3.2221 \(\int \frac{(d+e x)^5}{(a+b x+c x^2)^5} \, dx\)

Optimal. Leaf size=388 \[ \frac{(d+e x)^3 \left (-c x \left (-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right )-10 b c \left (3 a e^2+7 c d^2\right )+28 a c^2 d e+63 b^2 c d e-10 b^3 e^2\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (d+e x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

[Out]

-((b + 2*c*x)*(d + e*x)^5)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^4*(14*b*c*d - 5*b^2*e - 8*a*c*e
+ 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^3*(63*b^2*c*d*e + 28*a*c^2*d*e
- 10*b^3*e^2 - 10*b*c*(7*c*d^2 + 3*a*e^2) - c*(140*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(35*b*d - 8*a*e))*x))/(12*(b^2
- 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)*(b*d
- 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)
*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

________________________________________________________________________________________

Rubi [A]  time = 0.561369, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {736, 820, 804, 722, 618, 206} \[ \frac{(d+e x)^3 \left (-c x \left (-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right )-10 b c \left (3 a e^2+7 c d^2\right )+28 a c^2 d e+63 b^2 c d e-10 b^3 e^2\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (d+e x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b + 2*c*x)*(d + e*x)^5)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^4*(14*b*c*d - 5*b^2*e - 8*a*c*e
+ 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^3*(63*b^2*c*d*e + 28*a*c^2*d*e
- 10*b^3*e^2 - 10*b*c*(7*c*d^2 + 3*a*e^2) - c*(140*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(35*b*d - 8*a*e))*x))/(12*(b^2
- 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)*(b*d
- 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)
*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

Rule 736

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

Rule 820

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

Rule 804

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

Rule 722

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

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])

Rubi steps

\begin{align*} \int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{\int \frac{(d+e x)^4 (-14 c d+5 b e-4 c e x)}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{\int \frac{(d+e x)^3 \left (-2 \left (70 c^2 d^2+10 b^2 e^2-c e (63 b d-16 a e)\right )-14 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2}\\ &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{\left (5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )^3}\\ &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac{\left (5 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4}\\ &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{\left (10 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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}\\ &=-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (14 b c d-5 b^2 e-8 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (63 b^2 c d e+28 a c^2 d e-10 b^3 e^2-10 b c \left (7 c d^2+3 a e^2\right )-c \left (140 c^2 d^2+27 b^2 e^2-4 c e (35 b d-8 a e)\right ) x\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x) (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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}}\\ \end{align*}

Mathematica [B]  time = 2.69068, size = 985, normalized size = 2.54 \[ \frac{1}{12} \left (\frac{30 (2 c d-b e) \left (7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left (8 b^2 d^2-10 a b e d+3 a^2 e^2\right )\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{3 e^5 b^6+5 c d e^4 b^5+c e^3 \left (10 c d (e x-5 d)-41 a e^2\right ) b^4+10 c^2 e^2 \left (5 c (3 d-2 e x) d^2+a e^2 (6 d-e x)\right ) b^3+c^2 e \left (129 a^2 e^4-30 a c d (5 d-4 e x) e^2-25 c^2 d^3 (7 d-12 e x)\right ) b^2+10 c^3 \left (7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+3 a^2 e^4 (d-e x)\right ) b+4 c^3 \left (35 c^3 x d^5+50 a c^2 e^2 x d^3+15 a^2 c e^4 x d-48 a^3 e^5\right )}{c^3 \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}-\frac{3 \left (b^5 x e^5+b^4 (a e-5 c d x) e^4-5 b^3 c \left (a e (d+e x)-2 c d^2 x\right ) e^3-2 b^2 c \left (5 c^2 x d^3-5 a c e (d+2 e x) d+2 a^2 e^3\right ) e^2+2 c^2 \left (-c^3 x d^5+5 a c^2 e (d+2 e x) d^3-5 a^2 c e^3 (2 d+e x) d+a^3 e^5\right )+b c^2 \left (-c^2 (d-5 e x) d^4-10 a c e^2 (d+3 e x) d^2+5 a^2 e^4 (3 d+e x)\right )\right )}{c^4 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{-3 e^5 b^6+3 c e^4 (5 d+2 e x) b^5+c e^3 \left (27 a e^2-10 c d (3 d+e x)\right ) b^4-10 c^2 e^2 \left (c (2 e x-3 d) d^2+5 a e^2 (2 d+e x)\right ) b^3+c^2 e \left (-83 a^2 e^4+10 a c d (13 d+12 e x) e^2+5 c^2 d^3 (12 e x-7 d)\right ) b^2+2 c^3 \left (7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+5 a^2 e^4 (23 d+9 e x)\right ) b+4 c^3 \left (7 c^3 x d^5+10 a c^2 e^2 x d^3-5 a^2 c e^3 (16 d+9 e x) d+16 a^3 e^5\right )}{c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{120 (2 c d-b e) \left (7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left (8 b^2 d^2-10 a b e d+3 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a
*b*d*e + 3*a^2*e^2))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (5*b^5*c*d*e^4 + 3*b^6*e^5 + 4*c^3*(
-48*a^3*e^5 + 35*c^3*d^5*x + 50*a*c^2*d^3*e^2*x + 15*a^2*c*d*e^4*x) + b^2*c^2*e*(129*a^2*e^4 - 25*c^2*d^3*(7*d
- 12*e*x) - 30*a*c*d*e^2*(5*d - 4*e*x)) + 10*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 3*a^
2*e^4*(d - e*x)) + 10*b^3*c^2*e^2*(5*c*d^2*(3*d - 2*e*x) + a*e^2*(6*d - e*x)) + b^4*c*e^3*(-41*a*e^2 + 10*c*d*
(-5*d + e*x)))/(c^3*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) - (3*(b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*
e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*
e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a
^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x))))/(c^4*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (-3*b^6*e^5 + 3*
b^5*c*e^4*(5*d + 2*e*x) + b^4*c*e^3*(27*a*e^2 - 10*c*d*(3*d + e*x)) - 10*b^3*c^2*e^2*(5*a*e^2*(2*d + e*x) + c*
d^2*(-3*d + 2*e*x)) + 4*c^3*(16*a^3*e^5 + 7*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(16*d + 9*e*x)) + 2
*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(23*d + 9*e*x)) + b^2*c^2*e*(-83*a^2*e^
4 + 5*c^2*d^3*(-7*d + 12*e*x) + 10*a*c*d*e^2*(13*d + 12*e*x)))/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (12
0*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*
d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

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Maple [B]  time = 0.171, size = 3092, normalized size = 8. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10
*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*
b^6*c+b^8)*c^3*x^7-35/2*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*
d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+
96*a^2*b^4*c^2-16*a*b^6*c+b^8)*b*c^2*x^6-5/3*c*(11*a*c+13*b^2)*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b
^2*c*d*e^4+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-14
*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5-1/12*(768*a^4*c^4*e^5+882*a^3*b^2*c^
3*e^5-3300*a^3*b*c^4*d*e^4+1213*a^2*b^4*c^2*e^5-7350*a^2*b^3*c^3*d*e^4+16500*a^2*b^2*c^4*d^2*e^3-11000*a^2*b*c
^5*d^3*e^2+77*a*b^6*c*e^5-2050*a*b^5*c^2*d*e^4+9250*a*b^4*c^3*d^2*e^3-19000*a*b^3*c^4*d^3*e^2+19250*a*b^2*c^5*
d^4*e-7700*a*b*c^6*d^5+3*b^8*e^5-125*b^7*c*d*e^4+1250*b^6*c^2*d^2*e^3-3750*b^5*c^3*d^3*e^2+4375*b^4*c^4*d^4*e-
1750*b^3*c^5*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(219*a^4*b*c^3*e^5+330
*a^4*c^4*d*e^4+376*a^3*b^3*c^2*e^5-2250*a^3*b^2*c^3*d*e^4+2190*a^3*b*c^4*d^2*e^3-1460*a^3*c^5*d^3*e^2+110*a^2*
b^5*c*e^5-1015*a^2*b^4*c^2*d*e^4+3760*a^2*b^3*c^3*d^2*e^3-4210*a^2*b^2*c^4*d^3*e^2+2555*a^2*b*c^5*d^4*e-1022*a
^2*c^6*d^5+3*a*b^7*e^5-185*a*b^6*c*d*e^4+1100*a*b^5*c^2*d^2*e^3-3090*a*b^4*c^3*d^3*e^2+3535*a*b^3*c^4*d^4*e-14
14*a*b^2*c^5*d^5+30*b^7*c*d^2*e^3-90*b^6*c^2*d^3*e^2+105*b^5*c^3*d^4*e-42*b^4*c^4*d^5)/c/(256*a^4*c^4-256*a^3*
b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3-1/6*(256*a^5*c^3*e^5+401*a^4*b^2*c^2*e^5-1570*a^4*b*c^3*d*e^4+2560*
a^4*c^4*d^2*e^3+399*a^3*b^4*c*e^5-2540*a^3*b^3*c^2*d*e^4+4010*a^3*b^2*c^3*d^2*e^3-4380*a^3*b*c^4*d^3*e^2+9*a^2
*b^6*e^5-645*a^2*b^5*c*d*e^4+3990*a^2*b^4*c^2*d^2*e^3-7130*a^2*b^3*c^3*d^3*e^2+7665*a^2*b^2*c^4*d^4*e-3066*a^2
*b*c^5*d^5+90*a*b^6*c*d^2*e^3-820*a*b^5*c^2*d^3*e^2+980*a*b^4*c^3*d^4*e-392*a*b^3*c^4*d^5+30*b^7*c*d^3*e^2-35*
b^6*c^2*d^4*e+14*b^5*c^3*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^2-1/3*(83*a^5*b*
c^2*e^5+90*a^5*c^3*d*e^4+151*a^4*b^3*c*e^5-920*a^4*b^2*c^2*d*e^4+830*a^4*b*c^3*d^2*e^3+300*a^4*c^4*d^3*e^2+3*a
^3*b^5*e^5-235*a^3*b^4*c*d*e^4+1510*a^3*b^3*c^2*d^2*e^3-2790*a^3*b^2*c^3*d^3*e^2+1395*a^3*b*c^4*d^4*e-558*a^3*
c^5*d^5+30*a^2*b^5*c*d^2*e^3-280*a^2*b^4*c^2*d^3*e^2+870*a^2*b^3*c^3*d^4*e-348*a^2*b^2*c^4*d^5+10*a*b^6*c*d^3*
e^2-95*a*b^5*c^2*d^4*e+38*a*b^4*c^3*d^5+5*b^7*c*d^4*e-2*b^6*c^2*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4
*c^2-16*a*b^6*c+b^8)*x-1/12*(128*a^6*c^2*e^5+166*a^5*b^2*c*e^5-1100*a^5*b*c^2*d*e^4+1280*a^5*c^3*d^2*e^3+3*a^4
*b^4*e^5-250*a^4*b^3*c*d*e^4+1660*a^4*b^2*c^2*d^2*e^3-3240*a^4*b*c^3*d^3*e^2+1920*a^4*c^4*d^4*e+30*a^3*b^4*c*d
^2*e^3-280*a^3*b^3*c^2*d^3*e^2+870*a^3*b^2*c^3*d^4*e-1116*a^3*b*c^4*d^5+10*a^2*b^5*c*d^3*e^2-95*a^2*b^4*c^2*d^
4*e+326*a^2*b^3*c^3*d^5+5*a*b^6*c*d^4*e-50*a*b^5*c^2*d^5+3*b^7*c*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^
4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4-30/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*c*e^5+60/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*
b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d*a^2*c^2*e^4-10/(256*a^4*c^4-256*a^3*b^2*c^3
+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^5+120/(256*a^4*c
^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*
c*d*e^4-300/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*
a*c-b^2)^(1/2))*a*b*c^2*d^2*e^3+200/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1
/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^3*a*c^3*e^2+10/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*
c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d*e^4-100/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2
*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^2*b^3*c*e^3+300/(256*a^4*c^4-
256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^2*d
^3*e^2-350/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*b*c^3*d^4*e+140/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^5

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.91482, size = 18823, normalized size = 48.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(60*(14*(b^2*c^8 - 4*a*c^9)*d^5 - 35*(b^3*c^7 - 4*a*b*c^8)*d^4*e + 10*(3*b^4*c^6 - 10*a*b^2*c^7 - 8*a^2*
c^8)*d^3*e^2 - 10*(b^5*c^5 - a*b^3*c^6 - 12*a^2*b*c^7)*d^2*e^3 + (b^6*c^4 + 8*a*b^4*c^5 - 42*a^2*b^2*c^6 - 24*
a^3*c^7)*d*e^4 - (a*b^5*c^4 - a^2*b^3*c^5 - 12*a^3*b*c^6)*e^5)*x^7 + 210*(14*(b^3*c^7 - 4*a*b*c^8)*d^5 - 35*(b
^4*c^6 - 4*a*b^2*c^7)*d^4*e + 10*(3*b^5*c^5 - 10*a*b^3*c^6 - 8*a^2*b*c^7)*d^3*e^2 - 10*(b^6*c^4 - a*b^4*c^5 -
12*a^2*b^2*c^6)*d^2*e^3 + (b^7*c^3 + 8*a*b^5*c^4 - 42*a^2*b^3*c^5 - 24*a^3*b*c^6)*d*e^4 - (a*b^6*c^3 - a^2*b^4
*c^4 - 12*a^3*b^2*c^5)*e^5)*x^6 - (3*b^9*c - 62*a*b^7*c^2 + 526*a^2*b^5*c^3 - 2420*a^3*b^3*c^4 + 4464*a^4*b*c^
5)*d^5 - 5*(a*b^8*c - 23*a^2*b^6*c^2 + 250*a^3*b^4*c^3 - 312*a^4*b^2*c^4 - 1536*a^5*c^5)*d^4*e - 10*(a^2*b^7*c
- 32*a^3*b^5*c^2 - 212*a^4*b^3*c^3 + 1296*a^5*b*c^4)*d^3*e^2 - 10*(3*a^3*b^6*c + 154*a^4*b^4*c^2 - 536*a^5*b^
2*c^3 - 512*a^6*c^4)*d^2*e^3 + 50*(5*a^4*b^5*c + 2*a^5*b^3*c^2 - 88*a^6*b*c^3)*d*e^4 - (3*a^4*b^6 + 154*a^5*b^
4*c - 536*a^6*b^2*c^2 - 512*a^7*c^3)*e^5 + 20*(14*(13*b^4*c^6 - 41*a*b^2*c^7 - 44*a^2*c^8)*d^5 - 35*(13*b^5*c^
5 - 41*a*b^3*c^6 - 44*a^2*b*c^7)*d^4*e + 10*(39*b^6*c^4 - 97*a*b^4*c^5 - 214*a^2*b^2*c^6 - 88*a^3*c^7)*d^3*e^2
- 10*(13*b^7*c^3 - 2*a*b^5*c^4 - 167*a^2*b^3*c^5 - 132*a^3*b*c^6)*d^2*e^3 + (13*b^8*c^2 + 115*a*b^6*c^3 - 458
*a^2*b^4*c^4 - 774*a^3*b^2*c^5 - 264*a^4*c^6)*d*e^4 - (13*a*b^7*c^2 - 2*a^2*b^5*c^3 - 167*a^3*b^3*c^4 - 132*a^
4*b*c^5)*e^5)*x^5 + (350*(5*b^5*c^5 + 2*a*b^3*c^6 - 88*a^2*b*c^7)*d^5 - 875*(5*b^6*c^4 + 2*a*b^4*c^5 - 88*a^2*
b^2*c^6)*d^4*e + 250*(15*b^7*c^3 + 16*a*b^5*c^4 - 260*a^2*b^3*c^5 - 176*a^3*b*c^6)*d^3*e^2 - 250*(5*b^8*c^2 +
17*a*b^6*c^3 - 82*a^2*b^4*c^4 - 264*a^3*b^2*c^5)*d^2*e^3 + 25*(5*b^9*c + 62*a*b^7*c^2 - 34*a^2*b^5*c^3 - 1044*
a^3*b^3*c^4 - 528*a^4*b*c^5)*d*e^4 - (3*b^10 + 65*a*b^8*c + 905*a^2*b^6*c^2 - 3970*a^3*b^4*c^3 - 2760*a^4*b^2*
c^4 - 3072*a^5*c^5)*e^5)*x^4 + 4*(14*(3*b^6*c^4 + 89*a*b^4*c^5 - 331*a^2*b^2*c^6 - 292*a^3*c^7)*d^5 - 35*(3*b^
7*c^3 + 89*a*b^5*c^4 - 331*a^2*b^3*c^5 - 292*a^3*b*c^6)*d^4*e + 10*(9*b^8*c^2 + 273*a*b^6*c^3 - 815*a^2*b^4*c^
4 - 1538*a^3*b^2*c^5 - 584*a^4*c^6)*d^3*e^2 - 10*(3*b^9*c + 98*a*b^7*c^2 - 64*a^2*b^5*c^3 - 1285*a^3*b^3*c^4 -
876*a^4*b*c^5)*d^2*e^3 + 5*(37*a*b^8*c + 55*a^2*b^6*c^2 - 362*a^3*b^4*c^3 - 1866*a^4*b^2*c^4 + 264*a^5*c^5)*d
*e^4 - (3*a*b^9 + 98*a^2*b^7*c - 64*a^3*b^5*c^2 - 1285*a^4*b^3*c^3 - 876*a^5*b*c^4)*e^5)*x^3 - 2*(14*(b^7*c^3
- 32*a*b^5*c^4 - 107*a^2*b^3*c^5 + 876*a^3*b*c^6)*d^5 - 35*(b^8*c^2 - 32*a*b^6*c^3 - 107*a^2*b^4*c^4 + 876*a^3
*b^2*c^5)*d^4*e + 10*(3*b^9*c - 94*a*b^7*c^2 - 385*a^2*b^5*c^3 + 2414*a^3*b^3*c^4 + 1752*a^4*b*c^5)*d^3*e^2 +
10*(9*a*b^8*c + 363*a^2*b^6*c^2 - 1195*a^3*b^4*c^3 - 1348*a^4*b^2*c^4 - 1024*a^5*c^5)*d^2*e^3 - 5*(129*a^2*b^7
*c - 8*a^3*b^5*c^2 - 1718*a^4*b^3*c^3 - 1256*a^5*b*c^4)*d*e^4 + (9*a^2*b^8 + 363*a^3*b^6*c - 1195*a^4*b^4*c^2
- 1348*a^5*b^2*c^3 - 1024*a^6*c^4)*e^5)*x^2 - 60*(14*a^4*c^5*d^5 - 35*a^4*b*c^4*d^4*e + (14*c^9*d^5 - 35*b*c^8
*d^4*e + 10*(3*b^2*c^7 + 2*a*c^8)*d^3*e^2 - 10*(b^3*c^6 + 3*a*b*c^7)*d^2*e^3 + (b^4*c^5 + 12*a*b^2*c^6 + 6*a^2
*c^7)*d*e^4 - (a*b^3*c^5 + 3*a^2*b*c^6)*e^5)*x^8 + 4*(14*b*c^8*d^5 - 35*b^2*c^7*d^4*e + 10*(3*b^3*c^6 + 2*a*b*
c^7)*d^3*e^2 - 10*(b^4*c^5 + 3*a*b^2*c^6)*d^2*e^3 + (b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d*e^4 - (a*b^4*c^4
+ 3*a^2*b^2*c^5)*e^5)*x^7 + 2*(14*(3*b^2*c^7 + 2*a*c^8)*d^5 - 35*(3*b^3*c^6 + 2*a*b*c^7)*d^4*e + 10*(9*b^4*c^5
+ 12*a*b^2*c^6 + 4*a^2*c^7)*d^3*e^2 - 10*(3*b^5*c^4 + 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^2*e^3 + (3*b^6*c^3 + 38*a
*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^3*c^6)*d*e^4 - (3*a*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*e^5)*x^6 + 10*(3*
a^4*b^2*c^3 + 2*a^5*c^4)*d^3*e^2 - 10*(a^4*b^3*c^2 + 3*a^5*b*c^3)*d^2*e^3 + (a^4*b^4*c + 12*a^5*b^2*c^2 + 6*a^
6*c^3)*d*e^4 - (a^5*b^3*c + 3*a^6*b*c^2)*e^5 + 4*(14*(b^3*c^6 + 3*a*b*c^7)*d^5 - 35*(b^4*c^5 + 3*a*b^2*c^6)*d^
4*e + 10*(3*b^5*c^4 + 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^3*e^2 - 10*(b^6*c^3 + 6*a*b^4*c^4 + 9*a^2*b^2*c^5)*d^2*e^3
+ (b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5)*d*e^4 - (a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4
)*e^5)*x^5 + (14*(b^4*c^5 + 12*a*b^2*c^6 + 6*a^2*c^7)*d^5 - 35*(b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d^4*e +
10*(3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^3*c^6)*d^3*e^2 - 10*(b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*
c^4 + 18*a^3*b*c^5)*d^2*e^3 + (b^8*c + 24*a*b^6*c^2 + 156*a^2*b^4*c^3 + 144*a^3*b^2*c^4 + 36*a^4*c^5)*d*e^4 -
(a*b^7*c + 15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*e^5)*x^4 + 4*(14*(a*b^3*c^5 + 3*a^2*b*c^6)*d^5 - 35
*(a*b^4*c^4 + 3*a^2*b^2*c^5)*d^4*e + 10*(3*a*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*d^3*e^2 - 10*(a*b^6*c^2 +
6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*d^2*e^3 + (a*b^7*c + 15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*d*e^4 - (
a^2*b^6*c + 6*a^3*b^4*c^2 + 9*a^4*b^2*c^3)*e^5)*x^3 + 2*(14*(3*a^2*b^2*c^5 + 2*a^3*c^6)*d^5 - 35*(3*a^2*b^3*c^
4 + 2*a^3*b*c^5)*d^4*e + 10*(9*a^2*b^4*c^3 + 12*a^3*b^2*c^4 + 4*a^4*c^5)*d^3*e^2 - 10*(3*a^2*b^5*c^2 + 11*a^3*
b^3*c^3 + 6*a^4*b*c^4)*d^2*e^3 + (3*a^2*b^6*c + 38*a^3*b^4*c^2 + 42*a^4*b^2*c^3 + 12*a^5*c^4)*d*e^4 - (3*a^3*b
^5*c + 11*a^4*b^3*c^2 + 6*a^5*b*c^3)*e^5)*x^2 + 4*(14*a^3*b*c^5*d^5 - 35*a^3*b^2*c^4*d^4*e + 10*(3*a^3*b^3*c^3
+ 2*a^4*b*c^4)*d^3*e^2 - 10*(a^3*b^4*c^2 + 3*a^4*b^2*c^3)*d^2*e^3 + (a^3*b^5*c + 12*a^4*b^3*c^2 + 6*a^5*b*c^3
)*d*e^4 - (a^4*b^4*c + 3*a^5*b^2*c^2)*e^5)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(
b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 4*(2*(b^8*c^2 - 23*a*b^6*c^3 + 250*a^2*b^4*c^4 - 417*a^3*b^2*c^
5 - 1116*a^4*c^6)*d^5 - 5*(b^9*c - 23*a*b^7*c^2 + 250*a^2*b^5*c^3 - 417*a^3*b^3*c^4 - 1116*a^4*b*c^5)*d^4*e -
10*(a*b^8*c - 32*a^2*b^6*c^2 - 167*a^3*b^4*c^3 + 1146*a^4*b^2*c^4 - 120*a^5*c^5)*d^3*e^2 - 10*(3*a^2*b^7*c + 1
39*a^3*b^5*c^2 - 521*a^4*b^3*c^3 - 332*a^5*b*c^4)*d^2*e^3 + 5*(47*a^3*b^6*c - 4*a^4*b^4*c^2 - 754*a^5*b^2*c^3
+ 72*a^6*c^4)*d*e^4 - (3*a^3*b^7 + 139*a^4*b^5*c - 521*a^5*b^3*c^2 - 332*a^6*b*c^3)*e^5)*x)/(a^4*b^10*c - 20*a
^5*b^8*c^2 + 160*a^6*b^6*c^3 - 640*a^7*b^4*c^4 + 1280*a^8*b^2*c^5 - 1024*a^9*c^6 + (b^10*c^5 - 20*a*b^8*c^6 +
160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*x^8 + 4*(b^11*c^4 - 20*a*b^9*c^5 + 160*a
^2*b^7*c^6 - 640*a^3*b^5*c^7 + 1280*a^4*b^3*c^8 - 1024*a^5*b*c^9)*x^7 + 2*(3*b^12*c^3 - 58*a*b^10*c^4 + 440*a^
2*b^8*c^5 - 1600*a^3*b^6*c^6 + 2560*a^4*b^4*c^7 - 512*a^5*b^2*c^8 - 2048*a^6*c^9)*x^6 + 4*(b^13*c^2 - 17*a*b^1
1*c^3 + 100*a^2*b^9*c^4 - 160*a^3*b^7*c^5 - 640*a^4*b^5*c^6 + 2816*a^5*b^3*c^7 - 3072*a^6*b*c^8)*x^5 + (b^14*c
- 8*a*b^12*c^2 - 74*a^2*b^10*c^3 + 1160*a^3*b^8*c^4 - 5440*a^4*b^6*c^5 + 10496*a^5*b^4*c^6 - 4608*a^6*b^2*c^7
- 6144*a^7*c^8)*x^4 + 4*(a*b^13*c - 17*a^2*b^11*c^2 + 100*a^3*b^9*c^3 - 160*a^4*b^7*c^4 - 640*a^5*b^5*c^5 + 2
816*a^6*b^3*c^6 - 3072*a^7*b*c^7)*x^3 + 2*(3*a^2*b^12*c - 58*a^3*b^10*c^2 + 440*a^4*b^8*c^3 - 1600*a^5*b^6*c^4
+ 2560*a^6*b^4*c^5 - 512*a^7*b^2*c^6 - 2048*a^8*c^7)*x^2 + 4*(a^3*b^11*c - 20*a^4*b^9*c^2 + 160*a^5*b^7*c^3 -
640*a^6*b^5*c^4 + 1280*a^7*b^3*c^5 - 1024*a^8*b*c^6)*x), 1/12*(60*(14*(b^2*c^8 - 4*a*c^9)*d^5 - 35*(b^3*c^7 -
4*a*b*c^8)*d^4*e + 10*(3*b^4*c^6 - 10*a*b^2*c^7 - 8*a^2*c^8)*d^3*e^2 - 10*(b^5*c^5 - a*b^3*c^6 - 12*a^2*b*c^7
)*d^2*e^3 + (b^6*c^4 + 8*a*b^4*c^5 - 42*a^2*b^2*c^6 - 24*a^3*c^7)*d*e^4 - (a*b^5*c^4 - a^2*b^3*c^5 - 12*a^3*b*
c^6)*e^5)*x^7 + 210*(14*(b^3*c^7 - 4*a*b*c^8)*d^5 - 35*(b^4*c^6 - 4*a*b^2*c^7)*d^4*e + 10*(3*b^5*c^5 - 10*a*b^
3*c^6 - 8*a^2*b*c^7)*d^3*e^2 - 10*(b^6*c^4 - a*b^4*c^5 - 12*a^2*b^2*c^6)*d^2*e^3 + (b^7*c^3 + 8*a*b^5*c^4 - 42
*a^2*b^3*c^5 - 24*a^3*b*c^6)*d*e^4 - (a*b^6*c^3 - a^2*b^4*c^4 - 12*a^3*b^2*c^5)*e^5)*x^6 - (3*b^9*c - 62*a*b^7
*c^2 + 526*a^2*b^5*c^3 - 2420*a^3*b^3*c^4 + 4464*a^4*b*c^5)*d^5 - 5*(a*b^8*c - 23*a^2*b^6*c^2 + 250*a^3*b^4*c^
3 - 312*a^4*b^2*c^4 - 1536*a^5*c^5)*d^4*e - 10*(a^2*b^7*c - 32*a^3*b^5*c^2 - 212*a^4*b^3*c^3 + 1296*a^5*b*c^4)
*d^3*e^2 - 10*(3*a^3*b^6*c + 154*a^4*b^4*c^2 - 536*a^5*b^2*c^3 - 512*a^6*c^4)*d^2*e^3 + 50*(5*a^4*b^5*c + 2*a^
5*b^3*c^2 - 88*a^6*b*c^3)*d*e^4 - (3*a^4*b^6 + 154*a^5*b^4*c - 536*a^6*b^2*c^2 - 512*a^7*c^3)*e^5 + 20*(14*(13
*b^4*c^6 - 41*a*b^2*c^7 - 44*a^2*c^8)*d^5 - 35*(13*b^5*c^5 - 41*a*b^3*c^6 - 44*a^2*b*c^7)*d^4*e + 10*(39*b^6*c
^4 - 97*a*b^4*c^5 - 214*a^2*b^2*c^6 - 88*a^3*c^7)*d^3*e^2 - 10*(13*b^7*c^3 - 2*a*b^5*c^4 - 167*a^2*b^3*c^5 - 1
32*a^3*b*c^6)*d^2*e^3 + (13*b^8*c^2 + 115*a*b^6*c^3 - 458*a^2*b^4*c^4 - 774*a^3*b^2*c^5 - 264*a^4*c^6)*d*e^4 -
(13*a*b^7*c^2 - 2*a^2*b^5*c^3 - 167*a^3*b^3*c^4 - 132*a^4*b*c^5)*e^5)*x^5 + (350*(5*b^5*c^5 + 2*a*b^3*c^6 - 8
8*a^2*b*c^7)*d^5 - 875*(5*b^6*c^4 + 2*a*b^4*c^5 - 88*a^2*b^2*c^6)*d^4*e + 250*(15*b^7*c^3 + 16*a*b^5*c^4 - 260
*a^2*b^3*c^5 - 176*a^3*b*c^6)*d^3*e^2 - 250*(5*b^8*c^2 + 17*a*b^6*c^3 - 82*a^2*b^4*c^4 - 264*a^3*b^2*c^5)*d^2*
e^3 + 25*(5*b^9*c + 62*a*b^7*c^2 - 34*a^2*b^5*c^3 - 1044*a^3*b^3*c^4 - 528*a^4*b*c^5)*d*e^4 - (3*b^10 + 65*a*b
^8*c + 905*a^2*b^6*c^2 - 3970*a^3*b^4*c^3 - 2760*a^4*b^2*c^4 - 3072*a^5*c^5)*e^5)*x^4 + 4*(14*(3*b^6*c^4 + 89*
a*b^4*c^5 - 331*a^2*b^2*c^6 - 292*a^3*c^7)*d^5 - 35*(3*b^7*c^3 + 89*a*b^5*c^4 - 331*a^2*b^3*c^5 - 292*a^3*b*c^
6)*d^4*e + 10*(9*b^8*c^2 + 273*a*b^6*c^3 - 815*a^2*b^4*c^4 - 1538*a^3*b^2*c^5 - 584*a^4*c^6)*d^3*e^2 - 10*(3*b
^9*c + 98*a*b^7*c^2 - 64*a^2*b^5*c^3 - 1285*a^3*b^3*c^4 - 876*a^4*b*c^5)*d^2*e^3 + 5*(37*a*b^8*c + 55*a^2*b^6*
c^2 - 362*a^3*b^4*c^3 - 1866*a^4*b^2*c^4 + 264*a^5*c^5)*d*e^4 - (3*a*b^9 + 98*a^2*b^7*c - 64*a^3*b^5*c^2 - 128
5*a^4*b^3*c^3 - 876*a^5*b*c^4)*e^5)*x^3 - 2*(14*(b^7*c^3 - 32*a*b^5*c^4 - 107*a^2*b^3*c^5 + 876*a^3*b*c^6)*d^5
- 35*(b^8*c^2 - 32*a*b^6*c^3 - 107*a^2*b^4*c^4 + 876*a^3*b^2*c^5)*d^4*e + 10*(3*b^9*c - 94*a*b^7*c^2 - 385*a^
2*b^5*c^3 + 2414*a^3*b^3*c^4 + 1752*a^4*b*c^5)*d^3*e^2 + 10*(9*a*b^8*c + 363*a^2*b^6*c^2 - 1195*a^3*b^4*c^3 -
1348*a^4*b^2*c^4 - 1024*a^5*c^5)*d^2*e^3 - 5*(129*a^2*b^7*c - 8*a^3*b^5*c^2 - 1718*a^4*b^3*c^3 - 1256*a^5*b*c^
4)*d*e^4 + (9*a^2*b^8 + 363*a^3*b^6*c - 1195*a^4*b^4*c^2 - 1348*a^5*b^2*c^3 - 1024*a^6*c^4)*e^5)*x^2 - 120*(14
*a^4*c^5*d^5 - 35*a^4*b*c^4*d^4*e + (14*c^9*d^5 - 35*b*c^8*d^4*e + 10*(3*b^2*c^7 + 2*a*c^8)*d^3*e^2 - 10*(b^3*
c^6 + 3*a*b*c^7)*d^2*e^3 + (b^4*c^5 + 12*a*b^2*c^6 + 6*a^2*c^7)*d*e^4 - (a*b^3*c^5 + 3*a^2*b*c^6)*e^5)*x^8 + 4
*(14*b*c^8*d^5 - 35*b^2*c^7*d^4*e + 10*(3*b^3*c^6 + 2*a*b*c^7)*d^3*e^2 - 10*(b^4*c^5 + 3*a*b^2*c^6)*d^2*e^3 +
(b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d*e^4 - (a*b^4*c^4 + 3*a^2*b^2*c^5)*e^5)*x^7 + 2*(14*(3*b^2*c^7 + 2*a*c
^8)*d^5 - 35*(3*b^3*c^6 + 2*a*b*c^7)*d^4*e + 10*(9*b^4*c^5 + 12*a*b^2*c^6 + 4*a^2*c^7)*d^3*e^2 - 10*(3*b^5*c^4
+ 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^2*e^3 + (3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^3*c^6)*d*e^4 - (3*a
*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*e^5)*x^6 + 10*(3*a^4*b^2*c^3 + 2*a^5*c^4)*d^3*e^2 - 10*(a^4*b^3*c^2 +
3*a^5*b*c^3)*d^2*e^3 + (a^4*b^4*c + 12*a^5*b^2*c^2 + 6*a^6*c^3)*d*e^4 - (a^5*b^3*c + 3*a^6*b*c^2)*e^5 + 4*(14
*(b^3*c^6 + 3*a*b*c^7)*d^5 - 35*(b^4*c^5 + 3*a*b^2*c^6)*d^4*e + 10*(3*b^5*c^4 + 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^
3*e^2 - 10*(b^6*c^3 + 6*a*b^4*c^4 + 9*a^2*b^2*c^5)*d^2*e^3 + (b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3
*b*c^5)*d*e^4 - (a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*e^5)*x^5 + (14*(b^4*c^5 + 12*a*b^2*c^6 + 6*a^2*c^7
)*d^5 - 35*(b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d^4*e + 10*(3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a
^3*c^6)*d^3*e^2 - 10*(b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5)*d^2*e^3 + (b^8*c + 24*a*b^6*c^2
+ 156*a^2*b^4*c^3 + 144*a^3*b^2*c^4 + 36*a^4*c^5)*d*e^4 - (a*b^7*c + 15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*
b*c^4)*e^5)*x^4 + 4*(14*(a*b^3*c^5 + 3*a^2*b*c^6)*d^5 - 35*(a*b^4*c^4 + 3*a^2*b^2*c^5)*d^4*e + 10*(3*a*b^5*c^3
+ 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*d^3*e^2 - 10*(a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*d^2*e^3 + (a*b^7*c +
15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*d*e^4 - (a^2*b^6*c + 6*a^3*b^4*c^2 + 9*a^4*b^2*c^3)*e^5)*x^3
+ 2*(14*(3*a^2*b^2*c^5 + 2*a^3*c^6)*d^5 - 35*(3*a^2*b^3*c^4 + 2*a^3*b*c^5)*d^4*e + 10*(9*a^2*b^4*c^3 + 12*a^3*
b^2*c^4 + 4*a^4*c^5)*d^3*e^2 - 10*(3*a^2*b^5*c^2 + 11*a^3*b^3*c^3 + 6*a^4*b*c^4)*d^2*e^3 + (3*a^2*b^6*c + 38*a
^3*b^4*c^2 + 42*a^4*b^2*c^3 + 12*a^5*c^4)*d*e^4 - (3*a^3*b^5*c + 11*a^4*b^3*c^2 + 6*a^5*b*c^3)*e^5)*x^2 + 4*(1
4*a^3*b*c^5*d^5 - 35*a^3*b^2*c^4*d^4*e + 10*(3*a^3*b^3*c^3 + 2*a^4*b*c^4)*d^3*e^2 - 10*(a^3*b^4*c^2 + 3*a^4*b^
2*c^3)*d^2*e^3 + (a^3*b^5*c + 12*a^4*b^3*c^2 + 6*a^5*b*c^3)*d*e^4 - (a^4*b^4*c + 3*a^5*b^2*c^2)*e^5)*x)*sqrt(-
b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 4*(2*(b^8*c^2 - 23*a*b^6*c^3 + 250*a^2*b^
4*c^4 - 417*a^3*b^2*c^5 - 1116*a^4*c^6)*d^5 - 5*(b^9*c - 23*a*b^7*c^2 + 250*a^2*b^5*c^3 - 417*a^3*b^3*c^4 - 11
16*a^4*b*c^5)*d^4*e - 10*(a*b^8*c - 32*a^2*b^6*c^2 - 167*a^3*b^4*c^3 + 1146*a^4*b^2*c^4 - 120*a^5*c^5)*d^3*e^2
- 10*(3*a^2*b^7*c + 139*a^3*b^5*c^2 - 521*a^4*b^3*c^3 - 332*a^5*b*c^4)*d^2*e^3 + 5*(47*a^3*b^6*c - 4*a^4*b^4*
c^2 - 754*a^5*b^2*c^3 + 72*a^6*c^4)*d*e^4 - (3*a^3*b^7 + 139*a^4*b^5*c - 521*a^5*b^3*c^2 - 332*a^6*b*c^3)*e^5)
*x)/(a^4*b^10*c - 20*a^5*b^8*c^2 + 160*a^6*b^6*c^3 - 640*a^7*b^4*c^4 + 1280*a^8*b^2*c^5 - 1024*a^9*c^6 + (b^10
*c^5 - 20*a*b^8*c^6 + 160*a^2*b^6*c^7 - 640*a^3*b^4*c^8 + 1280*a^4*b^2*c^9 - 1024*a^5*c^10)*x^8 + 4*(b^11*c^4
- 20*a*b^9*c^5 + 160*a^2*b^7*c^6 - 640*a^3*b^5*c^7 + 1280*a^4*b^3*c^8 - 1024*a^5*b*c^9)*x^7 + 2*(3*b^12*c^3 -
58*a*b^10*c^4 + 440*a^2*b^8*c^5 - 1600*a^3*b^6*c^6 + 2560*a^4*b^4*c^7 - 512*a^5*b^2*c^8 - 2048*a^6*c^9)*x^6 +
4*(b^13*c^2 - 17*a*b^11*c^3 + 100*a^2*b^9*c^4 - 160*a^3*b^7*c^5 - 640*a^4*b^5*c^6 + 2816*a^5*b^3*c^7 - 3072*a^
6*b*c^8)*x^5 + (b^14*c - 8*a*b^12*c^2 - 74*a^2*b^10*c^3 + 1160*a^3*b^8*c^4 - 5440*a^4*b^6*c^5 + 10496*a^5*b^4*
c^6 - 4608*a^6*b^2*c^7 - 6144*a^7*c^8)*x^4 + 4*(a*b^13*c - 17*a^2*b^11*c^2 + 100*a^3*b^9*c^3 - 160*a^4*b^7*c^4
- 640*a^5*b^5*c^5 + 2816*a^6*b^3*c^6 - 3072*a^7*b*c^7)*x^3 + 2*(3*a^2*b^12*c - 58*a^3*b^10*c^2 + 440*a^4*b^8*
c^3 - 1600*a^5*b^6*c^4 + 2560*a^6*b^4*c^5 - 512*a^7*b^2*c^6 - 2048*a^8*c^7)*x^2 + 4*(a^3*b^11*c - 20*a^4*b^9*c
^2 + 160*a^5*b^7*c^3 - 640*a^6*b^5*c^4 + 1280*a^7*b^3*c^5 - 1024*a^8*b*c^6)*x)]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.15848, size = 3162, normalized size = 8.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*(14*c^4*d^5 - 35*b*c^3*d^4*e + 30*b^2*c^2*d^3*e^2 + 20*a*c^3*d^3*e^2 - 10*b^3*c*d^2*e^3 - 30*a*b*c^2*d^2*e^
3 + b^4*d*e^4 + 12*a*b^2*c*d*e^4 + 6*a^2*c^2*d*e^4 - a*b^3*e^5 - 3*a^2*b*c*e^5)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*
c^8*d^5*x^7 - 2100*b*c^7*d^4*x^7*e + 2940*b*c^7*d^5*x^6 + 1800*b^2*c^6*d^3*x^7*e^2 + 1200*a*c^7*d^3*x^7*e^2 -
7350*b^2*c^6*d^4*x^6*e + 3640*b^2*c^6*d^5*x^5 + 3080*a*c^7*d^5*x^5 - 600*b^3*c^5*d^2*x^7*e^3 - 1800*a*b*c^6*d^
2*x^7*e^3 + 6300*b^3*c^5*d^3*x^6*e^2 + 4200*a*b*c^6*d^3*x^6*e^2 - 9100*b^3*c^5*d^4*x^5*e - 7700*a*b*c^6*d^4*x^
5*e + 1750*b^3*c^5*d^5*x^4 + 7700*a*b*c^6*d^5*x^4 + 60*b^4*c^4*d*x^7*e^4 + 720*a*b^2*c^5*d*x^7*e^4 + 360*a^2*c
^6*d*x^7*e^4 - 2100*b^4*c^4*d^2*x^6*e^3 - 6300*a*b^2*c^5*d^2*x^6*e^3 + 7800*b^4*c^4*d^3*x^5*e^2 + 11800*a*b^2*
c^5*d^3*x^5*e^2 + 4400*a^2*c^6*d^3*x^5*e^2 - 4375*b^4*c^4*d^4*x^4*e - 19250*a*b^2*c^5*d^4*x^4*e + 168*b^4*c^4*
d^5*x^3 + 5656*a*b^2*c^5*d^5*x^3 + 4088*a^2*c^6*d^5*x^3 - 60*a*b^3*c^4*x^7*e^5 - 180*a^2*b*c^5*x^7*e^5 + 210*b
^5*c^3*d*x^6*e^4 + 2520*a*b^3*c^4*d*x^6*e^4 + 1260*a^2*b*c^5*d*x^6*e^4 - 2600*b^5*c^3*d^2*x^5*e^3 - 10000*a*b^
3*c^4*d^2*x^5*e^3 - 6600*a^2*b*c^5*d^2*x^5*e^3 + 3750*b^5*c^3*d^3*x^4*e^2 + 19000*a*b^3*c^4*d^3*x^4*e^2 + 1100
0*a^2*b*c^5*d^3*x^4*e^2 - 420*b^5*c^3*d^4*x^3*e - 14140*a*b^3*c^4*d^4*x^3*e - 10220*a^2*b*c^5*d^4*x^3*e - 28*b
^5*c^3*d^5*x^2 + 784*a*b^3*c^4*d^5*x^2 + 6132*a^2*b*c^5*d^5*x^2 - 210*a*b^4*c^3*x^6*e^5 - 630*a^2*b^2*c^4*x^6*
e^5 + 260*b^6*c^2*d*x^5*e^4 + 3340*a*b^4*c^3*d*x^5*e^4 + 4200*a^2*b^2*c^4*d*x^5*e^4 + 1320*a^3*c^5*d*x^5*e^4 -
1250*b^6*c^2*d^2*x^4*e^3 - 9250*a*b^4*c^3*d^2*x^4*e^3 - 16500*a^2*b^2*c^4*d^2*x^4*e^3 + 360*b^6*c^2*d^3*x^3*e
^2 + 12360*a*b^4*c^3*d^3*x^3*e^2 + 16840*a^2*b^2*c^4*d^3*x^3*e^2 + 5840*a^3*c^5*d^3*x^3*e^2 + 70*b^6*c^2*d^4*x
^2*e - 1960*a*b^4*c^3*d^4*x^2*e - 15330*a^2*b^2*c^4*d^4*x^2*e + 8*b^6*c^2*d^5*x - 152*a*b^4*c^3*d^5*x + 1392*a
^2*b^2*c^4*d^5*x + 2232*a^3*c^5*d^5*x - 260*a*b^5*c^2*x^5*e^5 - 1000*a^2*b^3*c^3*x^5*e^5 - 660*a^3*b*c^4*x^5*e
^5 + 125*b^7*c*d*x^4*e^4 + 2050*a*b^5*c^2*d*x^4*e^4 + 7350*a^2*b^3*c^3*d*x^4*e^4 + 3300*a^3*b*c^4*d*x^4*e^4 -
120*b^7*c*d^2*x^3*e^3 - 4400*a*b^5*c^2*d^2*x^3*e^3 - 15040*a^2*b^3*c^3*d^2*x^3*e^3 - 8760*a^3*b*c^4*d^2*x^3*e^
3 - 60*b^7*c*d^3*x^2*e^2 + 1640*a*b^5*c^2*d^3*x^2*e^2 + 14260*a^2*b^3*c^3*d^3*x^2*e^2 + 8760*a^3*b*c^4*d^3*x^2
*e^2 - 20*b^7*c*d^4*x*e + 380*a*b^5*c^2*d^4*x*e - 3480*a^2*b^3*c^3*d^4*x*e - 5580*a^3*b*c^4*d^4*x*e - 3*b^7*c*
d^5 + 50*a*b^5*c^2*d^5 - 326*a^2*b^3*c^3*d^5 + 1116*a^3*b*c^4*d^5 - 3*b^8*x^4*e^5 - 77*a*b^6*c*x^4*e^5 - 1213*
a^2*b^4*c^2*x^4*e^5 - 882*a^3*b^2*c^3*x^4*e^5 - 768*a^4*c^4*x^4*e^5 + 740*a*b^6*c*d*x^3*e^4 + 4060*a^2*b^4*c^2
*d*x^3*e^4 + 9000*a^3*b^2*c^3*d*x^3*e^4 - 1320*a^4*c^4*d*x^3*e^4 - 180*a*b^6*c*d^2*x^2*e^3 - 7980*a^2*b^4*c^2*
d^2*x^2*e^3 - 8020*a^3*b^2*c^3*d^2*x^2*e^3 - 5120*a^4*c^4*d^2*x^2*e^3 - 40*a*b^6*c*d^3*x*e^2 + 1120*a^2*b^4*c^
2*d^3*x*e^2 + 11160*a^3*b^2*c^3*d^3*x*e^2 - 1200*a^4*c^4*d^3*x*e^2 - 5*a*b^6*c*d^4*e + 95*a^2*b^4*c^2*d^4*e -
870*a^3*b^2*c^3*d^4*e - 1920*a^4*c^4*d^4*e - 12*a*b^7*x^3*e^5 - 440*a^2*b^5*c*x^3*e^5 - 1504*a^3*b^3*c^2*x^3*e
^5 - 876*a^4*b*c^3*x^3*e^5 + 1290*a^2*b^5*c*d*x^2*e^4 + 5080*a^3*b^3*c^2*d*x^2*e^4 + 3140*a^4*b*c^3*d*x^2*e^4
- 120*a^2*b^5*c*d^2*x*e^3 - 6040*a^3*b^3*c^2*d^2*x*e^3 - 3320*a^4*b*c^3*d^2*x*e^3 - 10*a^2*b^5*c*d^3*e^2 + 280
*a^3*b^3*c^2*d^3*e^2 + 3240*a^4*b*c^3*d^3*e^2 - 18*a^2*b^6*x^2*e^5 - 798*a^3*b^4*c*x^2*e^5 - 802*a^4*b^2*c^2*x
^2*e^5 - 512*a^5*c^3*x^2*e^5 + 940*a^3*b^4*c*d*x*e^4 + 3680*a^4*b^2*c^2*d*x*e^4 - 360*a^5*c^3*d*x*e^4 - 30*a^3
*b^4*c*d^2*e^3 - 1660*a^4*b^2*c^2*d^2*e^3 - 1280*a^5*c^3*d^2*e^3 - 12*a^3*b^5*x*e^5 - 604*a^4*b^3*c*x*e^5 - 33
2*a^5*b*c^2*x*e^5 + 250*a^4*b^3*c*d*e^4 + 1100*a^5*b*c^2*d*e^4 - 3*a^4*b^4*e^5 - 166*a^5*b^2*c*e^5 - 128*a^6*c
^2*e^5)/((b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*(c*x^2 + b*x + a)^4)