3.1690 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=206 \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{5 e^6 (d+e x)^5}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6 (d+e x)^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6 (d+e x)^8}+\frac{(b d-a e)^4 (B d-A e)}{9 e^6 (d+e x)^9}-\frac{b^4 B}{4 e^6 (d+e x)^4} \]

[Out]

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

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Rubi [A]  time = 0.572365, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{b^3 (-4 a B e-A b e+5 b B d)}{5 e^6 (d+e x)^5}-\frac{b^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{3 e^6 (d+e x)^6}+\frac{2 b (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{7 e^6 (d+e x)^7}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{8 e^6 (d+e x)^8}+\frac{(b d-a e)^4 (B d-A e)}{9 e^6 (d+e x)^9}-\frac{b^4 B}{4 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^10,x]

[Out]

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

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Rubi in Sympy [A]  time = 136.54, size = 204, normalized size = 0.99 \[ - \frac{B b^{4}}{4 e^{6} \left (d + e x\right )^{4}} - \frac{b^{3} \left (A b e + 4 B a e - 5 B b d\right )}{5 e^{6} \left (d + e x\right )^{5}} - \frac{b^{2} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{3 e^{6} \left (d + e x\right )^{6}} - \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{7 e^{6} \left (d + e x\right )^{7}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{8 e^{6} \left (d + e x\right )^{8}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{9 e^{6} \left (d + e x\right )^{9}} \]

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)**10,x)

[Out]

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

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Mathematica [A]  time = 0.338084, size = 322, normalized size = 1.56 \[ -\frac{35 a^4 e^4 (8 A e+B (d+9 e x))+20 a^3 b e^3 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+30 a^2 b^2 e^2 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a b^3 e \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+b^4 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )}{2520 e^6 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^10,x]

[Out]

-(35*a^4*e^4*(8*A*e + B*(d + 9*e*x)) + 20*a^3*b*e^3*(7*A*e*(d + 9*e*x) + 2*B*(d^
2 + 9*d*e*x + 36*e^2*x^2)) + 30*a^2*b^2*e^2*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2)
+ B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 4*a*b^3*e*(5*A*e*(d^3 + 9*d
^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84
*d*e^3*x^3 + 126*e^4*x^4)) + b^4*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d
*e^3*x^3 + 126*e^4*x^4) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3
+ 126*d*e^4*x^4 + 126*e^5*x^5)))/(2520*e^6*(d + e*x)^9)

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Maple [B]  time = 0.011, size = 430, normalized size = 2.1 \[ -{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{4}B}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{2\,b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,{b}^{2}B{d}^{2} \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{8\,{e}^{6} \left ( ex+d \right ) ^{8}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-B{a}^{4}d{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{9\,{e}^{6} \left ( ex+d \right ) ^{9}}} \]

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)^10,x)

[Out]

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

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Maxima [A]  time = 0.72136, size = 675, normalized size = 3.28 \[ -\frac{630 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 280 \, A a^{4} e^{5} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 126 \,{\left (5 \, B b^{4} d e^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 84 \,{\left (5 \, B b^{4} d^{2} e^{3} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 36 \,{\left (5 \, B b^{4} d^{3} e^{2} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 9 \,{\left (5 \, B b^{4} d^{4} e + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2520 \,{\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]

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)^10,x, algorithm="maxima")

[Out]

-1/2520*(630*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 280*A*a^4*e^5 + 4*(4*B*a*b^3 + A*b^4)
*d^4*e + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d^2
*e^3 + 35*(B*a^4 + 4*A*a^3*b)*d*e^4 + 126*(5*B*b^4*d*e^4 + 4*(4*B*a*b^3 + A*b^4)
*e^5)*x^4 + 84*(5*B*b^4*d^2*e^3 + 4*(4*B*a*b^3 + A*b^4)*d*e^4 + 10*(3*B*a^2*b^2
+ 2*A*a*b^3)*e^5)*x^3 + 36*(5*B*b^4*d^3*e^2 + 4*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 10
*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 9*(5*
B*b^4*d^4*e + 4*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 35*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^15*x
^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d
^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*e^6)

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Fricas [A]  time = 0.268621, size = 675, normalized size = 3.28 \[ -\frac{630 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 280 \, A a^{4} e^{5} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 126 \,{\left (5 \, B b^{4} d e^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 84 \,{\left (5 \, B b^{4} d^{2} e^{3} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 36 \,{\left (5 \, B b^{4} d^{3} e^{2} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 9 \,{\left (5 \, B b^{4} d^{4} e + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 20 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 35 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2520 \,{\left (e^{15} x^{9} + 9 \, d e^{14} x^{8} + 36 \, d^{2} e^{13} x^{7} + 84 \, d^{3} e^{12} x^{6} + 126 \, d^{4} e^{11} x^{5} + 126 \, d^{5} e^{10} x^{4} + 84 \, d^{6} e^{9} x^{3} + 36 \, d^{7} e^{8} x^{2} + 9 \, d^{8} e^{7} x + d^{9} e^{6}\right )}} \]

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)^10,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 280*A*a^4*e^5 + 4*(4*B*a*b^3 + A*b^4)
*d^4*e + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d^2
*e^3 + 35*(B*a^4 + 4*A*a^3*b)*d*e^4 + 126*(5*B*b^4*d*e^4 + 4*(4*B*a*b^3 + A*b^4)
*e^5)*x^4 + 84*(5*B*b^4*d^2*e^3 + 4*(4*B*a*b^3 + A*b^4)*d*e^4 + 10*(3*B*a^2*b^2
+ 2*A*a*b^3)*e^5)*x^3 + 36*(5*B*b^4*d^3*e^2 + 4*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 10
*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 9*(5*
B*b^4*d^4*e + 4*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 35*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^15*x
^9 + 9*d*e^14*x^8 + 36*d^2*e^13*x^7 + 84*d^3*e^12*x^6 + 126*d^4*e^11*x^5 + 126*d
^5*e^10*x^4 + 84*d^6*e^9*x^3 + 36*d^7*e^8*x^2 + 9*d^8*e^7*x + d^9*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)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28274, size = 594, normalized size = 2.88 \[ -\frac{{\left (630 \, B b^{4} x^{5} e^{5} + 630 \, B b^{4} d x^{4} e^{4} + 420 \, B b^{4} d^{2} x^{3} e^{3} + 180 \, B b^{4} d^{3} x^{2} e^{2} + 45 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 2016 \, B a b^{3} x^{4} e^{5} + 504 \, A b^{4} x^{4} e^{5} + 1344 \, B a b^{3} d x^{3} e^{4} + 336 \, A b^{4} d x^{3} e^{4} + 576 \, B a b^{3} d^{2} x^{2} e^{3} + 144 \, A b^{4} d^{2} x^{2} e^{3} + 144 \, B a b^{3} d^{3} x e^{2} + 36 \, A b^{4} d^{3} x e^{2} + 16 \, B a b^{3} d^{4} e + 4 \, A b^{4} d^{4} e + 2520 \, B a^{2} b^{2} x^{3} e^{5} + 1680 \, A a b^{3} x^{3} e^{5} + 1080 \, B a^{2} b^{2} d x^{2} e^{4} + 720 \, A a b^{3} d x^{2} e^{4} + 270 \, B a^{2} b^{2} d^{2} x e^{3} + 180 \, A a b^{3} d^{2} x e^{3} + 30 \, B a^{2} b^{2} d^{3} e^{2} + 20 \, A a b^{3} d^{3} e^{2} + 1440 \, B a^{3} b x^{2} e^{5} + 2160 \, A a^{2} b^{2} x^{2} e^{5} + 360 \, B a^{3} b d x e^{4} + 540 \, A a^{2} b^{2} d x e^{4} + 40 \, B a^{3} b d^{2} e^{3} + 60 \, A a^{2} b^{2} d^{2} e^{3} + 315 \, B a^{4} x e^{5} + 1260 \, A a^{3} b x e^{5} + 35 \, B a^{4} d e^{4} + 140 \, A a^{3} b d e^{4} + 280 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{2520 \,{\left (x e + d\right )}^{9}} \]

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)^10,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^4*x^5*e^5 + 630*B*b^4*d*x^4*e^4 + 420*B*b^4*d^2*x^3*e^3 + 180*B
*b^4*d^3*x^2*e^2 + 45*B*b^4*d^4*x*e + 5*B*b^4*d^5 + 2016*B*a*b^3*x^4*e^5 + 504*A
*b^4*x^4*e^5 + 1344*B*a*b^3*d*x^3*e^4 + 336*A*b^4*d*x^3*e^4 + 576*B*a*b^3*d^2*x^
2*e^3 + 144*A*b^4*d^2*x^2*e^3 + 144*B*a*b^3*d^3*x*e^2 + 36*A*b^4*d^3*x*e^2 + 16*
B*a*b^3*d^4*e + 4*A*b^4*d^4*e + 2520*B*a^2*b^2*x^3*e^5 + 1680*A*a*b^3*x^3*e^5 +
1080*B*a^2*b^2*d*x^2*e^4 + 720*A*a*b^3*d*x^2*e^4 + 270*B*a^2*b^2*d^2*x*e^3 + 180
*A*a*b^3*d^2*x*e^3 + 30*B*a^2*b^2*d^3*e^2 + 20*A*a*b^3*d^3*e^2 + 1440*B*a^3*b*x^
2*e^5 + 2160*A*a^2*b^2*x^2*e^5 + 360*B*a^3*b*d*x*e^4 + 540*A*a^2*b^2*d*x*e^4 + 4
0*B*a^3*b*d^2*e^3 + 60*A*a^2*b^2*d^2*e^3 + 315*B*a^4*x*e^5 + 1260*A*a^3*b*x*e^5
+ 35*B*a^4*d*e^4 + 140*A*a^3*b*d*e^4 + 280*A*a^4*e^5)*e^(-6)/(x*e + d)^9