3.1121 \(\int \frac{(A+B x) (b x+c x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=232 \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac{\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{B c^2 x^3}{3 e^3} \]

[Out]

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

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Rubi [A]  time = 0.320576, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{x \left (A c e (3 c d-2 b e)-B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )}{e^5}+\frac{\log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{e^6}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{c x^2 (-A c e-2 b B e+3 B c d)}{2 e^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{B c^2 x^3}{3 e^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^3,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^3} \, dx &=\int \left (\frac{-A c e (3 c d-2 b e)+B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )}{e^5}+\frac{c (-3 B c d+2 b B e+A c e) x}{e^4}+\frac{B c^2 x^2}{e^3}-\frac{d^2 (B d-A e) (c d-b e)^2}{e^5 (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))}{e^5 (d+e x)^2}+\frac{A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{\left (A c e (3 c d-2 b e)-B \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )\right ) x}{e^5}-\frac{c (3 B c d-2 b B e-A c e) x^2}{2 e^4}+\frac{B c^2 x^3}{3 e^3}+\frac{d^2 (B d-A e) (c d-b e)^2}{2 e^6 (d+e x)^2}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{e^6 (d+e x)}+\frac{\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.177888, size = 219, normalized size = 0.94 \[ \frac{6 e x \left (A c e (2 b e-3 c d)+B \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )\right )+6 \log (d+e x) \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )+B d \left (-3 b^2 e^2+12 b c d e-10 c^2 d^2\right )\right )+\frac{3 d^2 (B d-A e) (c d-b e)^2}{(d+e x)^2}+3 c e^2 x^2 (A c e+2 b B e-3 B c d)-\frac{6 d (c d-b e) (2 A e (b e-2 c d)+B d (5 c d-3 b e))}{d+e x}+2 B c^2 e^3 x^3}{6 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 420, normalized size = 1.8 \begin{align*}{\frac{B{c}^{2}{x}^{3}}{3\,{e}^{3}}}+{\frac{{b}^{2}Bx}{{e}^{3}}}+{\frac{A{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{4}Bbc}{{e}^{5} \left ( ex+d \right ) ^{2}}}-6\,{\frac{A{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) }}+8\,{\frac{Bcb{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-6\,{\frac{\ln \left ( ex+d \right ) Abcd}{{e}^{4}}}+12\,{\frac{\ln \left ( ex+d \right ) Bbc{d}^{2}}{{e}^{5}}}-6\,{\frac{Bcbdx}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) A{b}^{2}}{{e}^{3}}}+{\frac{A{c}^{2}{x}^{2}}{2\,{e}^{3}}}+2\,{\frac{Abcx}{{e}^{3}}}-3\,{\frac{A{c}^{2}dx}{{e}^{4}}}-{\frac{{d}^{2}A{b}^{2}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{4}A{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+6\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{2}}{{e}^{5}}}-3\,{\frac{\ln \left ( ex+d \right ) B{b}^{2}d}{{e}^{4}}}-10\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{3}}{{e}^{6}}}+4\,{\frac{A{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2}B{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+6\,{\frac{B{c}^{2}{d}^{2}x}{{e}^{5}}}+{\frac{B{x}^{2}bc}{{e}^{3}}}-{\frac{3\,B{c}^{2}{x}^{2}d}{2\,{e}^{4}}}-5\,{\frac{B{c}^{2}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{B{d}^{3}{b}^{2}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{B{c}^{2}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+2\,{\frac{dA{b}^{2}}{{e}^{3} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.0788, size = 408, normalized size = 1.76 \begin{align*} -\frac{9 \, B c^{2} d^{5} - 3 \, A b^{2} d^{2} e^{3} - 7 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 5 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 2 \,{\left (5 \, B c^{2} d^{4} e - 2 \, A b^{2} d e^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B c^{2} e^{2} x^{3} - 3 \,{\left (3 \, B c^{2} d e -{\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B c^{2} d^{2} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d e +{\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{{\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.41795, size = 1019, normalized size = 4.39 \begin{align*} \frac{2 \, B c^{2} e^{5} x^{5} - 27 \, B c^{2} d^{5} + 9 \, A b^{2} d^{2} e^{3} + 21 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 15 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} -{\left (5 \, B c^{2} d e^{4} - 3 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (10 \, B c^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B c^{2} d^{3} e^{2} - 11 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 4 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B c^{2} d^{4} e + 2 \, A b^{2} d e^{4} +{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 2 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 6 \,{\left (10 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} +{\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} + 2 \,{\left (10 \, B c^{2} d^{4} e - A b^{2} d e^{4} - 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\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^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/6*(2*B*c^2*e^5*x^5 - 27*B*c^2*d^5 + 9*A*b^2*d^2*e^3 + 21*(2*B*b*c + A*c^2)*d^4*e - 15*(B*b^2 + 2*A*b*c)*d^3*
e^2 - (5*B*c^2*d*e^4 - 3*(2*B*b*c + A*c^2)*e^5)*x^4 + 2*(10*B*c^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b
^2 + 2*A*b*c)*e^5)*x^3 + 3*(21*B*c^2*d^3*e^2 - 11*(2*B*b*c + A*c^2)*d^2*e^3 + 4*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 +
6*(B*c^2*d^4*e + 2*A*b^2*d*e^4 + (2*B*b*c + A*c^2)*d^3*e^2 - 2*(B*b^2 + 2*A*b*c)*d^2*e^3)*x - 6*(10*B*c^2*d^5
- A*b^2*d^2*e^3 - 6*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*A*b*c)*d^3*e^2 + (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 + 2*(10*B*c^2*d^4*e - A*b^2*d*e^4 - 6*(2*B*b*c + A
*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*x)*log(e*x + d))/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)

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Sympy [A]  time = 7.70473, size = 355, normalized size = 1.53 \begin{align*} \frac{B c^{2} x^{3}}{3 e^{3}} - \frac{- 3 A b^{2} d^{2} e^{3} + 10 A b c d^{3} e^{2} - 7 A c^{2} d^{4} e + 5 B b^{2} d^{3} e^{2} - 14 B b c d^{4} e + 9 B c^{2} d^{5} + x \left (- 4 A b^{2} d e^{4} + 12 A b c d^{2} e^{3} - 8 A c^{2} d^{3} e^{2} + 6 B b^{2} d^{2} e^{3} - 16 B b c d^{3} e^{2} + 10 B c^{2} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A c^{2} e + 2 B b c e - 3 B c^{2} d\right )}{2 e^{4}} + \frac{x \left (2 A b c e^{2} - 3 A c^{2} d e + B b^{2} e^{2} - 6 B b c d e + 6 B c^{2} d^{2}\right )}{e^{5}} - \frac{\left (- 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}\right ) \log{\left (d + e x \right )}}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**3,x)

[Out]

B*c**2*x**3/(3*e**3) - (-3*A*b**2*d**2*e**3 + 10*A*b*c*d**3*e**2 - 7*A*c**2*d**4*e + 5*B*b**2*d**3*e**2 - 14*B
*b*c*d**4*e + 9*B*c**2*d**5 + x*(-4*A*b**2*d*e**4 + 12*A*b*c*d**2*e**3 - 8*A*c**2*d**3*e**2 + 6*B*b**2*d**2*e*
*3 - 16*B*b*c*d**3*e**2 + 10*B*c**2*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2) + x**2*(A*c**2*e + 2*B*b
*c*e - 3*B*c**2*d)/(2*e**4) + x*(2*A*b*c*e**2 - 3*A*c**2*d*e + B*b**2*e**2 - 6*B*b*c*d*e + 6*B*c**2*d**2)/e**5
- (-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)*log(
d + e*x)/e**6

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Giac [A]  time = 1.16148, size = 414, normalized size = 1.78 \begin{align*} -{\left (10 \, B c^{2} d^{3} - 12 \, B b c d^{2} e - 6 \, A c^{2} d^{2} e + 3 \, B b^{2} d e^{2} + 6 \, A b c d e^{2} - A b^{2} e^{3}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c^{2} x^{3} e^{6} - 9 \, B c^{2} d x^{2} e^{5} + 36 \, B c^{2} d^{2} x e^{4} + 6 \, B b c x^{2} e^{6} + 3 \, A c^{2} x^{2} e^{6} - 36 \, B b c d x e^{5} - 18 \, A c^{2} d x e^{5} + 6 \, B b^{2} x e^{6} + 12 \, A b c x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B c^{2} d^{5} - 14 \, B b c d^{4} e - 7 \, A c^{2} d^{4} e + 5 \, B b^{2} d^{3} e^{2} + 10 \, A b c d^{3} e^{2} - 3 \, A b^{2} d^{2} e^{3} + 2 \,{\left (5 \, B c^{2} d^{4} e - 8 \, B b c d^{3} e^{2} - 4 \, A c^{2} d^{3} e^{2} + 3 \, B b^{2} d^{2} e^{3} + 6 \, A b c d^{2} e^{3} - 2 \, A b^{2} d e^{4}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

-(10*B*c^2*d^3 - 12*B*b*c*d^2*e - 6*A*c^2*d^2*e + 3*B*b^2*d*e^2 + 6*A*b*c*d*e^2 - A*b^2*e^3)*e^(-6)*log(abs(x*
e + d)) + 1/6*(2*B*c^2*x^3*e^6 - 9*B*c^2*d*x^2*e^5 + 36*B*c^2*d^2*x*e^4 + 6*B*b*c*x^2*e^6 + 3*A*c^2*x^2*e^6 -
36*B*b*c*d*x*e^5 - 18*A*c^2*d*x*e^5 + 6*B*b^2*x*e^6 + 12*A*b*c*x*e^6)*e^(-9) - 1/2*(9*B*c^2*d^5 - 14*B*b*c*d^4
*e - 7*A*c^2*d^4*e + 5*B*b^2*d^3*e^2 + 10*A*b*c*d^3*e^2 - 3*A*b^2*d^2*e^3 + 2*(5*B*c^2*d^4*e - 8*B*b*c*d^3*e^2
- 4*A*c^2*d^3*e^2 + 3*B*b^2*d^2*e^3 + 6*A*b*c*d^2*e^3 - 2*A*b^2*d*e^4)*x)*e^(-6)/(x*e + d)^2