3.1067 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx\)

Optimal. Leaf size=213 \[ -\frac{(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}+\frac{6 b^5 B (b d-a e)}{e^8 (d+e x)}-\frac{15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac{20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac{15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac{6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac{B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac{b^6 B \log (d+e x)}{e^8} \]

[Out]

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

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Rubi [A]  time = 0.256113, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 43} \[ -\frac{(a+b x)^7 (B d-A e)}{7 e (d+e x)^7 (b d-a e)}+\frac{6 b^5 B (b d-a e)}{e^8 (d+e x)}-\frac{15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac{20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac{15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac{6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac{B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac{b^6 B \log (d+e x)}{e^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^6 (A+B x)}{(d+e x)^8} \, dx &=-\frac{(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac{B \int \frac{(a+b x)^6}{(d+e x)^7} \, dx}{e}\\ &=-\frac{(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}+\frac{B \int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^7}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac{b^6}{e^6 (d+e x)}\right ) \, dx}{e}\\ &=-\frac{(B d-A e) (a+b x)^7}{7 e (b d-a e) (d+e x)^7}-\frac{B (b d-a e)^6}{6 e^8 (d+e x)^6}+\frac{6 b B (b d-a e)^5}{5 e^8 (d+e x)^5}-\frac{15 b^2 B (b d-a e)^4}{4 e^8 (d+e x)^4}+\frac{20 b^3 B (b d-a e)^3}{3 e^8 (d+e x)^3}-\frac{15 b^4 B (b d-a e)^2}{2 e^8 (d+e x)^2}+\frac{6 b^5 B (b d-a e)}{e^8 (d+e x)}+\frac{b^6 B \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [B]  time = 0.462472, size = 615, normalized size = 2.89 \[ -\frac{15 a^4 b^2 e^4 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (2 A e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )+12 a^5 b e^5 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+10 a^6 e^6 (6 A e+B (d+7 e x))+60 a b^5 e \left (A e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+6 B \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )\right )+b^6 \left (60 A e \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )-B d \left (20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+7203 d^5 e x+1089 d^6+13230 d e^5 x^5+2940 e^6 x^6\right )\right )-420 b^6 B (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^8,x]

[Out]

-(10*a^6*e^6*(6*A*e + B*(d + 7*e*x)) + 12*a^5*b*e^5*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 1
5*a^4*b^2*e^4*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 20*a^
3*b^3*e^3*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*
e^3*x^3 + 35*e^4*x^4)) + 30*a^2*b^4*e^2*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)
+ 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) + 60*a*b^5*e*(A*e*(d^5
+ 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 6*B*(d^6 + 7*d^5*e*x + 21*d^4*e^2
*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + b^6*(60*A*e*(d^6 + 7*d^5*e*x + 21*d^4*e^
2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - B*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4
*e^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) - 420*b^6*B*(d + e*x)^7*Lo
g[d + e*x])/(420*e^8*(d + e*x)^7)

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Maple [B]  time = 0.011, size = 1227, normalized size = 5.8 \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)^8,x)

[Out]

5/e^3/(e*x+d)^6*A*d*a^4*b^2+12*b^3/e^4/(e*x+d)^5*A*a^3*d-18*b^4/e^5/(e*x+d)^5*A*a^2*d^2+12*b^5/e^6/(e*x+d)^5*A
*a*d^3+9*b^2/e^4/(e*x+d)^5*B*a^4*d-24*b^3/e^5/(e*x+d)^5*B*a^3*d^2+30*b^4/e^6/(e*x+d)^5*B*a^2*d^3-18*b^5/e^7/(e
*x+d)^5*B*a*d^4+15*b^4/e^5/(e*x+d)^4*A*a^2*d-15*b^5/e^6/(e*x+d)^4*A*a*d^2+20*b^3/e^5/(e*x+d)^4*B*a^3*d-75/2*b^
4/e^6/(e*x+d)^4*B*a^2*d^2+30*b^5/e^7/(e*x+d)^4*B*a*d^3-10/e^4/(e*x+d)^6*A*d^2*a^3*b^3+10/e^5/(e*x+d)^6*A*d^3*a
^2*b^4+6/7/e^6/(e*x+d)^7*A*d^5*a*b^5-6/7/e^3/(e*x+d)^7*B*d^2*a^5*b+15/7/e^4/(e*x+d)^7*B*d^3*a^4*b^2-20/7/e^5/(
e*x+d)^7*B*d^4*a^3*b^3+15/7/e^6/(e*x+d)^7*B*d^5*a^2*b^4-6/7/e^7/(e*x+d)^7*B*d^6*a*b^5-15/7/e^3/(e*x+d)^7*A*d^2
*a^4*b^2+20/7/e^4/(e*x+d)^7*A*d^3*a^3*b^3-15/7/e^5/(e*x+d)^7*A*d^4*a^2*b^4-30*b^5/e^7/(e*x+d)^3*B*a*d^2+6/7/e^
2/(e*x+d)^7*A*d*a^5*b-15/2*b^4/e^6/(e*x+d)^2*B*a^2-21/2*b^6/e^8/(e*x+d)^2*B*d^2-3*b^2/e^3/(e*x+d)^5*A*a^4-6*b^
5/e^7/(e*x+d)*B*a+7*b^6/e^8/(e*x+d)*B*d-1/e^2/(e*x+d)^6*a^5*b*A+1/e^7/(e*x+d)^6*A*d^5*b^6-b^6/e^7/(e*x+d)*A-1/
7/e/(e*x+d)^7*a^6*A-1/6/e^2/(e*x+d)^6*B*a^6+21/5*b^6/e^8/(e*x+d)^5*B*d^5-3*b^6/e^7/(e*x+d)^5*A*d^4+5*b^6/e^7/(
e*x+d)^4*A*d^3-6/5*b/e^3/(e*x+d)^5*B*a^5+1/7/e^2/(e*x+d)^7*B*d*a^6+1/7/e^8/(e*x+d)^7*b^6*B*d^7-20/3*b^3/e^5/(e
*x+d)^3*B*a^3-15/4*b^2/e^4/(e*x+d)^4*B*a^4-35/4*b^6/e^8/(e*x+d)^4*B*d^4-5*b^3/e^4/(e*x+d)^4*A*a^3-7/6/e^8/(e*x
+d)^6*b^6*B*d^6-5*b^4/e^5/(e*x+d)^3*A*a^2-5*b^6/e^7/(e*x+d)^3*A*d^2-3*b^5/e^6/(e*x+d)^2*A*a+3*b^6/e^7/(e*x+d)^
2*A*d+35/3*b^6/e^8/(e*x+d)^3*B*d^3-1/7/e^7/(e*x+d)^7*A*d^6*b^6-5/e^6/(e*x+d)^6*A*d^4*a*b^5+2/e^3/(e*x+d)^6*B*d
*a^5*b-15/2/e^4/(e*x+d)^6*B*d^2*a^4*b^2+40/3/e^5/(e*x+d)^6*B*d^3*a^3*b^3-25/2/e^6/(e*x+d)^6*B*d^4*a^2*b^4+6/e^
7/(e*x+d)^6*B*d^5*a*b^5+10*b^5/e^6/(e*x+d)^3*A*a*d+25*b^4/e^6/(e*x+d)^3*B*a^2*d+18*b^5/e^7/(e*x+d)^2*B*a*d+b^6
*B*ln(e*x+d)/e^8

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Maxima [B]  time = 1.3296, size = 1137, normalized size = 5.34 \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)^8,x, algorithm="maxima")

[Out]

1/420*(1089*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(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 - 2
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 15*(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 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 + 420*(7*B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 630*(21*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 + 350*(77*B*b^6*d^3*e^4 - 6*(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 - 2*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 175*(175*B*b^
6*d^4*e^3 - 12*(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*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 21*(959*B*b^6*d^5*e^2 - 60*(6*B*a*b^5 + A*b^6)*d^4*e^3 - 3
0*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*
d*e^6 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 7*(1029*B*b^6*d^6*e - 60*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
5 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 10*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13
*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8) + B*b^6*log(e*x + d)/e^8

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Fricas [B]  time = 1.98316, size = 1987, normalized size = 9.33 \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)^8,x, algorithm="fricas")

[Out]

1/420*(1089*B*b^6*d^7 - 60*A*a^6*e^7 - 60*(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 - 2
0*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 15*(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 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 + 420*(7*B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 630*(21*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 + 350*(77*B*b^6*d^3*e^4 - 6*(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 - 2*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 175*(175*B*b^
6*d^4*e^3 - 12*(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*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 21*(959*B*b^6*d^5*e^2 - 60*(6*B*a*b^5 + A*b^6)*d^4*e^3 - 3
0*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*
d*e^6 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 7*(1029*B*b^6*d^6*e - 60*(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 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^
5 - 12*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 - 10*(B*a^6 + 6*A*a^5*b)*e^7)*x + 420*(B*b^6*e^7*x^7 + 7*B*b^6*d*e^6*x^
6 + 21*B*b^6*d^2*e^5*x^5 + 35*B*b^6*d^3*e^4*x^4 + 35*B*b^6*d^4*e^3*x^3 + 21*B*b^6*d^5*e^2*x^2 + 7*B*b^6*d^6*e*
x + B*b^6*d^7)*log(e*x + d))/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 +
21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*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)**8,x)

[Out]

Timed out

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Giac [B]  time = 2.58758, size = 1052, normalized size = 4.94 \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)^8,x, algorithm="giac")

[Out]

B*b^6*e^(-8)*log(abs(x*e + d)) + 1/420*(420*(7*B*b^6*d*e^5 - 6*B*a*b^5*e^6 - A*b^6*e^6)*x^6 + 630*(21*B*b^6*d^
2*e^4 - 12*B*a*b^5*d*e^5 - 2*A*b^6*d*e^5 - 5*B*a^2*b^4*e^6 - 2*A*a*b^5*e^6)*x^5 + 350*(77*B*b^6*d^3*e^3 - 36*B
*a*b^5*d^2*e^4 - 6*A*b^6*d^2*e^4 - 15*B*a^2*b^4*d*e^5 - 6*A*a*b^5*d*e^5 - 8*B*a^3*b^3*e^6 - 6*A*a^2*b^4*e^6)*x
^4 + 175*(175*B*b^6*d^4*e^2 - 72*B*a*b^5*d^3*e^3 - 12*A*b^6*d^3*e^3 - 30*B*a^2*b^4*d^2*e^4 - 12*A*a*b^5*d^2*e^
4 - 16*B*a^3*b^3*d*e^5 - 12*A*a^2*b^4*d*e^5 - 9*B*a^4*b^2*e^6 - 12*A*a^3*b^3*e^6)*x^3 + 21*(959*B*b^6*d^5*e -
360*B*a*b^5*d^4*e^2 - 60*A*b^6*d^4*e^2 - 150*B*a^2*b^4*d^3*e^3 - 60*A*a*b^5*d^3*e^3 - 80*B*a^3*b^3*d^2*e^4 - 6
0*A*a^2*b^4*d^2*e^4 - 45*B*a^4*b^2*d*e^5 - 60*A*a^3*b^3*d*e^5 - 24*B*a^5*b*e^6 - 60*A*a^4*b^2*e^6)*x^2 + 7*(10
29*B*b^6*d^6 - 360*B*a*b^5*d^5*e - 60*A*b^6*d^5*e - 150*B*a^2*b^4*d^4*e^2 - 60*A*a*b^5*d^4*e^2 - 80*B*a^3*b^3*
d^3*e^3 - 60*A*a^2*b^4*d^3*e^3 - 45*B*a^4*b^2*d^2*e^4 - 60*A*a^3*b^3*d^2*e^4 - 24*B*a^5*b*d*e^5 - 60*A*a^4*b^2
*d*e^5 - 10*B*a^6*e^6 - 60*A*a^5*b*e^6)*x + (1089*B*b^6*d^7 - 360*B*a*b^5*d^6*e - 60*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 - 80*B*a^3*b^3*d^4*e^3 - 60*A*a^2*b^4*d^4*e^3 - 45*B*a^4*b^2*d^3*e^4 - 60*A*a^
3*b^3*d^3*e^4 - 24*B*a^5*b*d^2*e^5 - 60*A*a^4*b^2*d^2*e^5 - 10*B*a^6*d*e^6 - 60*A*a^5*b*d*e^6 - 60*A*a^6*e^7)*
e^(-1))*e^(-7)/(x*e + d)^7