3.1046 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^4} \, dx\)

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

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Rubi [A]  time = 1.20712, 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 \[ -\frac{b^5 (d+e x)^3 (-6 a B e-A b e+7 b B d)}{3 e^8}+\frac{3 b^4 (d+e x)^2 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{2 e^8}-\frac{5 b^3 x (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{e^7}+\frac{5 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{e^8 (d+e x)}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{2 e^8 (d+e x)^2}+\frac{(b d-a e)^6 (B d-A e)}{3 e^8 (d+e x)^3}+\frac{b^6 B (d+e x)^4}{4 e^8} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b^{6} \left (d + e x\right )^{4}}{4 e^{8}} + \frac{b^{5} \left (d + e x\right )^{3} \left (A b e + 6 B a e - 7 B b d\right )}{3 e^{8}} + \frac{3 b^{4} \left (d + e x\right )^{2} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{2 e^{8}} + \frac{5 b^{2} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{3 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{e^{8} \left (d + e x\right )} + \frac{5 \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right ) \int b^{3}\, dx}{e^{7}} - \frac{\left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{2 e^{8} \left (d + e x\right )^{2}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{6}}{3 e^{8} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**6*(d + e*x)**4/(4*e**8) + b**5*(d + e*x)**3*(A*b*e + 6*B*a*e - 7*B*b*d)/(3*
e**8) + 3*b**4*(d + e*x)**2*(a*e - b*d)*(2*A*b*e + 5*B*a*e - 7*B*b*d)/(2*e**8) +
5*b**2*(a*e - b*d)**3*(4*A*b*e + 3*B*a*e - 7*B*b*d)*log(d + e*x)/e**8 - 3*b*(a*
e - b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)/(e**8*(d + e*x)) + 5*(a*e - b*d)**2*(3
*A*b*e + 4*B*a*e - 7*B*b*d)*Integral(b**3, x)/e**7 - (a*e - b*d)**5*(6*A*b*e + B
*a*e - 7*B*b*d)/(2*e**8*(d + e*x)**2) - (A*e - B*d)*(a*e - b*d)**6/(3*e**8*(d +
e*x)**3)

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Mathematica [A]  time = 0.324283, size = 297, normalized size = 1.06 \[ \frac{-6 b^4 e^2 x^2 \left (-15 a^2 B e^2-6 a b e (A e-4 B d)+2 b^2 d (2 A e-5 B d)\right )+12 b^3 e x \left (20 a^3 B e^3+15 a^2 b e^2 (A e-4 B d)+12 a b^2 d e (5 B d-2 A e)+10 b^3 d^2 (A e-2 B d)\right )+4 b^5 e^3 x^3 (6 a B e+A b e-4 b B d)+60 b^2 (b d-a e)^3 \log (d+e x) (-3 a B e-4 A b e+7 b B d)+\frac{36 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{d+e x}-\frac{6 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{(d+e x)^2}+\frac{4 (b d-a e)^6 (B d-A e)}{(d+e x)^3}+3 b^6 B e^4 x^4}{12 e^8} \]

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)

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Maple [B]  time = 0.026, size = 1143, normalized size = 4.1 \[ \text{result too large to display} \]

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]

-1/3/e/(e*x+d)^3*a^6*A-1/2/e^2/(e*x+d)^2*B*a^6+1/4*b^6/e^4*B*x^4+1/3*b^6/e^4*A*x
^3+1/3/e^8/(e*x+d)^3*b^6*B*d^7-15*b^2/e^3/(e*x+d)*A*a^4-15*b^6/e^7/(e*x+d)*A*d^4
-6*b/e^3/(e*x+d)*B*a^5+21*b^6/e^8/(e*x+d)*B*d^5-3/e^2/(e*x+d)^2*A*a^5*b+3/e^7/(e
*x+d)^2*A*b^6*d^5+20*b^3/e^4*B*a^3*x-20*b^6/e^7*B*d^3*x+20*b^3/e^4*ln(e*x+d)*A*a
^3-7/2/e^8/(e*x+d)^2*b^6*B*d^6-20*b^6/e^7*ln(e*x+d)*A*d^3+15*b^2/e^4*ln(e*x+d)*B
*a^4+35*b^6/e^8*ln(e*x+d)*B*d^4+2*b^5/e^4*B*x^3*a-4/3*b^6/e^5*B*x^3*d+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+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+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^3-80*b^3/e^5*ln(e*x+d)*B*a^3*d+15
0*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^3-12*b^5/e^5*B*x^2*a*d
-24*b^5/e^5*A*a*d*x-60*b^4/e^5*B*a^2*d*x+60*b^5/e^6*B*a*d^2*x-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^2-75/2/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+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+1
50*b^4/e^6/(e*x+d)*B*a^2*d^3-90*b^5/e^7/(e*x+d)*B*a*d^4+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^3-15/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+3*b^5/e^4*A*x^2*a-2*b^6/e^5*A*x^2*d+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-1/3/e^7/(e*x+d)^3
*A*b^6*d^6+1/3/e^2/(e*x+d)^3*B*d*a^6

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Maxima [A]  time = 1.40138, size = 1071, normalized size = 3.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(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

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Fricas [A]  time = 0.210869, size = 1654, normalized size = 5.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(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)

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

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

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GIAC/XCAS [A]  time = 0.233239, size = 1075, normalized size = 3.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(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)*ln(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^12 - 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 + 2
0*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 - 3
24*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 + 18
0*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