3.1689 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=185 \[ \frac{b^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac{b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((5*b*B*d + 3*A*b*e -
8*a*B*e)*(a + b*x)^5)/(56*e*(b*d - a*e)^2*(d + e*x)^7) + (b*(5*b*B*d + 3*A*b*e
- 8*a*B*e)*(a + b*x)^5)/(168*e*(b*d - a*e)^3*(d + e*x)^6) + (b^2*(5*b*B*d + 3*A*
b*e - 8*a*B*e)*(a + b*x)^5)/(840*e*(b*d - a*e)^4*(d + e*x)^5)

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Rubi [A]  time = 0.242069, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{b^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac{b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac{(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((5*b*B*d + 3*A*b*e -
8*a*B*e)*(a + b*x)^5)/(56*e*(b*d - a*e)^2*(d + e*x)^7) + (b*(5*b*B*d + 3*A*b*e
- 8*a*B*e)*(a + b*x)^5)/(168*e*(b*d - a*e)^3*(d + e*x)^6) + (b^2*(5*b*B*d + 3*A*
b*e - 8*a*B*e)*(a + b*x)^5)/(840*e*(b*d - a*e)^4*(d + e*x)^5)

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Rubi in Sympy [A]  time = 74.798, size = 172, normalized size = 0.93 \[ \frac{b^{2} \left (a + b x\right )^{5} \left (3 A b e - 8 B a e + 5 B b d\right )}{840 e \left (d + e x\right )^{5} \left (a e - b d\right )^{4}} - \frac{b \left (a + b x\right )^{5} \left (3 A b e - 8 B a e + 5 B b d\right )}{168 e \left (d + e x\right )^{6} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{5} \left (3 A b e - 8 B a e + 5 B b d\right )}{56 e \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{5} \left (A e - B d\right )}{8 e \left (d + e x\right )^{8} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**9,x)

[Out]

b**2*(a + b*x)**5*(3*A*b*e - 8*B*a*e + 5*B*b*d)/(840*e*(d + e*x)**5*(a*e - b*d)*
*4) - b*(a + b*x)**5*(3*A*b*e - 8*B*a*e + 5*B*b*d)/(168*e*(d + e*x)**6*(a*e - b*
d)**3) + (a + b*x)**5*(3*A*b*e - 8*B*a*e + 5*B*b*d)/(56*e*(d + e*x)**7*(a*e - b*
d)**2) - (a + b*x)**5*(A*e - B*d)/(8*e*(d + e*x)**8*(a*e - b*d))

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Mathematica [A]  time = 0.303475, size = 320, normalized size = 1.73 \[ -\frac{15 a^4 e^4 (7 A e+B (d+8 e x))+20 a^3 b e^3 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+12 a b^3 e \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+b^4 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^9,x]

[Out]

-(15*a^4*e^4*(7*A*e + B*(d + 8*e*x)) + 20*a^3*b*e^3*(3*A*e*(d + 8*e*x) + B*(d^2
+ 8*d*e*x + 28*e^2*x^2)) + 6*a^2*b^2*e^2*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3
*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 12*a*b^3*e*(A*e*(d^3 + 8*d^2
*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e
^3*x^3 + 70*e^4*x^4)) + b^4*(3*A*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*
x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*
d*e^4*x^4 + 56*e^5*x^5)))/(840*e^6*(d + e*x)^8)

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Maple [B]  time = 0.011, size = 430, normalized size = 2.3 \[ -{\frac{2\,{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,{b}^{2}B{d}^{2} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{4}B}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-B{a}^{4}d{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}} \]

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)^9,x)

[Out]

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

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Maxima [A]  time = 0.72239, size = 660, normalized size = 3.57 \[ -\frac{280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \,{\left (5 \, B b^{4} d e^{4} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B b^{4} d^{2} e^{3} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B b^{4} d^{3} e^{2} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} d^{4} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A*b^4)*
d^4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e
^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4*B*a*b^3 + A*b^4)*e^
5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 6*(3*B*a^2*b^2 + 2*
A*a*b^3)*e^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B
*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4
*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 1
0*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8
*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x
^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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Fricas [A]  time = 0.286919, size = 660, normalized size = 3.57 \[ -\frac{280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \,{\left (5 \, B b^{4} d e^{4} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \,{\left (5 \, B b^{4} d^{2} e^{3} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \,{\left (5 \, B b^{4} d^{3} e^{2} + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \,{\left (5 \, B b^{4} d^{4} e + 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \,{\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A*b^4)*
d^4*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e
^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4*B*a*b^3 + A*b^4)*e^
5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 6*(3*B*a^2*b^2 + 2*
A*a*b^3)*e^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B
*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4
*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 1
0*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8
*d*e^13*x^7 + 28*d^2*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x
^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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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)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280782, size = 594, normalized size = 3.21 \[ -\frac{{\left (280 \, B b^{4} x^{5} e^{5} + 350 \, B b^{4} d x^{4} e^{4} + 280 \, B b^{4} d^{2} x^{3} e^{3} + 140 \, B b^{4} d^{3} x^{2} e^{2} + 40 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 840 \, B a b^{3} x^{4} e^{5} + 210 \, A b^{4} x^{4} e^{5} + 672 \, B a b^{3} d x^{3} e^{4} + 168 \, A b^{4} d x^{3} e^{4} + 336 \, B a b^{3} d^{2} x^{2} e^{3} + 84 \, A b^{4} d^{2} x^{2} e^{3} + 96 \, B a b^{3} d^{3} x e^{2} + 24 \, A b^{4} d^{3} x e^{2} + 12 \, B a b^{3} d^{4} e + 3 \, A b^{4} d^{4} e + 1008 \, B a^{2} b^{2} x^{3} e^{5} + 672 \, A a b^{3} x^{3} e^{5} + 504 \, B a^{2} b^{2} d x^{2} e^{4} + 336 \, A a b^{3} d x^{2} e^{4} + 144 \, B a^{2} b^{2} d^{2} x e^{3} + 96 \, A a b^{3} d^{2} x e^{3} + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} + 560 \, B a^{3} b x^{2} e^{5} + 840 \, A a^{2} b^{2} x^{2} e^{5} + 160 \, B a^{3} b d x e^{4} + 240 \, A a^{2} b^{2} d x e^{4} + 20 \, B a^{3} b d^{2} e^{3} + 30 \, A a^{2} b^{2} d^{2} e^{3} + 120 \, B a^{4} x e^{5} + 480 \, A a^{3} b x e^{5} + 15 \, B a^{4} d e^{4} + 60 \, A a^{3} b d e^{4} + 105 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{840 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/840*(280*B*b^4*x^5*e^5 + 350*B*b^4*d*x^4*e^4 + 280*B*b^4*d^2*x^3*e^3 + 140*B*
b^4*d^3*x^2*e^2 + 40*B*b^4*d^4*x*e + 5*B*b^4*d^5 + 840*B*a*b^3*x^4*e^5 + 210*A*b
^4*x^4*e^5 + 672*B*a*b^3*d*x^3*e^4 + 168*A*b^4*d*x^3*e^4 + 336*B*a*b^3*d^2*x^2*e
^3 + 84*A*b^4*d^2*x^2*e^3 + 96*B*a*b^3*d^3*x*e^2 + 24*A*b^4*d^3*x*e^2 + 12*B*a*b
^3*d^4*e + 3*A*b^4*d^4*e + 1008*B*a^2*b^2*x^3*e^5 + 672*A*a*b^3*x^3*e^5 + 504*B*
a^2*b^2*d*x^2*e^4 + 336*A*a*b^3*d*x^2*e^4 + 144*B*a^2*b^2*d^2*x*e^3 + 96*A*a*b^3
*d^2*x*e^3 + 18*B*a^2*b^2*d^3*e^2 + 12*A*a*b^3*d^3*e^2 + 560*B*a^3*b*x^2*e^5 + 8
40*A*a^2*b^2*x^2*e^5 + 160*B*a^3*b*d*x*e^4 + 240*A*a^2*b^2*d*x*e^4 + 20*B*a^3*b*
d^2*e^3 + 30*A*a^2*b^2*d^2*e^3 + 120*B*a^4*x*e^5 + 480*A*a^3*b*x*e^5 + 15*B*a^4*
d*e^4 + 60*A*a^3*b*d*e^4 + 105*A*a^4*e^5)*e^(-6)/(x*e + d)^8