3.1688 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=135 \[ \frac{b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac{(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(7*e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(42*e*(
b*d - a*e)^2*(d + e*x)^6) + (b*(5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(210*e*(b*d - a*e)^3*(d + e*x)^5)

________________________________________________________________________________________

Rubi [A]  time = 0.0587619, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {27, 78, 45, 37} \[ \frac{b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac{(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(7*e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(42*e*(
b*d - a*e)^2*(d + e*x)^6) + (b*(5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(210*e*(b*d - a*e)^3*(d + e*x)^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 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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac{(a+b x)^4 (A+B x)}{(d+e x)^8} \, dx\\ &=-\frac{(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac{(5 b B d+2 A b e-7 a B e) \int \frac{(a+b x)^4}{(d+e x)^7} \, dx}{7 e (b d-a e)}\\ &=-\frac{(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac{(5 b B d+2 A b e-7 a B e) (a+b x)^5}{42 e (b d-a e)^2 (d+e x)^6}+\frac{(b (5 b B d+2 A b e-7 a B e)) \int \frac{(a+b x)^4}{(d+e x)^6} \, dx}{42 e (b d-a e)^2}\\ &=-\frac{(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac{(5 b B d+2 A b e-7 a B e) (a+b x)^5}{42 e (b d-a e)^2 (d+e x)^6}+\frac{b (5 b B d+2 A b e-7 a B e) (a+b x)^5}{210 e (b d-a e)^3 (d+e x)^5}\\ \end{align*}

Mathematica [B]  time = 0.139883, size = 323, normalized size = 2.39 \[ -\frac{3 a^2 b^2 e^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+4 a^3 b e^3 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+5 a^4 e^4 (6 A e+B (d+7 e x))+2 a b^3 e \left (3 A e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+b^4 \left (2 A e \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-(5*a^4*e^4*(6*A*e + B*(d + 7*e*x)) + 4*a^3*b*e^3*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 3*a
^2*b^2*e^2*(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)) + 2*a*b^3*
e*(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)) + b^4*(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)))/(210*e^6*(d + e*x)^7)

________________________________________________________________________________________

Maple [B]  time = 0.007, size = 430, normalized size = 3.2 \begin{align*} -{\frac{{b}^{4}B}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,B{b}^{2}{d}^{2} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-Bd{a}^{4}{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{2\,b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x)

[Out]

-1/2*b^4*B/e^6/(e*x+d)^2-1/2*b^2*(2*A*a*b*e^2-2*A*b^2*d*e+3*B*a^2*e^2-8*B*a*b*d*e+5*B*b^2*d^2)/e^6/(e*x+d)^4-1
/6*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+B*a^4*e^4-8*B*a^3*b*d*e^3+18*B*a^2*b^2*d
^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/e^6/(e*x+d)^6-1/7*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3-4*A*a*b^
3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^6/(e*x+d)
^7-2/5*b*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-9*B*a^2*b*d*e^2+12*B*a*b^2*d^2*e-5*B*b^3*d^3
)/e^6/(e*x+d)^5-1/3*b^3*(A*b*e+4*B*a*e-5*B*b*d)/e^6/(e*x+d)^3

________________________________________________________________________________________

Maxima [B]  time = 1.16433, size = 645, normalized size = 4.78 \begin{align*} -\frac{105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \,{\left (5 \, B b^{4} d e^{4} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B b^{4} d^{2} e^{3} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \,{\left (5 \, B b^{4} d^{3} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \,{\left (5 \, B b^{4} d^{4} e + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*
a*b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)
*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b
+ 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*
x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

________________________________________________________________________________________

Fricas [B]  time = 1.45341, size = 1002, normalized size = 7.42 \begin{align*} -\frac{105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \,{\left (5 \, B b^{4} d e^{4} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B b^{4} d^{2} e^{3} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \,{\left (5 \, B b^{4} d^{3} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \,{\left (5 \, B b^{4} d^{4} e + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*
a*b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)
*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b
+ 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*
x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

________________________________________________________________________________________

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)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.13786, size = 594, normalized size = 4.4 \begin{align*} -\frac{{\left (105 \, B b^{4} x^{5} e^{5} + 175 \, B b^{4} d x^{4} e^{4} + 175 \, B b^{4} d^{2} x^{3} e^{3} + 105 \, B b^{4} d^{3} x^{2} e^{2} + 35 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 280 \, B a b^{3} x^{4} e^{5} + 70 \, A b^{4} x^{4} e^{5} + 280 \, B a b^{3} d x^{3} e^{4} + 70 \, A b^{4} d x^{3} e^{4} + 168 \, B a b^{3} d^{2} x^{2} e^{3} + 42 \, A b^{4} d^{2} x^{2} e^{3} + 56 \, B a b^{3} d^{3} x e^{2} + 14 \, A b^{4} d^{3} x e^{2} + 8 \, B a b^{3} d^{4} e + 2 \, A b^{4} d^{4} e + 315 \, B a^{2} b^{2} x^{3} e^{5} + 210 \, A a b^{3} x^{3} e^{5} + 189 \, B a^{2} b^{2} d x^{2} e^{4} + 126 \, A a b^{3} d x^{2} e^{4} + 63 \, B a^{2} b^{2} d^{2} x e^{3} + 42 \, A a b^{3} d^{2} x e^{3} + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 168 \, B a^{3} b x^{2} e^{5} + 252 \, A a^{2} b^{2} x^{2} e^{5} + 56 \, B a^{3} b d x e^{4} + 84 \, A a^{2} b^{2} d x e^{4} + 8 \, B a^{3} b d^{2} e^{3} + 12 \, A a^{2} b^{2} d^{2} e^{3} + 35 \, B a^{4} x e^{5} + 140 \, A a^{3} b x e^{5} + 5 \, B a^{4} d e^{4} + 20 \, A a^{3} b d e^{4} + 30 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{210 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/210*(105*B*b^4*x^5*e^5 + 175*B*b^4*d*x^4*e^4 + 175*B*b^4*d^2*x^3*e^3 + 105*B*b^4*d^3*x^2*e^2 + 35*B*b^4*d^4
*x*e + 5*B*b^4*d^5 + 280*B*a*b^3*x^4*e^5 + 70*A*b^4*x^4*e^5 + 280*B*a*b^3*d*x^3*e^4 + 70*A*b^4*d*x^3*e^4 + 168
*B*a*b^3*d^2*x^2*e^3 + 42*A*b^4*d^2*x^2*e^3 + 56*B*a*b^3*d^3*x*e^2 + 14*A*b^4*d^3*x*e^2 + 8*B*a*b^3*d^4*e + 2*
A*b^4*d^4*e + 315*B*a^2*b^2*x^3*e^5 + 210*A*a*b^3*x^3*e^5 + 189*B*a^2*b^2*d*x^2*e^4 + 126*A*a*b^3*d*x^2*e^4 +
63*B*a^2*b^2*d^2*x*e^3 + 42*A*a*b^3*d^2*x*e^3 + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 168*B*a^3*b*x^2*e^5
+ 252*A*a^2*b^2*x^2*e^5 + 56*B*a^3*b*d*x*e^4 + 84*A*a^2*b^2*d*x*e^4 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3
+ 35*B*a^4*x*e^5 + 140*A*a^3*b*x*e^5 + 5*B*a^4*d*e^4 + 20*A*a^3*b*d*e^4 + 30*A*a^4*e^5)*e^(-6)/(x*e + d)^7