3.63 \(\int \frac{A+B x+C x^2}{(d+e x)^3 (a+c x^2)^3} \, dx\)

Optimal. Leaf size=753 \[ \frac{c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )+4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}-\frac{e^3 \log \left (a+c x^2\right ) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^5}+\frac{e^3 \log (d+e x) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A c d \left (35 a^2 c d^2 e^4-35 a^3 e^6+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-5 a^2 c d^2 e^4 (25 C d-27 B e)+15 a^3 e^6 (3 C d-B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^5}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}+\frac{e^3 \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right )}{(d+e x) \left (a e^2+c d^2\right )^4}-\frac{e^3 \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^3} \]

[Out]

-(e^3*(C*d^2 - B*d*e + A*e^2))/(2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (e^3*(a*e^2*(2*C*d - B*e) - c*d*(4*C*d^2 -
e*(5*B*d - 6*A*e))))/((c*d^2 + a*e^2)^4*(d + e*x)) - (a*(B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*
e^2)) - c*(A*c*d*(c*d^2 - 3*a*e^2) - a*(c*d^2*(C*d - 3*B*e) - a*e^2*(3*C*d - B*e)))*x)/(4*a*(c*d^2 + a*e^2)^3*
(a + c*x^2)^2) + (4*a^2*e*(a^2*C*e^4 + c^2*d^2*(3*C*d^2 - 2*e*(3*B*d - 5*A*e)) - 2*a*c*e^2*(4*C*d^2 - e*(3*B*d
- A*e))) + c*(3*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 - 11*a^2*e^4) - a*(2*a*c*d^2*e^2*(13*C*d - 19*B*e) - c^2*d^4*(
C*d - 3*B*e) - 7*a^2*e^4*(3*C*d - B*e)))*x)/(8*a^2*(c*d^2 + a*e^2)^4*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6
+ 7*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25
*C*d - 27*B*e) + c^3*d^6*(C*d - 3*B*e) + 15*a^3*e^6*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c
*d^2 + a*e^2)^5) + (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*d^2*(10*C*d^2 - 3*e*(5*B*d -
7*A*e)))*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*d^2
*(10*C*d^2 - 3*e*(5*B*d - 7*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^5)

________________________________________________________________________________________

Rubi [A]  time = 3.14256, antiderivative size = 753, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1647, 1629, 635, 205, 260} \[ \frac{c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )+4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}-\frac{e^3 \log \left (a+c x^2\right ) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^5}+\frac{e^3 \log (d+e x) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A c d \left (35 a^2 c d^2 e^4-35 a^3 e^6+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-5 a^2 c d^2 e^4 (25 C d-27 B e)+15 a^3 e^6 (3 C d-B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^5}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}-\frac{e^3 \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{(d+e x) \left (a e^2+c d^2\right )^4}-\frac{e^3 \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/((d + e*x)^3*(a + c*x^2)^3),x]

[Out]

-(e^3*(C*d^2 - B*d*e + A*e^2))/(2*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (e^3*(4*c*C*d^3 - c*d*e*(5*B*d - 6*A*e) - a
*e^2*(2*C*d - B*e)))/((c*d^2 + a*e^2)^4*(d + e*x)) - (a*(B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*
e^2)) - c*(A*c*d*(c*d^2 - 3*a*e^2) - a*(c*d^2*(C*d - 3*B*e) - a*e^2*(3*C*d - B*e)))*x)/(4*a*(c*d^2 + a*e^2)^3*
(a + c*x^2)^2) + (4*a^2*e*(a^2*C*e^4 + c^2*(3*C*d^4 - 2*d^2*e*(3*B*d - 5*A*e)) - 2*a*c*e^2*(4*C*d^2 - e*(3*B*d
- A*e))) + c*(3*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 - 11*a^2*e^4) - a*(2*a*c*d^2*e^2*(13*C*d - 19*B*e) - c^2*d^4*(
C*d - 3*B*e) - 7*a^2*e^4*(3*C*d - B*e)))*x)/(8*a^2*(c*d^2 + a*e^2)^4*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6
+ 7*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25
*C*d - 27*B*e) + c^3*d^6*(C*d - 3*B*e) + 15*a^3*e^6*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c
*d^2 + a*e^2)^5) + (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*(10*C*d^4 - 3*d^2*e*(5*B*d -
7*A*e)))*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*(10
*C*d^4 - 3*d^2*e*(5*B*d - 7*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^5)

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx &=-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}-\frac{\int \frac{-\frac{c \left (A \left (3 c^3 d^6+15 a c^2 d^4 e^2+12 a^2 c d^2 e^4+4 a^3 e^6\right )+a c d^3 \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\left (c d^2+a e^2\right )^3}-\frac{c e \left (A c^2 d^3 \left (9 c d^2+5 a e^2\right )+a \left (4 a^2 B e^5-3 c^2 d^4 (3 C d-5 B e)-5 a c d^2 e^2 (C d-3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}-\frac{c e^2 \left (A c \left (9 c^2 d^4-15 a c d^2 e^2-4 a^2 e^4\right )+a \left (4 a^2 C e^4-c^2 d^3 (9 C d-23 B e)+3 a c d e^2 (5 C d+B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}-\frac{3 c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac{4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac{\int \frac{\frac{c^2 \left (A \left (3 c^4 d^8+18 a c^3 d^6 e^2+87 a^2 c^2 d^4 e^4+32 a^3 c d^2 e^6+8 a^4 e^8\right )+a c d^3 \left (2 a c d^2 e^2 (11 C d-21 B e)+c^2 d^4 (C d-3 B e)-9 a^2 e^4 (3 C d-B e)\right )\right )}{\left (c d^2+a e^2\right )^4}+\frac{c^2 e \left (3 A c^2 d^3 \left (3 c^2 d^4+18 a c d^2 e^2+31 a^2 e^4\right )+a \left (8 a^3 B e^7-a^2 c d^2 e^4 (65 C d-59 B e)+3 c^3 d^6 (C d-3 B e)-2 a c^2 d^4 e^2 (7 C d+3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^4}+\frac{c^2 e^2 \left (A c \left (9 c^3 d^6+54 a c^2 d^4 e^2-19 a^2 c d^2 e^4-16 a^3 e^6\right )+a \left (8 a^3 C e^6-a^2 c d e^4 (C d-27 B e)-6 a c^2 d^3 e^2 (9 C d-11 B e)+3 c^3 d^5 (C d-3 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^4}+\frac{c^3 e^3 \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^4}}{(d+e x)^3 \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac{4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac{\int \left (\frac{8 a^2 c^2 e^4 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^3}+\frac{8 a^2 c^2 e^4 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)^2}+\frac{8 a^2 c^2 e^4 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (c d^2+a e^2\right )^5 (d+e x)}+\frac{c^3 \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )-8 a^2 e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) x\right )}{\left (c d^2+a e^2\right )^5 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c^2}\\ &=-\frac{e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac{4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac{e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}+\frac{c \int \frac{3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )-8 a^2 e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^5}\\ &=-\frac{e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac{4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac{e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{\left (c e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^5}+\frac{\left (c \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )\right )\right ) \int \frac{1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^5}\\ &=-\frac{e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac{e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac{4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac{\sqrt{c} \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^5}+\frac{e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^5}\\ \end{align*}

Mathematica [A]  time = 1.22617, size = 672, normalized size = 0.89 \[ \frac{\frac{2 \left (a e^2+c d^2\right )^2 \left (-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))+a^3 C e^3-a c^2 d \left (3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right )+A c^3 d^3 x\right )}{a \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \left (a^2 c^2 d e \left (e \left (40 A d e-33 A e^2 x-24 B d^2+38 B d e x\right )+2 C d^2 (6 d-13 e x)\right )+a^3 c e^3 (e (-8 A e+24 B d-7 B e x)+C d (21 e x-32 d))+4 a^4 C e^5+a c^3 d^3 x \left (3 e (6 A e-B d)+C d^2\right )+3 A c^4 d^5 x\right )}{a^2 \left (a+c x^2\right )}-4 \log \left (a+c x^2\right ) \left (a^2 C e^7+a c e^5 \left (-3 A e^2+9 B d e-13 C d^2\right )+c^2 d^2 e^3 \left (3 e (7 A e-5 B d)+10 C d^2\right )\right )+8 \log (d+e x) \left (a^2 C e^7+a c e^5 \left (-3 A e^2+9 B d e-13 C d^2\right )+c^2 d^2 e^3 \left (3 e (7 A e-5 B d)+10 C d^2\right )\right )+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A c d \left (35 a^2 c d^2 e^4-35 a^3 e^6+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-5 a^2 c d^2 e^4 (25 C d-27 B e)-15 a^3 e^6 (B e-3 C d)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{a^{5/2}}-\frac{8 e^3 \left (a e^2+c d^2\right ) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )}{d+e x}-\frac{4 e^3 \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}}{8 \left (a e^2+c d^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/((d + e*x)^3*(a + c*x^2)^3),x]

[Out]

((-4*e^3*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 - (8*e^3*(c*d^2 + a*e^2)*(4*c*C*d^3 + c*d*e
*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^2*(a^3*C*e^3 + A*c^3*d^3*x - a*c^2*d
*(C*d^2*x + B*d*(d - 3*e*x) + 3*A*e*(-d + e*x)) - a^2*c*e*(3*C*d*(d - e*x) + e*(-3*B*d + A*e + B*e*x))))/(a*(a
+ c*x^2)^2) + ((c*d^2 + a*e^2)*(4*a^4*C*e^5 + 3*A*c^4*d^5*x + a*c^3*d^3*(C*d^2 + 3*e*(-(B*d) + 6*A*e))*x + a^
3*c*e^3*(C*d*(-32*d + 21*e*x) + e*(24*B*d - 8*A*e - 7*B*e*x)) + a^2*c^2*d*e*(2*C*d^2*(6*d - 13*e*x) + e*(-24*B
*d^2 + 40*A*d*e + 38*B*d*e*x - 33*A*e^2*x))))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6 + 7*a*c^2*d^4*e^2
+ 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25*C*d - 27*B*e) + c
^3*d^6*(C*d - 3*B*e) - 15*a^3*e^6*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*(a^2*C*e^7 + a*c*e
^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[d + e*x] - 4*(a^2*C*e^
7 + a*c*e^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[a + c*x^2])/(
8*(c*d^2 + a*e^2)^5)

________________________________________________________________________________________

Maple [B]  time = 0.081, size = 2737, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,x)

[Out]

-e^6/(a*e^2+c*d^2)^4/(e*x+d)*B*a+1/2*e^4/(a*e^2+c*d^2)^3/(e*x+d)^2*B*d-1/2*e^3/(a*e^2+c*d^2)^3/(e*x+d)^2*C*d^2
+e^7/(a*e^2+c*d^2)^5*ln(e*x+d)*a^2*C+3/4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*a^4*e^7+9*e^6/(a*e^2+c*d^2)^5*ln(e*x+d)
*B*a*c*d+1/2*c/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^2*a^3*e^7+15/4*c/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*a^3*d*e^6+3/8*c^
6/(a*e^2+c*d^2)^5/(c*x^2+a)^2/a^2*x^3*A*d^7+1/8*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^2/a*x^3*C*d^7+3/2*c^4/(a*e^2+c*d
^2)^5/(c*x^2+a)^2*C*x^2*d^6*e+19/8*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*d^5*e^2*x+3/8*c^4/(a*e^2+c*d^2)^5/(c*x^2+
a)^2*B*d^6*e*x+5*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*x^2*d^4*e^3-3*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*x^2*d^5*e^2
-25/8*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^3*d^5*e^2+5/8*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^2*x/a*A*d^7+17/4*c^2/(a*
e^2+c*d^2)^5/(c*x^2+a)^2*A*a^2*d^2*e^5+25/4*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*a*d^4*e^3+5/4*c^2/(a*e^2+c*d^2)^
5/(c*x^2+a)^2*B*a^2*d^3*e^4-11/4*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*a*d^5*e^2-15/4*c^2/(a*e^2+c*d^2)^5/(c*x^2+a
)^2*C*a^2*d^4*e^3+3/4*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*a*d^6*e-9/2*c/(a*e^2+c*d^2)^5*a*ln(c*x^2+a)*B*d*e^6+13
/2*c/(a*e^2+c*d^2)^5*a*ln(c*x^2+a)*C*d^2*e^5-15/4*c/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*a^3*d^2*e^5-15/8*c/(a*e^2+c*
d^2)^5*a^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*e^7+105/8*c^3/(a*e^2+c*d^2)^5/(a*c)^(1/2)*arctan(x*c/(a*c)^(1
/2))*A*d^3*e^4+23/8*c^3/(a*e^2+c*d^2)^5/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d^5*e^2-45/8*c^3/(a*e^2+c*d^2)^5
/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d^4*e^3+3/8*c^5/(a*e^2+c*d^2)^5/a^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))
*A*d^7+1/8*c^4/(a*e^2+c*d^2)^5/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d^7-13*e^5/(a*e^2+c*d^2)^5*ln(e*x+d)*C*
a*c*d^2+21/8*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^3*a^2*d*e^6+5/8*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*a^2*d^3*e^4
*x+31/8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*x^3*a*d^2*e^5-7/2*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^2*a^2*d^2*e^5-
5/8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^3*a*d^3*e^4-39/8*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*a^2*d*e^6*x+3*c^2/(
a*e^2+c*d^2)^5/(c*x^2+a)^2*B*x^2*a^2*d*e^6-33/8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*x^3*a*d*e^6-23/8*c^3/(a*e^2+
c*d^2)^5/(c*x^2+a)^2*C*a*d^5*e^2*x+21/8*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^2/a*x^3*A*d^5*e^2-25/8*c^3/(a*e^2+c*d^2)
^5/(c*x^2+a)^2*A*a*d^3*e^4*x+33/8*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*a^2*d^2*e^5*x+4*c^3/(a*e^2+c*d^2)^5/(c*x^2
+a)^2*A*x^2*a*d^2*e^5-5/2*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*x^2*a*d^4*e^3+21/8*c^4/(a*e^2+c*d^2)^5/a/(a*c)^(1/
2)*arctan(x*c/(a*c)^(1/2))*A*d^5*e^2+27/8*c/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*a^3*d*e^6*x+45/8*c/(a*e^2+c*d^2)^5*a
^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d*e^6-3/8*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^2/a*x^3*B*d^6*e+45/8*c^3/(a*e
^2+c*d^2)^5/(c*x^2+a)^2*B*a*d^4*e^3*x-125/8*c^2/(a*e^2+c*d^2)^5*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d^3*e^
4-105/8*c^2/(a*e^2+c*d^2)^5*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d*e^6+135/8*c^2/(a*e^2+c*d^2)^5*a/(a*c)^(1
/2)*arctan(x*c/(a*c)^(1/2))*B*d^2*e^5-3/8*c^4/(a*e^2+c*d^2)^5/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d^6*e-1/
2*e^5/(a*e^2+c*d^2)^3/(e*x+d)^2*A-c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*x^2*a^2*e^7-1/4*c^4/(a*e^2+c*d^2)^5/(c*x^2
+a)^2*B*d^7-1/2/(a*e^2+c*d^2)^5*a^2*ln(c*x^2+a)*C*e^7+35/8*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*x^3*d^4*e^3-15/8*
c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*x^3*d^3*e^4-7/8*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^2*B*x^3*a^2*e^7-9/8*c/(a*e^2+c
*d^2)^5/(c*x^2+a)^2*a^3*e^7*B*x+21*e^5/(a*e^2+c*d^2)^5*ln(e*x+d)*A*c^2*d^2-15*e^4/(a*e^2+c*d^2)^5*ln(e*x+d)*B*
c^2*d^3+10*e^3/(a*e^2+c*d^2)^5*ln(e*x+d)*C*c^2*d^4-4*e^3/(a*e^2+c*d^2)^4/(e*x+d)*C*c*d^3-3*e^7/(a*e^2+c*d^2)^5
*ln(e*x+d)*A*a*c+2*e^5/(a*e^2+c*d^2)^4/(e*x+d)*C*a*d+15/2*c^2/(a*e^2+c*d^2)^5*ln(c*x^2+a)*B*d^3*e^4-5*c^2/(a*e
^2+c*d^2)^5*ln(c*x^2+a)*C*d^4*e^3-6*e^5/(a*e^2+c*d^2)^4/(e*x+d)*A*c*d+5*e^4/(a*e^2+c*d^2)^4/(e*x+d)*B*c*d^2-21
/2*c^2/(a*e^2+c*d^2)^5*ln(c*x^2+a)*A*d^2*e^5-5/4*c/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*a^3*e^7+3/2*c/(a*e^2+c*d^2)^5
*a*ln(c*x^2+a)*A*e^7-1/8*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*C*d^7*x+3/4*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^2*A*d^6*e

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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((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,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((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,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((C*x**2+B*x+A)/(e*x+d)**3/(c*x**2+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.2198, size = 2068, normalized size = 2.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,x, algorithm="giac")

[Out]

-1/2*(10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 - 13*C*a*c*d^2*e^5 + 21*A*c^2*d^2*e^5 + 9*B*a*c*d*e^6 + C*a^2*e^7 -
3*A*a*c*e^7)*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^
2*e^8 + a^5*e^10) + (10*C*c^2*d^4*e^4 - 15*B*c^2*d^3*e^5 - 13*C*a*c*d^2*e^6 + 21*A*c^2*d^2*e^6 + 9*B*a*c*d*e^7
+ C*a^2*e^8 - 3*A*a*c*e^8)*log(abs(x*e + d))/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 + 10*a^3*c^2*
d^4*e^7 + 5*a^4*c*d^2*e^9 + a^5*e^11) + 1/8*(C*a*c^4*d^7 + 3*A*c^5*d^7 - 3*B*a*c^4*d^6*e + 23*C*a^2*c^3*d^5*e^
2 + 21*A*a*c^4*d^5*e^2 - 45*B*a^2*c^3*d^4*e^3 - 125*C*a^3*c^2*d^3*e^4 + 105*A*a^2*c^3*d^3*e^4 + 135*B*a^3*c^2*
d^2*e^5 + 45*C*a^4*c*d*e^6 - 105*A*a^3*c^2*d*e^6 - 15*B*a^4*c*e^7)*arctan(c*x/sqrt(a*c))/((a^2*c^5*d^10 + 5*a^
3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6 + 5*a^6*c*d^2*e^8 + a^7*e^10)*sqrt(a*c)) + 1/8*(C*a*c^
4*d^5*x^5*e^2 + 3*A*c^5*d^5*x^5*e^2 + 2*C*a*c^4*d^6*x^4*e + 6*A*c^5*d^6*x^4*e + C*a*c^4*d^7*x^3 + 3*A*c^5*d^7*
x^3 - 3*B*a*c^4*d^4*x^5*e^3 - 6*B*a*c^4*d^5*x^4*e^2 - 3*B*a*c^4*d^6*x^3*e - 58*C*a^2*c^3*d^3*x^5*e^4 + 18*A*a*
c^4*d^3*x^5*e^4 - 76*C*a^2*c^3*d^4*x^4*e^3 + 36*A*a*c^4*d^4*x^4*e^3 - 3*C*a^2*c^3*d^5*x^3*e^2 + 23*A*a*c^4*d^5
*x^3*e^2 + 10*C*a^2*c^3*d^6*x^2*e + 10*A*a*c^4*d^6*x^2*e - C*a^2*c^3*d^7*x + 5*A*a*c^4*d^7*x + 78*B*a^2*c^3*d^
2*x^5*e^5 + 96*B*a^2*c^3*d^3*x^4*e^4 - 7*B*a^2*c^3*d^4*x^3*e^3 - 20*B*a^2*c^3*d^5*x^2*e^2 - B*a^2*c^3*d^6*x*e
- 2*B*a^2*c^3*d^7 + 37*C*a^3*c^2*d*x^5*e^6 - 81*A*a^2*c^3*d*x^5*e^6 + 22*C*a^3*c^2*d^2*x^4*e^5 - 78*A*a^2*c^3*
d^2*x^4*e^5 - 129*C*a^3*c^2*d^3*x^3*e^4 + 61*A*a^2*c^3*d^3*x^3*e^4 - 142*C*a^3*c^2*d^4*x^2*e^3 + 74*A*a^2*c^3*
d^4*x^2*e^3 - 10*C*a^3*c^2*d^5*x*e^2 + 26*A*a^2*c^3*d^5*x*e^2 + 6*C*a^3*c^2*d^6*e + 6*A*a^2*c^3*d^6*e - 15*B*a
^3*c^2*x^5*e^7 + 6*B*a^3*c^2*d*x^4*e^6 + 163*B*a^3*c^2*d^2*x^3*e^5 + 176*B*a^3*c^2*d^3*x^2*e^4 + 2*B*a^3*c^2*d
^4*x*e^3 - 20*B*a^3*c^2*d^5*e^2 + 4*C*a^4*c*x^4*e^7 - 12*A*a^3*c^2*x^4*e^7 + 67*C*a^4*c*d*x^3*e^6 - 151*A*a^3*
c^2*d*x^3*e^6 + 46*C*a^4*c*d^2*x^2*e^5 - 146*A*a^3*c^2*d^2*x^2*e^5 - 77*C*a^4*c*d^3*x*e^4 + 49*A*a^3*c^2*d^3*x
*e^4 - 72*C*a^4*c*d^4*e^3 + 44*A*a^3*c^2*d^4*e^3 - 25*B*a^4*c*x^3*e^7 + 4*B*a^4*c*d*x^2*e^6 + 91*B*a^4*c*d^2*x
*e^5 + 74*B*a^4*c*d^3*e^4 + 6*C*a^5*x^2*e^7 - 18*A*a^4*c*x^2*e^7 + 28*C*a^5*d*x*e^6 - 68*A*a^4*c*d*x*e^6 + 18*
C*a^5*d^2*e^5 - 62*A*a^4*c*d^2*e^5 - 8*B*a^5*x*e^7 - 4*B*a^5*d*e^6 - 4*A*a^5*e^7)/((a^2*c^4*d^8 + 4*a^3*c^3*d^
6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*(c*x^3*e + c*d*x^2 + a*x*e + a*d)^2)