3.1063 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=279 \[ -\frac{b^5 (d+e x)^3 (-6 a B e-A b e+7 b B d)}{3 e^8}+\frac{3 b^4 (d+e x)^2 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{2 e^8}-\frac{5 b^3 x (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^7}+\frac{5 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}+\frac{b^6 B (d+e x)^4}{4 e^8} \]

[Out]

(-5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d - A*e))/(3*e^8*(d + e*x)^3) -
((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(2*e^8*(d + e*x)^2) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a
*B*e))/(e^8*(d + e*x)) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^2)/(2*e^8) - (b^5*(7*b*B*d
- A*b*e - 6*a*B*e)*(d + e*x)^3)/(3*e^8) + (b^6*B*(d + e*x)^4)/(4*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b
*e - 3*a*B*e)*Log[d + e*x])/e^8

________________________________________________________________________________________

Rubi [A]  time = 0.449756, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b^5 (d+e x)^3 (-6 a B e-A b e+7 b B d)}{3 e^8}+\frac{3 b^4 (d+e x)^2 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{2 e^8}-\frac{5 b^3 x (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^7}+\frac{5 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}+\frac{b^6 B (d+e x)^4}{4 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]

[Out]

(-5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d - A*e))/(3*e^8*(d + e*x)^3) -
((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(2*e^8*(d + e*x)^2) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a
*B*e))/(e^8*(d + e*x)) + (3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^2)/(2*e^8) - (b^5*(7*b*B*d
- A*b*e - 6*a*B*e)*(d + e*x)^3)/(3*e^8) + (b^6*B*(d + e*x)^4)/(4*e^8) + (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b
*e - 3*a*B*e)*Log[d + e*x])/e^8

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx &=\int \left (\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7}+\frac{(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^4}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^3}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)}-\frac{3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e) (d+e x)}{e^7}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)^2}{e^7}+\frac{b^6 B (d+e x)^3}{e^7}\right ) \, dx\\ &=-\frac{5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) x}{e^7}+\frac{(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}-\frac{(b d-a e)^5 (7 b B d-6 A b e-a B e)}{2 e^8 (d+e x)^2}+\frac{3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{e^8 (d+e x)}+\frac{3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) (d+e x)^2}{2 e^8}-\frac{b^5 (7 b B d-A b e-6 a B e) (d+e x)^3}{3 e^8}+\frac{b^6 B (d+e x)^4}{4 e^8}+\frac{5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.191015, size = 297, normalized size = 1.06 \[ \frac{12 b^3 e x \left (15 a^2 b e^2 (A e-4 B d)+20 a^3 B e^3+12 a b^2 d e (5 B d-2 A e)+10 b^3 d^2 (A e-2 B d)\right )-6 b^4 e^2 x^2 \left (-15 a^2 B e^2-6 a b e (A e-4 B d)+2 b^2 d (2 A e-5 B d)\right )+4 b^5 e^3 x^3 (6 a B e+A b e-4 b B d)+60 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)+\frac{36 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{d+e x}-\frac{6 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{(d+e x)^2}+\frac{4 (b d-a e)^6 (B d-A e)}{(d+e x)^3}+3 b^6 B e^4 x^4}{12 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^4,x]

[Out]

(12*b^3*e*(20*a^3*B*e^3 + 12*a*b^2*d*e*(5*B*d - 2*A*e) + 15*a^2*b*e^2*(-4*B*d + A*e) + 10*b^3*d^2*(-2*B*d + A*
e))*x - 6*b^4*e^2*(-15*a^2*B*e^2 - 6*a*b*e*(-4*B*d + A*e) + 2*b^2*d*(-5*B*d + 2*A*e))*x^2 + 4*b^5*e^3*(-4*b*B*
d + A*b*e + 6*a*B*e)*x^3 + 3*b^6*B*e^4*x^4 + (4*(b*d - a*e)^6*(B*d - A*e))/(d + e*x)^3 - (6*(b*d - a*e)^5*(7*b
*B*d - 6*A*b*e - a*B*e))/(d + e*x)^2 + (36*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(d + e*x) + 60*b^2*(
b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*Log[d + e*x])/(12*e^8)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 1143, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^4,x)

[Out]

150*b^4/e^6*ln(e*x+d)*B*a^2*d^2-120*b^5/e^7*ln(e*x+d)*B*a*d^3+150*b^4/e^6/(e*x+d)*B*a^2*d^3-90*b^5/e^7/(e*x+d)
*B*a*d^4-60*b^4/e^5*ln(e*x+d)*A*a^2*d+60*b^5/e^6*ln(e*x+d)*A*a*d^2-80*b^3/e^5*ln(e*x+d)*B*a^3*d+2*b^5/e^4*B*x^
3*a-4/3*b^6/e^5*B*x^3*d-2*b^6/e^5*A*x^2*d-20*b^6/e^7*ln(e*x+d)*A*d^3+15*b^2/e^4*ln(e*x+d)*B*a^4+1/4*b^6/e^4*B*
x^4+1/3*b^6/e^4*A*x^3-1/3/e/(e*x+d)^3*a^6*A-1/2/e^2/(e*x+d)^2*B*a^6+35*b^6/e^8*ln(e*x+d)*B*d^4+15/2*b^4/e^4*B*
x^2*a^2+5*b^6/e^6*B*x^2*d^2+15*b^4/e^4*A*a^2*x+10*b^6/e^6*A*d^2*x+20*b^3/e^4*a^3*B*x-20*b^6/e^7*B*d^3*x-1/3/e^
7/(e*x+d)^3*A*b^6*d^6+3*b^5/e^4*A*x^2*a+1/3/e^2/(e*x+d)^3*B*d*a^6+1/3/e^8/(e*x+d)^3*b^6*B*d^7-3/e^2/(e*x+d)^2*
A*a^5*b+3/e^7/(e*x+d)^2*A*b^6*d^5-7/2/e^8/(e*x+d)^2*b^6*B*d^6-15*b^2/e^3/(e*x+d)*A*a^4-15*b^6/e^7/(e*x+d)*A*d^
4-15/e^6/(e*x+d)^2*A*a*b^5*d^4+6/e^3/(e*x+d)^2*B*a^5*b*d-45/2/e^4/(e*x+d)^2*B*a^4*b^2*d^2+40/e^5/(e*x+d)^2*B*a
^3*b^3*d^3-75/2/e^6/(e*x+d)^2*B*a^2*b^4*d^4+18/e^7/(e*x+d)^2*B*a*b^5*d^5+60*b^3/e^4/(e*x+d)*A*a^3*d-90*b^4/e^5
/(e*x+d)*A*a^2*d^2+60*b^5/e^6/(e*x+d)*A*a*d^3+45*b^2/e^4/(e*x+d)*B*a^4*d-120*b^3/e^5/(e*x+d)*B*a^3*d^2-60*b^4/
e^5*B*a^2*d*x+60*b^5/e^6*B*a*d^2*x+2/e^2/(e*x+d)^3*A*d*a^5*b-5/e^3/(e*x+d)^3*A*d^2*a^4*b^2-6*b/e^3/(e*x+d)*B*a
^5+21*b^6/e^8/(e*x+d)*B*d^5+20*b^3/e^4*ln(e*x+d)*A*a^3+20/3/e^4/(e*x+d)^3*A*d^3*a^3*b^3-5/e^5/(e*x+d)^3*A*a^2*
b^4*d^4+2/e^6/(e*x+d)^3*A*a*b^5*d^5-2/e^3/(e*x+d)^3*B*d^2*a^5*b+5/e^4/(e*x+d)^3*B*d^3*a^4*b^2-20/3/e^5/(e*x+d)
^3*B*a^3*b^3*d^4+5/e^6/(e*x+d)^3*B*a^2*b^4*d^5-2/e^7/(e*x+d)^3*B*a*b^5*d^6+15/e^3/(e*x+d)^2*A*a^4*b^2*d-30/e^4
/(e*x+d)^2*A*a^3*b^3*d^2+30/e^5/(e*x+d)^2*A*a^2*b^4*d^3-12*b^5/e^5*B*x^2*a*d-24*b^5/e^5*A*a*d*x

________________________________________________________________________________________

Maxima [B]  time = 1.22664, size = 1071, normalized size = 3.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(107*B*b^6*d^7 - 2*A*a^6*e^7 - 74*(6*B*a*b^5 + A*b^6)*d^6*e + 141*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 130*
(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 55*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d^2
*e^5 - (B*a^6 + 6*A*a^5*b)*d*e^6 + 18*(7*B*b^6*d^5*e^2 - 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A
*a*b^5)*d^3*e^4 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - (2*B*a^5*b +
5*A*a^4*b^2)*e^7)*x^2 + 3*(77*B*b^6*d^6*e - 54*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 105*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4
*e^3 - 100*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 45*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a
^4*b^2)*d*e^6 - (B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*b^6*
e^3*x^4 - 4*(4*B*b^6*d*e^2 - (6*B*a*b^5 + A*b^6)*e^3)*x^3 + 6*(10*B*b^6*d^2*e - 4*(6*B*a*b^5 + A*b^6)*d*e^2 +
3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^3)*x^2 - 12*(20*B*b^6*d^3 - 10*(6*B*a*b^5 + A*b^6)*d^2*e + 12*(5*B*a^2*b^4 + 2*A
*a*b^5)*d*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^3)*x)/e^7 + 5*(7*B*b^6*d^4 - 4*(6*B*a*b^5 + A*b^6)*d^3*e + 6*(
5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^4)*log(
e*x + d)/e^8

________________________________________________________________________________________

Fricas [B]  time = 1.99482, size = 2535, normalized size = 9.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/12*(3*B*b^6*e^7*x^7 + 214*B*b^6*d^7 - 4*A*a^6*e^7 - 148*(6*B*a*b^5 + A*b^6)*d^6*e + 282*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^5*e^2 - 260*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 110*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 12*(2*B*a^
5*b + 5*A*a^4*b^2)*d^2*e^5 - 2*(B*a^6 + 6*A*a^5*b)*d*e^6 - (7*B*b^6*d*e^6 - 4*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 3
*(7*B*b^6*d^2*e^5 - 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 15*(7*B*b^6*d^3*e^4 -
4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 -
2*(278*B*b^6*d^4*e^3 - 146*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 189*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 90*(4*B*a^3*b
^3 + 3*A*a^2*b^4)*d*e^6)*x^3 - 6*(68*B*b^6*d^5*e^2 - 26*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 9*(5*B*a^2*b^4 + 2*A*a*b
^5)*d^3*e^4 + 30*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 30*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 6*(2*B*a^5*b + 5
*A*a^4*b^2)*e^7)*x^2 + 6*(37*B*b^6*d^6*e - 34*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 81*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e
^3 - 90*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 45*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^4*
b^2)*d*e^6 - (B*a^6 + 6*A*a^5*b)*e^7)*x + 60*(7*B*b^6*d^7 - 4*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A
*a*b^5)*d^5*e^2 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + (7*B*b^6*d^4*e
^3 - 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6
+ (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 3*(7*B*b^6*d^5*e^2 - 4*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^3*e^4 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6)*x^2 + 3*(7*B
*b^6*d^6*e - 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d^3*e^4 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^
3*e^8)

________________________________________________________________________________________

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((b*x+a)**6*(B*x+A)/(e*x+d)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.28955, size = 1075, normalized size = 3.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^4,x, algorithm="giac")

[Out]

5*(7*B*b^6*d^4 - 24*B*a*b^5*d^3*e - 4*A*b^6*d^3*e + 30*B*a^2*b^4*d^2*e^2 + 12*A*a*b^5*d^2*e^2 - 16*B*a^3*b^3*d
*e^3 - 12*A*a^2*b^4*d*e^3 + 3*B*a^4*b^2*e^4 + 4*A*a^3*b^3*e^4)*e^(-8)*log(abs(x*e + d)) + 1/12*(3*B*b^6*x^4*e^
12 - 16*B*b^6*d*x^3*e^11 + 60*B*b^6*d^2*x^2*e^10 - 240*B*b^6*d^3*x*e^9 + 24*B*a*b^5*x^3*e^12 + 4*A*b^6*x^3*e^1
2 - 144*B*a*b^5*d*x^2*e^11 - 24*A*b^6*d*x^2*e^11 + 720*B*a*b^5*d^2*x*e^10 + 120*A*b^6*d^2*x*e^10 + 90*B*a^2*b^
4*x^2*e^12 + 36*A*a*b^5*x^2*e^12 - 720*B*a^2*b^4*d*x*e^11 - 288*A*a*b^5*d*x*e^11 + 240*B*a^3*b^3*x*e^12 + 180*
A*a^2*b^4*x*e^12)*e^(-16) + 1/6*(107*B*b^6*d^7 - 444*B*a*b^5*d^6*e - 74*A*b^6*d^6*e + 705*B*a^2*b^4*d^5*e^2 +
282*A*a*b^5*d^5*e^2 - 520*B*a^3*b^3*d^4*e^3 - 390*A*a^2*b^4*d^4*e^3 + 165*B*a^4*b^2*d^3*e^4 + 220*A*a^3*b^3*d^
3*e^4 - 12*B*a^5*b*d^2*e^5 - 30*A*a^4*b^2*d^2*e^5 - B*a^6*d*e^6 - 6*A*a^5*b*d*e^6 - 2*A*a^6*e^7 + 18*(7*B*b^6*
d^5*e^2 - 30*B*a*b^5*d^4*e^3 - 5*A*b^6*d^4*e^3 + 50*B*a^2*b^4*d^3*e^4 + 20*A*a*b^5*d^3*e^4 - 40*B*a^3*b^3*d^2*
e^5 - 30*A*a^2*b^4*d^2*e^5 + 15*B*a^4*b^2*d*e^6 + 20*A*a^3*b^3*d*e^6 - 2*B*a^5*b*e^7 - 5*A*a^4*b^2*e^7)*x^2 +
3*(77*B*b^6*d^6*e - 324*B*a*b^5*d^5*e^2 - 54*A*b^6*d^5*e^2 + 525*B*a^2*b^4*d^4*e^3 + 210*A*a*b^5*d^4*e^3 - 400
*B*a^3*b^3*d^3*e^4 - 300*A*a^2*b^4*d^3*e^4 + 135*B*a^4*b^2*d^2*e^5 + 180*A*a^3*b^3*d^2*e^5 - 12*B*a^5*b*d*e^6
- 30*A*a^4*b^2*d*e^6 - B*a^6*e^7 - 6*A*a^5*b*e^7)*x)*e^(-8)/(x*e + d)^3