3.1122 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=238 \[ -\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^3}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

[Out]

-((c*(4*B*c*d - 2*b*B*e - A*c*e)*x)/e^5) + (B*c^2*x^2)/(2*e^4) + (d^2*(B*d - A*e
)*(c*d - b*e)^2)/(3*e^6*(d + e*x)^3) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A
*e*(2*c*d - b*e)))/(2*e^6*(d + e*x)^2) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)
- B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(e^6*(d + e*x)) - ((2*A*c*e*(2*c*d
- b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*Log[d + e*x])/e^6

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Rubi [A]  time = 0.734743, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\log (d+e x) \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{3 e^6 (d+e x)^3}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{2 e^6 (d+e x)^2}-\frac{c x (-A c e-2 b B e+4 B c d)}{e^5}+\frac{B c^2 x^2}{2 e^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

-((c*(4*B*c*d - 2*b*B*e - A*c*e)*x)/e^5) + (B*c^2*x^2)/(2*e^4) + (d^2*(B*d - A*e
)*(c*d - b*e)^2)/(3*e^6*(d + e*x)^3) - (d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A
*e*(2*c*d - b*e)))/(2*e^6*(d + e*x)^2) - (A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)
- B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(e^6*(d + e*x)) - ((2*A*c*e*(2*c*d
- b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*Log[d + e*x])/e^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{2} \int x\, dx}{e^{4}} - \frac{d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{3 e^{6} \left (d + e x\right )^{3}} + \frac{d \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{2 e^{6} \left (d + e x\right )^{2}} + \frac{\left (A c e + 2 B b e - 4 B c d\right ) \int c\, dx}{e^{5}} + \frac{\left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}}{e^{6} \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**4,x)

[Out]

B*c**2*Integral(x, x)/e**4 - d**2*(A*e - B*d)*(b*e - c*d)**2/(3*e**6*(d + e*x)**
3) + d*(b*e - c*d)*(2*A*b*e**2 - 4*A*c*d*e - 3*B*b*d*e + 5*B*c*d**2)/(2*e**6*(d
+ e*x)**2) + (A*c*e + 2*B*b*e - 4*B*c*d)*Integral(c, x)/e**5 + (2*A*b*c*e**2 - 4
*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)*log(d + e*x)/e**6 - (A
*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*
e - 10*B*c**2*d**3)/(e**6*(d + e*x))

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Mathematica [A]  time = 0.454207, size = 220, normalized size = 0.92 \[ \frac{-\frac{6 \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )+B d \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )}{d+e x}+6 \log (d+e x) \left (2 A c e (b e-2 c d)+B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )+\frac{2 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^3}-\frac{3 d (c d-b e) (2 A e (b e-2 c d)+B d (5 c d-3 b e))}{(d+e x)^2}+6 c e x (A c e+2 b B e-4 B c d)+3 B c^2 e^2 x^2}{6 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^4,x]

[Out]

(6*c*e*(-4*B*c*d + 2*b*B*e + A*c*e)*x + 3*B*c^2*e^2*x^2 + (2*d^2*(B*d - A*e)*(c*
d - b*e)^2)/(d + e*x)^3 - (3*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) + 2*A*e*(-2*c*d
+ b*e)))/(d + e*x)^2 - (6*(B*d*(-10*c^2*d^2 + 12*b*c*d*e - 3*b^2*e^2) + A*e*(6*c
^2*d^2 - 6*b*c*d*e + b^2*e^2)))/(d + e*x) + 6*(2*A*c*e*(-2*c*d + b*e) + B*(10*c^
2*d^2 - 8*b*c*d*e + b^2*e^2))*Log[d + e*x])/(6*e^6)

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Maple [A]  time = 0.017, size = 446, normalized size = 1.9 \[{\frac{{b}^{2}B\ln \left ( ex+d \right ) }{{e}^{4}}}+2\,{\frac{Bxbc}{{e}^{4}}}-4\,{\frac{B{c}^{2}dx}{{e}^{5}}}-{\frac{3\,B{d}^{2}{b}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{5\,B{c}^{2}{d}^{4}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+2\,{\frac{\ln \left ( ex+d \right ) Abc}{{e}^{4}}}+{\frac{2\,A{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{2\,B{d}^{4}bc}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-8\,{\frac{\ln \left ( ex+d \right ) Bbcd}{{e}^{5}}}-3\,{\frac{A{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}+4\,{\frac{Bbc{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+6\,{\frac{Abcd}{{e}^{4} \left ( ex+d \right ) }}-12\,{\frac{Bbc{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{B{c}^{2}{x}^{2}}{2\,{e}^{4}}}-{\frac{{b}^{2}A}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Ax{c}^{2}}{{e}^{4}}}+10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{2}}{{e}^{6}}}-6\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+3\,{\frac{{b}^{2}Bd}{{e}^{4} \left ( ex+d \right ) }}+10\,{\frac{B{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{d}^{2}A{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{4}A{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{B{d}^{3}{b}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{2}{d}^{5}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{dA{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}d}{{e}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^4,x)

[Out]

b^2*B*ln(e*x+d)/e^4+2*c/e^4*b*B*x-4*c^2/e^5*B*d*x-3/2*d^2/e^4/(e*x+d)^2*B*b^2-5/
2*d^4/e^6/(e*x+d)^2*B*c^2+2/e^4*ln(e*x+d)*A*b*c+2/3*d^3/e^4/(e*x+d)^3*A*b*c-2/3*
d^4/e^5/(e*x+d)^3*B*b*c-8/e^5*ln(e*x+d)*B*b*c*d-3*d^2/e^4/(e*x+d)^2*A*b*c+4*d^3/
e^5/(e*x+d)^2*B*b*c+6/e^4/(e*x+d)*A*b*c*d-12/e^5/(e*x+d)*B*b*c*d^2+1/2*B*c^2*x^2
/e^4-1/e^3/(e*x+d)*A*b^2+c^2/e^4*A*x+10/e^6*ln(e*x+d)*B*c^2*d^2-6/e^5/(e*x+d)*A*
c^2*d^2+3/e^4/(e*x+d)*B*b^2*d+10/e^6/(e*x+d)*B*c^2*d^3-1/3*d^2/e^3/(e*x+d)^3*A*b
^2-1/3*d^4/e^5/(e*x+d)^3*A*c^2+1/3*d^3/e^4/(e*x+d)^3*B*b^2+1/3*d^5/e^6/(e*x+d)^3
*B*c^2+d/e^3/(e*x+d)^2*A*b^2+2*d^3/e^5/(e*x+d)^2*A*c^2-4/e^5*ln(e*x+d)*A*c^2*d

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Maxima [A]  time = 0.698537, size = 420, normalized size = 1.76 \[ \frac{47 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 6 \,{\left (10 \, B c^{2} d^{3} e^{2} - A b^{2} e^{5} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 20 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{B c^{2} e x^{2} - 2 \,{\left (4 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )} x}{2 \, e^{5}} + \frac{{\left (10 \, B c^{2} d^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")

[Out]

1/6*(47*B*c^2*d^5 - 2*A*b^2*d^2*e^3 - 26*(2*B*b*c + A*c^2)*d^4*e + 11*(B*b^2 + 2
*A*b*c)*d^3*e^2 + 6*(10*B*c^2*d^3*e^2 - A*b^2*e^5 - 6*(2*B*b*c + A*c^2)*d^2*e^3
+ 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 3*(35*B*c^2*d^4*e - 2*A*b^2*d*e^4 - 20*(2*B*b
*c + A*c^2)*d^3*e^2 + 9*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)/(e^9*x^3 + 3*d*e^8*x^2 + 3
*d^2*e^7*x + d^3*e^6) + 1/2*(B*c^2*e*x^2 - 2*(4*B*c^2*d - (2*B*b*c + A*c^2)*e)*x
)/e^5 + (10*B*c^2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*log(e*x
+ d)/e^6

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Fricas [A]  time = 0.308014, size = 682, normalized size = 2.87 \[ \frac{3 \, B c^{2} e^{5} x^{5} + 47 \, B c^{2} d^{5} - 2 \, A b^{2} d^{2} e^{3} - 26 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 11 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 3 \,{\left (5 \, B c^{2} d e^{4} - 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} - 9 \,{\left (7 \, B c^{2} d^{2} e^{3} - 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, B c^{2} d^{3} e^{2} + 2 \, A b^{2} e^{5} + 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 6 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 3 \,{\left (27 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 9 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x + 6 \,{\left (10 \, B c^{2} d^{5} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} +{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 3 \,{\left (10 \, B c^{2} d^{3} e^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 3 \,{\left (10 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/6*(3*B*c^2*e^5*x^5 + 47*B*c^2*d^5 - 2*A*b^2*d^2*e^3 - 26*(2*B*b*c + A*c^2)*d^4
*e + 11*(B*b^2 + 2*A*b*c)*d^3*e^2 - 3*(5*B*c^2*d*e^4 - 2*(2*B*b*c + A*c^2)*e^5)*
x^4 - 9*(7*B*c^2*d^2*e^3 - 2*(2*B*b*c + A*c^2)*d*e^4)*x^3 - 3*(3*B*c^2*d^3*e^2 +
2*A*b^2*e^5 + 6*(2*B*b*c + A*c^2)*d^2*e^3 - 6*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 3*
(27*B*c^2*d^4*e - 2*A*b^2*d*e^4 - 18*(2*B*b*c + A*c^2)*d^3*e^2 + 9*(B*b^2 + 2*A*
b*c)*d^2*e^3)*x + 6*(10*B*c^2*d^5 - 4*(2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c
)*d^3*e^2 + (10*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*A*b*c)*e^
5)*x^3 + 3*(10*B*c^2*d^3*e^2 - 4*(2*B*b*c + A*c^2)*d^2*e^3 + (B*b^2 + 2*A*b*c)*d
*e^4)*x^2 + 3*(10*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 + (B*b^2 + 2*A*b*c)*
d^2*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [A]  time = 54.6485, size = 371, normalized size = 1.56 \[ \frac{B c^{2} x^{2}}{2 e^{4}} + \frac{- 2 A b^{2} d^{2} e^{3} + 22 A b c d^{3} e^{2} - 26 A c^{2} d^{4} e + 11 B b^{2} d^{3} e^{2} - 52 B b c d^{4} e + 47 B c^{2} d^{5} + x^{2} \left (- 6 A b^{2} e^{5} + 36 A b c d e^{4} - 36 A c^{2} d^{2} e^{3} + 18 B b^{2} d e^{4} - 72 B b c d^{2} e^{3} + 60 B c^{2} d^{3} e^{2}\right ) + x \left (- 6 A b^{2} d e^{4} + 54 A b c d^{2} e^{3} - 60 A c^{2} d^{3} e^{2} + 27 B b^{2} d^{2} e^{3} - 120 B b c d^{3} e^{2} + 105 B c^{2} d^{4} e\right )}{6 d^{3} e^{6} + 18 d^{2} e^{7} x + 18 d e^{8} x^{2} + 6 e^{9} x^{3}} + \frac{x \left (A c^{2} e + 2 B b c e - 4 B c^{2} d\right )}{e^{5}} + \frac{\left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**4,x)

[Out]

B*c**2*x**2/(2*e**4) + (-2*A*b**2*d**2*e**3 + 22*A*b*c*d**3*e**2 - 26*A*c**2*d**
4*e + 11*B*b**2*d**3*e**2 - 52*B*b*c*d**4*e + 47*B*c**2*d**5 + x**2*(-6*A*b**2*e
**5 + 36*A*b*c*d*e**4 - 36*A*c**2*d**2*e**3 + 18*B*b**2*d*e**4 - 72*B*b*c*d**2*e
**3 + 60*B*c**2*d**3*e**2) + x*(-6*A*b**2*d*e**4 + 54*A*b*c*d**2*e**3 - 60*A*c**
2*d**3*e**2 + 27*B*b**2*d**2*e**3 - 120*B*b*c*d**3*e**2 + 105*B*c**2*d**4*e))/(6
*d**3*e**6 + 18*d**2*e**7*x + 18*d*e**8*x**2 + 6*e**9*x**3) + x*(A*c**2*e + 2*B*
b*c*e - 4*B*c**2*d)/e**5 + (2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*
d*e + 10*B*c**2*d**2)*log(d + e*x)/e**6

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GIAC/XCAS [A]  time = 0.281597, size = 404, normalized size = 1.7 \[{\left (10 \, B c^{2} d^{2} - 8 \, B b c d e - 4 \, A c^{2} d e + B b^{2} e^{2} + 2 \, A b c e^{2}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{2} x^{2} e^{4} - 8 \, B c^{2} d x e^{3} + 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, B c^{2} d^{5} - 52 \, B b c d^{4} e - 26 \, A c^{2} d^{4} e + 11 \, B b^{2} d^{3} e^{2} + 22 \, A b c d^{3} e^{2} - 2 \, A b^{2} d^{2} e^{3} + 6 \,{\left (10 \, B c^{2} d^{3} e^{2} - 12 \, B b c d^{2} e^{3} - 6 \, A c^{2} d^{2} e^{3} + 3 \, B b^{2} d e^{4} + 6 \, A b c d e^{4} - A b^{2} e^{5}\right )} x^{2} + 3 \,{\left (35 \, B c^{2} d^{4} e - 40 \, B b c d^{3} e^{2} - 20 \, A c^{2} d^{3} e^{2} + 9 \, B b^{2} d^{2} e^{3} + 18 \, A b c d^{2} e^{3} - 2 \, A b^{2} d e^{4}\right )} x\right )} e^{\left (-6\right )}}{6 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^4,x, algorithm="giac")

[Out]

(10*B*c^2*d^2 - 8*B*b*c*d*e - 4*A*c^2*d*e + B*b^2*e^2 + 2*A*b*c*e^2)*e^(-6)*ln(a
bs(x*e + d)) + 1/2*(B*c^2*x^2*e^4 - 8*B*c^2*d*x*e^3 + 4*B*b*c*x*e^4 + 2*A*c^2*x*
e^4)*e^(-8) + 1/6*(47*B*c^2*d^5 - 52*B*b*c*d^4*e - 26*A*c^2*d^4*e + 11*B*b^2*d^3
*e^2 + 22*A*b*c*d^3*e^2 - 2*A*b^2*d^2*e^3 + 6*(10*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e
^3 - 6*A*c^2*d^2*e^3 + 3*B*b^2*d*e^4 + 6*A*b*c*d*e^4 - A*b^2*e^5)*x^2 + 3*(35*B*
c^2*d^4*e - 40*B*b*c*d^3*e^2 - 20*A*c^2*d^3*e^2 + 9*B*b^2*d^2*e^3 + 18*A*b*c*d^2
*e^3 - 2*A*b^2*d*e^4)*x)*e^(-6)/(x*e + d)^3