### 3.1514 $$\int \frac{1}{(d+e x)^4 (a^2+2 a b x+b^2 x^2)} \, dx$$

Optimal. Leaf size=133 $-\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2}$

[Out]

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

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Rubi [A]  time = 0.107646, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 44} $-\frac{b^3}{(a+b x) (b d-a e)^4}-\frac{3 b^2 e}{(d+e x) (b d-a e)^4}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}-\frac{b e}{(d+e x)^2 (b d-a e)^3}-\frac{e}{3 (d+e x)^3 (b d-a e)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^4} \, dx\\ &=\int \left (\frac{b^4}{(b d-a e)^4 (a+b x)^2}-\frac{4 b^4 e}{(b d-a e)^5 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)^4}+\frac{2 b e^2}{(b d-a e)^3 (d+e x)^3}+\frac{3 b^2 e^2}{(b d-a e)^4 (d+e x)^2}+\frac{4 b^3 e^2}{(b d-a e)^5 (d+e x)}\right ) \, dx\\ &=-\frac{b^3}{(b d-a e)^4 (a+b x)}-\frac{e}{3 (b d-a e)^2 (d+e x)^3}-\frac{b e}{(b d-a e)^3 (d+e x)^2}-\frac{3 b^2 e}{(b d-a e)^4 (d+e x)}-\frac{4 b^3 e \log (a+b x)}{(b d-a e)^5}+\frac{4 b^3 e \log (d+e x)}{(b d-a e)^5}\\ \end{align*}

Mathematica [A]  time = 0.124966, size = 120, normalized size = 0.9 $\frac{-\frac{3 b^3 (b d-a e)}{a+b x}-\frac{9 b^2 e (b d-a e)}{d+e x}-12 b^3 e \log (a+b x)-\frac{3 b e (b d-a e)^2}{(d+e x)^2}+\frac{e (a e-b d)^3}{(d+e x)^3}+12 b^3 e \log (d+e x)}{3 (b d-a e)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^4*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

((-3*b^3*(b*d - a*e))/(a + b*x) + (e*(-(b*d) + a*e)^3)/(d + e*x)^3 - (3*b*e*(b*d - a*e)^2)/(d + e*x)^2 - (9*b^
2*e*(b*d - a*e))/(d + e*x) - 12*b^3*e*Log[a + b*x] + 12*b^3*e*Log[d + e*x])/(3*(b*d - a*e)^5)

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Maple [A]  time = 0.055, size = 131, normalized size = 1. \begin{align*} -{\frac{e}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-4\,{\frac{e{b}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{5}}}-3\,{\frac{{b}^{2}e}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}+{\frac{be}{ \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{3}}{ \left ( ae-bd \right ) ^{4} \left ( bx+a \right ) }}+4\,{\frac{e{b}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-1/3*e/(a*e-b*d)^2/(e*x+d)^3-4*e/(a*e-b*d)^5*b^3*ln(e*x+d)-3*e/(a*e-b*d)^4*b^2/(e*x+d)+e/(a*e-b*d)^3*b/(e*x+d)
^2-b^3/(a*e-b*d)^4/(b*x+a)+4*e/(a*e-b*d)^5*b^3*ln(b*x+a)

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Maxima [B]  time = 1.33082, size = 809, normalized size = 6.08 \begin{align*} -\frac{4 \, b^{3} e \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{4 \, b^{3} e \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{12 \, b^{3} e^{3} x^{3} + 3 \, b^{3} d^{3} + 13 \, a b^{2} d^{2} e - 5 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (5 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 2 \,{\left (11 \, b^{3} d^{2} e + 8 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{3 \,{\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} +{\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} +{\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \,{\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} +{\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-4*b^3*e*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5
*e^5) + 4*b^3*e*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^
4 - a^5*e^5) - 1/3*(12*b^3*e^3*x^3 + 3*b^3*d^3 + 13*a*b^2*d^2*e - 5*a^2*b*d*e^2 + a^3*e^3 + 6*(5*b^3*d*e^2 + a
*b^2*e^3)*x^2 + 2*(11*b^3*d^2*e + 8*a*b^2*d*e^2 - a^2*b*e^3)*x)/(a*b^4*d^7 - 4*a^2*b^3*d^6*e + 6*a^3*b^2*d^5*e
^2 - 4*a^4*b*d^4*e^3 + a^5*d^3*e^4 + (b^5*d^4*e^3 - 4*a*b^4*d^3*e^4 + 6*a^2*b^3*d^2*e^5 - 4*a^3*b^2*d*e^6 + a^
4*b*e^7)*x^4 + (3*b^5*d^5*e^2 - 11*a*b^4*d^4*e^3 + 14*a^2*b^3*d^3*e^4 - 6*a^3*b^2*d^2*e^5 - a^4*b*d*e^6 + a^5*
e^7)*x^3 + 3*(b^5*d^6*e - 3*a*b^4*d^5*e^2 + 2*a^2*b^3*d^4*e^3 + 2*a^3*b^2*d^3*e^4 - 3*a^4*b*d^2*e^5 + a^5*d*e^
6)*x^2 + (b^5*d^7 - a*b^4*d^6*e - 6*a^2*b^3*d^5*e^2 + 14*a^3*b^2*d^4*e^3 - 11*a^4*b*d^3*e^4 + 3*a^5*d^2*e^5)*x
)

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Fricas [B]  time = 1.85781, size = 1501, normalized size = 11.29 \begin{align*} -\frac{3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a b^{3} d^{3} e +{\left (3 \, b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (b^{4} d^{2} e^{2} + a b^{3} d e^{3}\right )} x^{2} +{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a b^{5} d^{8} - 5 \, a^{2} b^{4} d^{7} e + 10 \, a^{3} b^{3} d^{6} e^{2} - 10 \, a^{4} b^{2} d^{5} e^{3} + 5 \, a^{5} b d^{4} e^{4} - a^{6} d^{3} e^{5} +{\left (b^{6} d^{5} e^{3} - 5 \, a b^{5} d^{4} e^{4} + 10 \, a^{2} b^{4} d^{3} e^{5} - 10 \, a^{3} b^{3} d^{2} e^{6} + 5 \, a^{4} b^{2} d e^{7} - a^{5} b e^{8}\right )} x^{4} +{\left (3 \, b^{6} d^{6} e^{2} - 14 \, a b^{5} d^{5} e^{3} + 25 \, a^{2} b^{4} d^{4} e^{4} - 20 \, a^{3} b^{3} d^{3} e^{5} + 5 \, a^{4} b^{2} d^{2} e^{6} + 2 \, a^{5} b d e^{7} - a^{6} e^{8}\right )} x^{3} + 3 \,{\left (b^{6} d^{7} e - 4 \, a b^{5} d^{6} e^{2} + 5 \, a^{2} b^{4} d^{5} e^{3} - 5 \, a^{4} b^{2} d^{3} e^{5} + 4 \, a^{5} b d^{2} e^{6} - a^{6} d e^{7}\right )} x^{2} +{\left (b^{6} d^{8} - 2 \, a b^{5} d^{7} e - 5 \, a^{2} b^{4} d^{6} e^{2} + 20 \, a^{3} b^{3} d^{5} e^{3} - 25 \, a^{4} b^{2} d^{4} e^{4} + 14 \, a^{5} b d^{3} e^{5} - 3 \, a^{6} d^{2} e^{6}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^3 - a^4*e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x
^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 2*(11*b^4*d^3*e - 3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3
+ a^3*b*e^4)*x + 12*(b^4*e^4*x^4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 + a*b^3*d*e^3
)*x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(b*x + a) - 12*(b^4*e^4*x^4 + a*b^3*d^3*e + (3*b^4*d*e^3 + a*b^3*e
^4)*x^3 + 3*(b^4*d^2*e^2 + a*b^3*d*e^3)*x^2 + (b^4*d^3*e + 3*a*b^3*d^2*e^2)*x)*log(e*x + d))/(a*b^5*d^8 - 5*a^
2*b^4*d^7*e + 10*a^3*b^3*d^6*e^2 - 10*a^4*b^2*d^5*e^3 + 5*a^5*b*d^4*e^4 - a^6*d^3*e^5 + (b^6*d^5*e^3 - 5*a*b^5
*d^4*e^4 + 10*a^2*b^4*d^3*e^5 - 10*a^3*b^3*d^2*e^6 + 5*a^4*b^2*d*e^7 - a^5*b*e^8)*x^4 + (3*b^6*d^6*e^2 - 14*a*
b^5*d^5*e^3 + 25*a^2*b^4*d^4*e^4 - 20*a^3*b^3*d^3*e^5 + 5*a^4*b^2*d^2*e^6 + 2*a^5*b*d*e^7 - a^6*e^8)*x^3 + 3*(
b^6*d^7*e - 4*a*b^5*d^6*e^2 + 5*a^2*b^4*d^5*e^3 - 5*a^4*b^2*d^3*e^5 + 4*a^5*b*d^2*e^6 - a^6*d*e^7)*x^2 + (b^6*
d^8 - 2*a*b^5*d^7*e - 5*a^2*b^4*d^6*e^2 + 20*a^3*b^3*d^5*e^3 - 25*a^4*b^2*d^4*e^4 + 14*a^5*b*d^3*e^5 - 3*a^6*d
^2*e^6)*x)

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Sympy [B]  time = 3.7665, size = 881, normalized size = 6.62 \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)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

-4*b**3*e*log(x + (-4*a**6*b**3*e**7/(a*e - b*d)**5 + 24*a**5*b**4*d*e**6/(a*e - b*d)**5 - 60*a**4*b**5*d**2*e
**5/(a*e - b*d)**5 + 80*a**3*b**6*d**3*e**4/(a*e - b*d)**5 - 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 + 24*a*b**8
*d**5*e**2/(a*e - b*d)**5 + 4*a*b**3*e**2 - 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d*e)/(8*b**4*e**2))/(a*e - b
*d)**5 + 4*b**3*e*log(x + (4*a**6*b**3*e**7/(a*e - b*d)**5 - 24*a**5*b**4*d*e**6/(a*e - b*d)**5 + 60*a**4*b**5
*d**2*e**5/(a*e - b*d)**5 - 80*a**3*b**6*d**3*e**4/(a*e - b*d)**5 + 60*a**2*b**7*d**4*e**3/(a*e - b*d)**5 - 24
*a*b**8*d**5*e**2/(a*e - b*d)**5 + 4*a*b**3*e**2 + 4*b**9*d**6*e/(a*e - b*d)**5 + 4*b**4*d*e)/(8*b**4*e**2))/(
a*e - b*d)**5 - (a**3*e**3 - 5*a**2*b*d*e**2 + 13*a*b**2*d**2*e + 3*b**3*d**3 + 12*b**3*e**3*x**3 + x**2*(6*a*
b**2*e**3 + 30*b**3*d*e**2) + x*(-2*a**2*b*e**3 + 16*a*b**2*d*e**2 + 22*b**3*d**2*e))/(3*a**5*d**3*e**4 - 12*a
**4*b*d**4*e**3 + 18*a**3*b**2*d**5*e**2 - 12*a**2*b**3*d**6*e + 3*a*b**4*d**7 + x**4*(3*a**4*b*e**7 - 12*a**3
*b**2*d*e**6 + 18*a**2*b**3*d**2*e**5 - 12*a*b**4*d**3*e**4 + 3*b**5*d**4*e**3) + x**3*(3*a**5*e**7 - 3*a**4*b
*d*e**6 - 18*a**3*b**2*d**2*e**5 + 42*a**2*b**3*d**3*e**4 - 33*a*b**4*d**4*e**3 + 9*b**5*d**5*e**2) + x**2*(9*
a**5*d*e**6 - 27*a**4*b*d**2*e**5 + 18*a**3*b**2*d**3*e**4 + 18*a**2*b**3*d**4*e**3 - 27*a*b**4*d**5*e**2 + 9*
b**5*d**6*e) + x*(9*a**5*d**2*e**5 - 33*a**4*b*d**3*e**4 + 42*a**3*b**2*d**4*e**3 - 18*a**2*b**3*d**5*e**2 - 3
*a*b**4*d**6*e + 3*b**5*d**7))

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Giac [B]  time = 1.12195, size = 455, normalized size = 3.42 \begin{align*} -\frac{4 \, b^{4} e \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac{4 \, b^{3} e^{2} \log \left ({\left | x e + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{3 \, b^{4} d^{4} + 10 \, a b^{3} d^{3} e - 18 \, a^{2} b^{2} d^{2} e^{2} + 6 \, a^{3} b d e^{3} - a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (5 \, b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} - a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (11 \, b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} - 9 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \,{\left (b d - a e\right )}^{5}{\left (b x + a\right )}{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

-4*b^4*e*log(abs(b*x + a))/(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^
4 - a^5*b*e^5) + 4*b^3*e^2*log(abs(x*e + d))/(b^5*d^5*e - 5*a*b^4*d^4*e^2 + 10*a^2*b^3*d^3*e^3 - 10*a^3*b^2*d^
2*e^4 + 5*a^4*b*d*e^5 - a^5*e^6) - 1/3*(3*b^4*d^4 + 10*a*b^3*d^3*e - 18*a^2*b^2*d^2*e^2 + 6*a^3*b*d*e^3 - a^4*
e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x^3 + 6*(5*b^4*d^2*e^2 - 4*a*b^3*d*e^3 - a^2*b^2*e^4)*x^2 + 2*(11*b^4*d^3*e -
3*a*b^3*d^2*e^2 - 9*a^2*b^2*d*e^3 + a^3*b*e^4)*x)/((b*d - a*e)^5*(b*x + a)*(x*e + d)^3)