3.1066 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^7} \, dx\)

Optimal. Leaf size=278 \[ -\frac{3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac{5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}-\frac{b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac{(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac{b^6 B x}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.326039, antiderivative size = 278, 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} \[ -\frac{3 b^4 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8 (d+e x)}+\frac{5 b^3 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{3 e^8 (d+e x)^3}-\frac{b^5 \log (d+e x) (-6 a B e-A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{4 e^8 (d+e x)^4}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{5 e^8 (d+e x)^5}+\frac{(b d-a e)^6 (B d-A e)}{6 e^8 (d+e x)^6}+\frac{b^6 B x}{e^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]

[Out]

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

Mathematica [B]  time = 0.331132, size = 619, normalized size = 2.23 \[ -\frac{30 a^2 b^4 e^2 \left (A e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+5 B \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )+20 a^3 b^3 e^3 \left (A e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+15 a^4 b^2 e^4 \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )+6 a^5 b e^5 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 a^6 e^6 (5 A e+B (d+6 e x))-6 a b^5 e \left (B d \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )-10 A e \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )+60 b^5 (d+e x)^6 \log (d+e x) (-6 a B e-A b e+7 b B d)+b^6 \left (-\left (A d e \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )-B \left (7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5+3594 d^6 e x+669 d^7-360 d e^6 x^6-60 e^7 x^7\right )\right )\right )}{60 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^7,x]

[Out]

-(2*a^6*e^6*(5*A*e + B*(d + 6*e*x)) + 6*a^5*b*e^5*(2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 15*a^
4*b^2*e^4*(A*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 20*a^3*b^3*e^
3*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 +
15*e^4*x^4)) + 30*a^2*b^4*e^2*(A*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*B*(d^5 +
6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) - 6*a*b^5*e*(-10*A*e*(d^5 + 6*d^4*e*
x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + B*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^
2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) - b^6*(A*d*e*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 +
2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - B*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^
3*x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*b^5*(7*b*B*d - A*b*e - 6*a*B*e)
*(d + e*x)^6*Log[d + e*x])/(60*e^8*(d + e*x)^6)

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Maple [B]  time = 0.015, size = 1217, normalized size = 4.4 \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)^7,x)

[Out]

-15/2*b^6/e^7/(e*x+d)^2*A*d^2-10*b^3/e^5/(e*x+d)^2*B*a^3+35/2*b^6/e^8/(e*x+d)^2*B*d^3-5*b^2/e^4/(e*x+d)^3*B*a^
4-35/3*b^6/e^8/(e*x+d)^3*B*d^4-15/2*b^4/e^5/(e*x+d)^2*A*a^2+6*b^6/e^7/(e*x+d)*A*d-15*b^4/e^6/(e*x+d)*B*a^2-21*
b^6/e^8/(e*x+d)*B*d^2+6/5/e^7/(e*x+d)^5*A*b^6*d^5-7/5/e^8/(e*x+d)^5*b^6*B*d^6-6/5/e^2/(e*x+d)^5*A*a^5*b+1/6/e^
2/(e*x+d)^6*B*d*a^6+1/6/e^8/(e*x+d)^6*b^6*B*d^7-6*b^5/e^6/(e*x+d)*A*a-20/3*b^3/e^4/(e*x+d)^3*A*a^3+20/3*b^6/e^
7/(e*x+d)^3*A*d^3-15/4*b^2/e^3/(e*x+d)^4*A*a^4-15/4*b^6/e^7/(e*x+d)^4*A*d^4-3/2*b/e^3/(e*x+d)^4*B*a^5+21/4*b^6
/e^8/(e*x+d)^4*B*d^5-1/6/e^7/(e*x+d)^6*A*d^6*b^6+6*b^5/e^7*ln(e*x+d)*B*a-7*b^6/e^8*ln(e*x+d)*B*d-1/5/e^2/(e*x+
d)^5*B*a^6+b^6/e^7*ln(e*x+d)*A-1/6/e/(e*x+d)^6*a^6*A+b^6*B*x/e^7+40*b^5/e^7/(e*x+d)^3*B*a*d^3+15*b^5/e^6/(e*x+
d)^2*A*a*d+75/2*b^4/e^6/(e*x+d)^2*B*a^2*d-45/2*b^4/e^5/(e*x+d)^4*A*a^2*d^2-45*b^5/e^7/(e*x+d)^2*B*a*d^2+36*b^5
/e^7/(e*x+d)*B*a*d+6/e^3/(e*x+d)^5*A*a^4*b^2*d-12/e^4/(e*x+d)^5*A*a^3*b^3*d^2+12/e^5/(e*x+d)^5*A*a^2*b^4*d^3-6
/e^6/(e*x+d)^5*A*a*b^5*d^4+12/5/e^3/(e*x+d)^5*B*a^5*b*d-9/e^4/(e*x+d)^5*B*a^4*b^2*d^2+16/e^5/(e*x+d)^5*B*a^3*b
^3*d^3-15/e^6/(e*x+d)^5*B*a^2*b^4*d^4+36/5/e^7/(e*x+d)^5*B*a*b^5*d^5+1/e^6/(e*x+d)^6*A*d^5*a*b^5-1/e^3/(e*x+d)
^6*B*d^2*a^5*b+5/2/e^4/(e*x+d)^6*B*d^3*a^4*b^2-10/3/e^5/(e*x+d)^6*B*d^4*a^3*b^3+5/2/e^6/(e*x+d)^6*B*d^5*a^2*b^
4-1/e^7/(e*x+d)^6*B*d^6*a*b^5+20*b^4/e^5/(e*x+d)^3*A*a^2*d-20*b^5/e^6/(e*x+d)^3*A*a*d^2+80/3*b^3/e^5/(e*x+d)^3
*B*a^3*d-50*b^4/e^6/(e*x+d)^3*B*a^2*d^2+10/3/e^4/(e*x+d)^6*A*d^3*a^3*b^3-5/2/e^5/(e*x+d)^6*A*d^4*a^2*b^4-5/2/e
^3/(e*x+d)^6*A*d^2*a^4*b^2+1/e^2/(e*x+d)^6*A*d*a^5*b+15*b^5/e^6/(e*x+d)^4*A*a*d^3+45/4*b^2/e^4/(e*x+d)^4*B*a^4
*d-30*b^3/e^5/(e*x+d)^4*B*a^3*d^2+75/2*b^4/e^6/(e*x+d)^4*B*a^2*d^3-45/2*b^5/e^7/(e*x+d)^4*B*a*d^4+15*b^3/e^4/(
e*x+d)^4*A*a^3*d

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Maxima [B]  time = 2.79342, size = 1112, normalized size = 4. \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)^7,x, algorithm="maxima")

[Out]

B*b^6*x/e^7 - 1/60*(669*B*b^6*d^7 + 10*A*a^6*e^7 - 147*(6*B*a*b^5 + A*b^6)*d^6*e + 30*(5*B*a^2*b^4 + 2*A*a*b^5
)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 3*(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 + 180*(7*B*b^6*d^2*e^5 - 2*(6*B*a*b^5 + A*b^6)*d*e^6 + (5*B*a
^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 150*(35*B*b^6*d^3*e^4 - 9*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*
b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 100*(91*B*b^6*d^4*e^3 - 22*(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 + 2*(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 + 15*(539*B*b^6*d^5*e^2 - 125*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 30*(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 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 6
*(609*B*b^6*d^6*e - 137*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10*(4*B*a^3*b^3 +
3*A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 2*(B*a^6 +
6*A*a^5*b)*e^7)*x)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9
*x + d^6*e^8) - (7*B*b^6*d - (6*B*a*b^5 + A*b^6)*e)*log(e*x + d)/e^8

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Fricas [B]  time = 2.00932, size = 2210, normalized size = 7.95 \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)^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^6*e^7*x^7 + 360*B*b^6*d*e^6*x^6 - 669*B*b^6*d^7 - 10*A*a^6*e^7 + 147*(6*B*a*b^5 + A*b^6)*d^6*e -
30*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*
d^3*e^4 - 3*(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 - 180*(2*B*b^6*d^2*e^5 - 2*(6*B*a*
b^5 + A*b^6)*d*e^6 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 - 150*(27*B*b^6*d^3*e^4 - 9*(6*B*a*b^5 + A*b^6)*d^2*e^
5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 - 100*(82*B*b^6*d^4*e^3 - 22*(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 + 2*(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 - 15*(515*B*b^6*d^5*e^2 - 125*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 30*(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 + 3*(2*B*a^5*b
+ 5*A*a^4*b^2)*e^7)*x^2 - 6*(599*B*b^6*d^6*e - 137*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 30*(5*B*a^2*b^4 + 2*A*a*b^5)*
d^4*e^3 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5*A*
a^4*b^2)*d*e^6 + 2*(B*a^6 + 6*A*a^5*b)*e^7)*x - 60*(7*B*b^6*d^7 - (6*B*a*b^5 + A*b^6)*d^6*e + (7*B*b^6*d*e^6 -
(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 6*(7*B*b^6*d^2*e^5 - (6*B*a*b^5 + A*b^6)*d*e^6)*x^5 + 15*(7*B*b^6*d^3*e^4 - (6
*B*a*b^5 + A*b^6)*d^2*e^5)*x^4 + 20*(7*B*b^6*d^4*e^3 - (6*B*a*b^5 + A*b^6)*d^3*e^4)*x^3 + 15*(7*B*b^6*d^5*e^2
- (6*B*a*b^5 + A*b^6)*d^4*e^3)*x^2 + 6*(7*B*b^6*d^6*e - (6*B*a*b^5 + A*b^6)*d^5*e^2)*x)*log(e*x + d))/(e^14*x^
6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8)

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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((b*x+a)**6*(B*x+A)/(e*x+d)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.88545, size = 1046, normalized size = 3.76 \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)^7,x, algorithm="giac")

[Out]

B*b^6*x*e^(-7) - (7*B*b^6*d - 6*B*a*b^5*e - A*b^6*e)*e^(-8)*log(abs(x*e + d)) - 1/60*(669*B*b^6*d^7 - 882*B*a*
b^5*d^6*e - 147*A*b^6*d^6*e + 150*B*a^2*b^4*d^5*e^2 + 60*A*a*b^5*d^5*e^2 + 40*B*a^3*b^3*d^4*e^3 + 30*A*a^2*b^4
*d^4*e^3 + 15*B*a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 + 6*B*a^5*b*d^2*e^5 + 15*A*a^4*b^2*d^2*e^5 + 2*B*a^6*d*
e^6 + 12*A*a^5*b*d*e^6 + 10*A*a^6*e^7 + 180*(7*B*b^6*d^2*e^5 - 12*B*a*b^5*d*e^6 - 2*A*b^6*d*e^6 + 5*B*a^2*b^4*
e^7 + 2*A*a*b^5*e^7)*x^5 + 150*(35*B*b^6*d^3*e^4 - 54*B*a*b^5*d^2*e^5 - 9*A*b^6*d^2*e^5 + 15*B*a^2*b^4*d*e^6 +
6*A*a*b^5*d*e^6 + 4*B*a^3*b^3*e^7 + 3*A*a^2*b^4*e^7)*x^4 + 100*(91*B*b^6*d^4*e^3 - 132*B*a*b^5*d^3*e^4 - 22*A
*b^6*d^3*e^4 + 30*B*a^2*b^4*d^2*e^5 + 12*A*a*b^5*d^2*e^5 + 8*B*a^3*b^3*d*e^6 + 6*A*a^2*b^4*d*e^6 + 3*B*a^4*b^2
*e^7 + 4*A*a^3*b^3*e^7)*x^3 + 15*(539*B*b^6*d^5*e^2 - 750*B*a*b^5*d^4*e^3 - 125*A*b^6*d^4*e^3 + 150*B*a^2*b^4*
d^3*e^4 + 60*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 + 6*B*a^5*b*e^7 + 15*A*a^4*b^2*e^7)*x^2 + 6*(609*B*b^6*d^6*e - 822*B*a*b^5*d^5*e^2 - 137*A*b^6*d^5*e^2
+ 150*B*a^2*b^4*d^4*e^3 + 60*A*a*b^5*d^4*e^3 + 40*B*a^3*b^3*d^3*e^4 + 30*A*a^2*b^4*d^3*e^4 + 15*B*a^4*b^2*d^2*
e^5 + 20*A*a^3*b^3*d^2*e^5 + 6*B*a^5*b*d*e^6 + 15*A*a^4*b^2*d*e^6 + 2*B*a^6*e^7 + 12*A*a^5*b*e^7)*x)*e^(-8)/(x
*e + d)^6