d^9 - 1116*a^3*b*c^8*d^9 - 15*b^8*c^
4*d^8*e + 249*a*b^6*c^5*d^8*e - 1611*a^2*b^4*c^6*d^8*e + 5406*a^3*b^2*c^7*d^8*e - 384*a^4*c^8*d^8*e + 30*b^9*c
^3*d^7*e^2 - 480*a*b^7*c^4*d^7*e^2 + 2916*a^2*b^5*c^5*d^7*e^2 - 8688*a^3*b^3*c^6*d^7*e^2 - 3984*a^4*b*c^7*d^7*
e^2 - 30*b^10*c^2*d^6*e^3 + 430*a*b^8*c^3*d^6*e^3 - 2080*a^2*b^6*c^4*d^6*e^3 + 3132*a^3*b^4*c^5*d^6*e^3 + 1906
4*a^4*b^2*c^6*d^6*e^3 - 2048*a^5*c^7*d^6*e^3 + 15*b^11*c*d^5*e^4 - 150*a*b^9*c^2*d^5*e^4 + 4800*a^3*b^5*c^4*d^
5*e^4 - 25500*a^4*b^3*c^5*d^5*e^4 - 4680*a^5*b*c^6*d^5*e^4 - 3*b^12*d^4*e^5 - 15*a*b^10*c*d^4*e^5 + 639*a^2*b^
8*c^2*d^4*e^5 - 3984*a^3*b^6*c^3*d^4*e^5 + 7962*a^4*b^4*c^4*d^4*e^5 + 25524*a^5*b^2*c^5*d^4*e^5 - 4608*a^6*c^6
*d^4*e^5 + 16*a*b^11*d^3*e^6 - 154*a^2*b^9*c*d^3*e^6 - 96*a^3*b^7*c^2*d^3*e^6 + 5780*a^4*b^5*c^3*d^3*e^6 - 243
68*a^5*b^3*c^4*d^3*e^6 - 1104*a^6*b*c^5*d^3*e^6 - 36*a^2*b^10*d^2*e^7 + 498*a^3*b^8*c*d^2*e^7 - 2172*a^4*b^6*c
^2*d^2*e^7 + 1044*a^5*b^4*c^3*d^2*e^7 + 17016*a^6*b^2*c^4*d^2*e^7 - 6144*a^7*c^5*d^2*e^7 + 48*a^3*b^9*d*e^8 -
741*a^4*b^7*c*d*e^8 + 4158*a^5*b^5*c^2*d*e^8 - 9354*a^6*b^3*c^3*d*e^8 + 2244*a^7*b*c^4*d*e^8 - 25*a^4*b^8*e^9
+ 385*a^5*b^6*c*e^9 - 2175*a^6*b^4*c^2*e^9 + 5150*a^7*b^2*c^3*e^9 - 3200*a^8*c^4*e^9 - 12*(70*c^12*d^9 - 315*b
*c^11*d^8*e + 540*b^2*c^10*d^7*e^2 + 360*a*c^11*d^7*e^2 - 420*b^3*c^9*d^6*e^3 - 1260*a*b*c^10*d^6*e^3 + 126*b^
4*c^8*d^5*e^4 + 1512*a*b^2*c^9*d^5*e^4 + 756*a^2*c^10*d^5*e^4 - 630*a*b^3*c^8*d^4*e^5 - 1890*a^2*b*c^9*d^4*e^5
+ 1260*a^2*b^2*c^8*d^3*e^6 + 840*a^3*c^9*d^3*e^6 - 1260*a^3*b*c^8*d^2*e^7 - b^8*c^4*d*e^8 + 16*a*b^6*c^5*d*e^
8 - 96*a^2*b^4*c^6*d*e^8 + 256*a^3*b^2*c^7*d*e^8 + 374*a^4*c^8*d*e^8 + a*b^7*c^4*e^9 - 15*a^2*b^5*c^5*e^9 + 82
*a^3*b^3*c^6*e^9 - 187*a^4*b*c^7*e^9)*x^7 - 6*(490*b*c^11*d^9 - 2205*b^2*c^10*d^8*e + 3780*b^3*c^9*d^7*e^2 + 2
520*a*b*c^10*d^7*e^2 - 2940*b^4*c^8*d^6*e^3 - 8820*a*b^2*c^9*d^6*e^3 + 882*b^5*c^7*d^5*e^4 + 10584*a*b^3*c^8*d
^5*e^4 + 5292*a^2*b*c^9*d^5*e^4 - 4410*a*b^4*c^7*d^4*e^5 - 13230*a^2*b^2*c^8*d^4*e^5 + 8820*a^2*b^3*c^7*d^3*e^
6 + 5880*a^3*b*c^8*d^3*e^6 + b^8*c^4*d^2*e^7 - 16*a*b^6*c^5*d^2*e^7 + 96*a^2*b^4*c^6*d^2*e^7 - 9076*a^3*b^2*c^
7*d^2*e^7 + 256*a^4*c^8*d^2*e^7 - 8*b^9*c^3*d*e^8 + 128*a*b^7*c^4*d*e^8 - 768*a^2*b^5*c^5*d*e^8 + 2048*a^3*b^3
*c^6*d*e^8 + 2362*a^4*b*c^7*d*e^8 + 8*a*b^8*c^3*e^9 - 121*a^2*b^6*c^4*e^9 + 670*a^3*b^4*c^5*e^9 - 1565*a^4*b^2
*c^6*e^9 + 256*a^5*c^7*e^9)*x^6 - 4*(910*b^2*c^10*d^9 + 770*a*c^11*d^9 - 4095*b^3*c^9*d^8*e - 3465*a*b*c^10*d^
8*e + 7020*b^4*c^8*d^7*e^2 + 10620*a*b^2*c^9*d^7*e^2 + 3960*a^2*c^10*d^7*e^2 - 5460*b^5*c^7*d^6*e^3 - 21000*a*
b^3*c^8*d^6*e^3 - 13860*a^2*b*c^9*d^6*e^3 + 1638*b^6*c^6*d^5*e^4 + 21042*a*b^4*c^7*d^5*e^4 + 26460*a^2*b^2*c^8
*d^5*e^4 + 8316*a^3*c^9*d^5*e^4 - 8190*a*b^5*c^6*d^4*e^5 - 31500*a^2*b^3*c^7*d^4*e^5 - 20790*a^3*b*c^8*d^4*e^5
- b^8*c^4*d^3*e^6 + 16*a*b^6*c^5*d^3*e^6 + 16284*a^2*b^4*c^6*d^3*e^6 + 25036*a^3*b^2*c^7*d^3*e^6 + 8984*a^4*c
^8*d^3*e^6 + 6*b^9*c^3*d^2*e^7 - 96*a*b^7*c^4*d^2*e^7 + 576*a^2*b^5*c^5*d^2*e^7 - 17916*a^3*b^3*c^6*d^2*e^7 -
12324*a^4*b*c^7*d^2*e^7 - 18*b^10*c^2*d*e^8 + 276*a*b^8*c^3*d*e^8 - 1536*a^2*b^6*c^4*d*e^8 + 3456*a^3*b^4*c^5*
d*e^8 + 6654*a^4*b^2*c^6*d*e^8 + 3858*a^5*c^7*d*e^8 + 18*a*b^9*c^2*e^9 - 264*a^2*b^7*c^3*e^9 + 1381*a^3*b^5*c^
4*e^9 - 2809*a^4*b^3*c^5*e^9 - 777*a^5*b*c^6*e^9)*x^5 - (1750*b^3*c^9*d^9 + 7700*a*b*c^10*d^9 - 7875*b^4*c^8*d
^8*e - 34650*a*b^2*c^9*d^8*e + 13500*b^5*c^7*d^7*e^2 + 68400*a*b^3*c^8*d^7*e^2 + 39600*a^2*b*c^9*d^7*e^2 - 105
00*b^6*c^6*d^6*e^3 - 77700*a*b^4*c^7*d^6*e^3 - 138600*a^2*b^2*c^8*d^6*e^3 + 3150*b^7*c^5*d^5*e^4 + 51660*a*b^5
*c^6*d^5*e^4 + 185220*a^2*b^3*c^7*d^5*e^4 + 83160*a^3*b*c^8*d^5*e^4 + 3*b^8*c^4*d^4*e^5 - 15798*a*b^6*c^5*d^4*
e^5 - 116262*a^2*b^4*c^6*d^4*e^5 - 208668*a^3*b^2*c^7*d^4*e^5 + 768*a^4*c^8*d^4*e^5 - 16*b^9*c^3*d^3*e^6 + 256
*a*b^7*c^4*d^3*e^6 + 29964*a^2*b^5*c^5*d^3*e^6 + 163696*a^3*b^3*c^6*d^3*e^6 + 88304*a^4*b*c^7*d^3*e^6 + 36*b^1
0*c^2*d^2*e^7 - 552*a*b^8*c^3*d^2*e^7 + 3072*a^2*b^6*c^4*d^2*e^7 - 38412*a^3*b^4*c^5*d^2*e^7 - 135528*a^4*b^2*
c^6*d^2*e^7 + 6144*a^5*c^7*d^2*e^7 - 48*b^11*c*d*e^8 + 624*a*b^9*c^2*d*e^8 - 2304*a^2*b^7*c^3*d*e^8 - 1536*a^3
*b^5*c^4*d*e^8 + 40326*a^4*b^3*c^5*d*e^8 + 32436*a^5*b*c^6*d*e^8 + 48*a*b^10*c*e^9 - 612*a^2*b^8*c^2*e^9 + 227
2*a^3*b^6*c^3*e^9 + 473*a^4*b^4*c^4*e^9 - 20058*a^5*b^2*c^5*e^9 + 5376*a^6*c^6*e^9)*x^4 - 4*(42*b^4*c^8*d^9 +
1414*a*b^2*c^9*d^9 + 1022*a^2*c^10*d^9 - 189*b^5*c^7*d^8*e - 6363*a*b^3*c^8*d^8*e - 4599*a^2*b*c^9*d^8*e + 324
*b^6*c^6*d^7*e^2 + 11124*a*b^4*c^7*d^7*e^2 + 15156*a^2*b^2*c^8*d^7*e^2 + 5256*a^3*c^9*d^7*e^2 - 252*b^7*c^5*d^
6*e^3 - 9240*a*b^5*c^6*d^6*e^3 - 31584*a^2*b^3*c^7*d^6*e^3 - 18396*a^3*b*c^8*d^6*e^3 + 75*b^8*c^4*d^5*e^4 + 34
62*a*b^6*c^5*d^5*e^4 + 32778*a^2*b^4*c^6*d^5*e^4 + 37500*a^3*b^2*c^7*d^5*e^4 + 10884*a^4*c^8*d^5*e^4 + 3*b^9*c
^3*d^4*e^5 - 426*a*b^7*c^4*d^4*e^5 - 13572*a^2*b^5*c^5*d^4*e^5 - 48144*a^3*b^3*c^6*d^4*e^5 - 26826*a^4*b*c^7*d
^4*e^5 - 6*b^10*c^2*d^3*e^6 + 92*a*b^8*c^3*d^3*e^6 + 244*a^2*b^6*c^4*d^3*e^6 + 27108*a^3*b^4*c^5*d^3*e^6 + 348
52*a^4*b^2*c^6*d^3*e^6 + 11240*a^5*c^7*d^3*e^6 + 6*b^11*c*d^2*e^7 - 78*a*b^9*c^2*d^2*e^7 + 288*a^2*b^7*c^3*d^2
*e^7 - 564*a^3*b^5*c^4*d^2*e^7 - 28524*a^4*b^3*c^5*d^2*e^7 - 13788*a^5*b*c^6*d^2*e^7 - 3*b^12*d*e^8 + 12*a*b^1
0*c*d*e^8 + 270*a^2*b^8*c^2*d*e^8 - 2400*a^3*b^6*c^3*d*e^8 + 7098*a^4*b^4*c^4*d*e^8 + 8118*a^5*b^2*c^5*d*e^8 +
4590*a^6*c^6*d*e^8 + 3*a*b^11*e^9 - 15*a^2*b^9*c*e^9 - 204*a^3*b^7*c^2*e^9 + 1882*a^4*b^5*c^3*e^9 - 5089*a^5*
b^3*c^4*e^9 + 393*a^6*b*c^5*e^9)*x^3 + 2*(14*b^5*c^7*d^9 - 392*a*b^3*c^8*d^9 - 3066*a^2*b*c^9*d^9 - 63*b^6*c^6
*d^8*e + 1764*a*b^4*c^7*d^8*e + 13797*a^2*b^2*c^8*d^8*e + 108*b^7*c^5*d^7*e^2 - 2952*a*b^5*c^6*d^7*e^2 - 25668
*a^2*b^3*c^7*d^7*e^2 - 15768*a^3*b*c^8*d^7*e^2 - 85*b^8*c^4*d^6*e^3 + 2116*a*b^6*c^5*d^6*e^3 + 25356*a^2*b^4*c
^6*d^6*e^3 + 55444*a^3*b^2*c^7*d^6*e^3 - 256*a^4*c^8*d^6*e^3 + 30*b^9*c^3*d^5*e^4 - 480*a*b^7*c^4*d^5*e^4 - 13
374*a^2*b^5*c^5*d^5*e^4 - 71688*a^3*b^3*c^6*d^5*e^4 - 31884*a^4*b*c^7*d^5*e^4 - 9*b^10*c^2*d^4*e^5 + 12*a*b^8*
c^3*d^4*e^5 + 2382*a^2*b^6*c^4*d^4*e^5 + 39906*a^3*b^4*c^5*d^4*e^5 + 82014*a^4*b^2*c^6*d^4*e^5 - 1536*a^5*c^7*
d^4*e^5 + 8*b^11*c*d^3*e^6 - 104*a*b^9*c^2*d^3*e^6 + 636*a^2*b^7*c^3*d^3*e^6 - 6632*a^3*b^5*c^4*d^3*e^6 - 6398
8*a^4*b^3*c^5*d^3*e^6 - 30648*a^5*b*c^6*d^3*e^6 - 3*b^12*d^2*e^7 + 12*a*b^10*c*d^2*e^7 + 270*a^2*b^8*c^2*d^2*e
^7 - 2652*a^3*b^6*c^3*d^2*e^7 + 13776*a^4*b^4*c^4*d^2*e^7 + 50580*a^5*b^2*c^5*d^2*e^7 - 4608*a^6*c^6*d^2*e^7 +
24*a*b^11*d*e^8 - 312*a^2*b^9*c*d*e^8 + 1152*a^3*b^7*c^2*d*e^8 + 894*a^4*b^5*c^3*d*e^8 - 15816*a^5*b^3*c^4*d*
e^8 - 9162*a^6*b*c^5*d*e^8 - 21*a^2*b^10*e^9 + 274*a^3*b^8*c*e^9 - 1078*a^4*b^6*c^2*e^9 + 150*a^5*b^4*c^3*e^9
+ 7525*a^6*b^2*c^4*e^9 - 3328*a^7*c^5*e^9)*x^2 - 4*(2*b^6*c^6*d^9 - 38*a*b^4*c^7*d^9 + 348*a^2*b^2*c^8*d^9 + 5
58*a^3*c^9*d^9 - 9*b^7*c^5*d^8*e + 171*a*b^5*c^6*d^8*e - 1566*a^2*b^3*c^7*d^8*e - 2511*a^3*b*c^8*d^8*e + 15*b^
8*c^4*d^7*e^2 - 276*a*b^6*c^5*d^7*e^2 + 2448*a^2*b^4*c^6*d^7*e^2 + 6204*a^3*b^2*c^7*d^7*e^2 + 2760*a^4*c^8*d^7
*e^2 - 10*b^9*c^3*d^6*e^3 + 160*a*b^7*c^4*d^6*e^3 - 1212*a^2*b^5*c^5*d^6*e^3 - 10124*a^3*b^3*c^6*d^6*e^3 - 953
2*a^4*b*c^7*d^6*e^3 + 30*a*b^8*c^3*d^5*e^4 - 480*a^2*b^6*c^4*d^5*e^4 + 8802*a^3*b^4*c^5*d^5*e^4 + 15504*a^4*b^
2*c^6*d^5*e^4 + 5412*a^5*c^7*d^5*e^4 + 3*b^11*c*d^4*e^5 - 57*a*b^9*c^2*d^4*e^5 + 432*a^2*b^7*c^3*d^4*e^5 - 201
0*a^3*b^5*c^4*d^4*e^5 - 15954*a^4*b^3*c^5*d^4*e^5 - 12762*a^5*b*c^6*d^4*e^5 - b^12*d^3*e^6 + 4*a*b^10*c*d^3*e^
6 + 126*a^2*b^8*c^2*d^3*e^6 - 1460*a^3*b^6*c^3*d^3*e^6 + 8048*a^4*b^4*c^4*d^3*e^6 + 12684*a^5*b^2*c^5*d^3*e^6
+ 5160*a^6*c^6*d^3*e^6 + 6*a*b^11*d^2*e^7 - 78*a^2*b^9*c*d^2*e^7 + 252*a^3*b^7*c^2*d^2*e^7 + 876*a^4*b^5*c^3*d
^2*e^7 - 9336*a^5*b^3*c^4*d^2*e^7 - 5436*a^6*b*c^5*d^2*e^7 - 18*a^2*b^10*d*e^8 + 276*a^3*b^8*c*d*e^8 - 1518*a^
4*b^6*c^2*d*e^8 + 3114*a^5*b^4*c^3*d*e^8 + 1596*a^6*b^2*c^4*d*e^8 + 1950*a^7*c^5*d*e^8 + 13*a^3*b^9*e^9 - 196*
a^4*b^7*c*e^9 + 1068*a^5*b^5*c^2*e^9 - 2324*a^6*b^3*c^3*e^9 + 689*a^7*b*c^4*e^9)*x)/((c*d^2 - b*d*e + a*e^2)^5
*(c*x^2 + b*x + a)^4*(b^2 - 4*a*c)^4)