3.2329 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=297 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{4 e^6 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6 (d+e x)^5}+\frac{c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^6*(d + e*x)^5) + ((c*d^2 - b*d*e + a*e^2)*(2*A*e*(2*c*d - b*e) -
B*(5*c*d^2 - e*(3*b*d - a*e))))/(4*e^6*(d + e*x)^4) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d
- a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(3*e^6*(d + e*x)^3) + (2*A*c*e*(2*c*d - b*e) - B*(1
0*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(2*e^6*(d + e*x)^2) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(e^6*(d + e*
x)) + (B*c^2*Log[d + e*x])/e^6

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Rubi [A]  time = 0.344283, antiderivative size = 295, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^6 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6 (d+e x)^5}+\frac{c (-A c e-2 b B e+5 B c d)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^6*(d + e*x)^5) - ((c*d^2 - b*d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d
- a*e) - 2*A*e*(2*c*d - b*e)))/(4*e^6*(d + e*x)^4) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d
- a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(3*e^6*(d + e*x)^3) + (2*A*c*e*(2*c*d - b*e) - B*(1
0*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(2*e^6*(d + e*x)^2) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(e^6*(d + e*
x)) + (B*c^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 (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^6}+\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^5}+\frac{-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^4}+\frac{-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)^3}+\frac{c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^2}+\frac{B c^2}{e^5 (d+e x)}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{5 e^6 (d+e x)^5}-\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{4 e^6 (d+e x)^4}+\frac{B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{3 e^6 (d+e x)^3}+\frac{2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{2 e^6 (d+e x)^2}+\frac{c (5 B c d-2 b B e-A c e)}{e^6 (d+e x)}+\frac{B c^2 \log (d+e x)}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.241684, size = 386, normalized size = 1.3 \[ \frac{-2 A e \left (e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )+6 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+B \left (-e^2 \left (3 a^2 e^2 (d+5 e x)+4 a b e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b^2 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )-6 c e \left (a e \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )+4 b \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 d \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 B c^2 (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^6,x]

[Out]

(-2*A*e*(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5
*e*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 1
0*d*e^2*x^2 + 10*e^3*x^3))) + B*(c^2*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4
) - e^2*(3*a^2*e^2*(d + 5*e*x) + 4*a*b*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2
+ 10*e^3*x^3)) - 6*c*e*(a*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b*(d^4 + 5*d^3*e*x + 10*d^2*e^2*
x^2 + 10*d*e^3*x^3 + 5*e^4*x^4))) + 60*B*c^2*(d + e*x)^5*Log[d + e*x])/(60*e^6*(d + e*x)^5)

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Maple [B]  time = 0.007, size = 715, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.05495, size = 591, normalized size = 1.99 \begin{align*} \frac{137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \,{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \,{\left (30 \, 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 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e^{2} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac{B c^{2} \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+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B
*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B*a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 +
30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2
- 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 +
5*(125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2
+ 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*
d^4*e^7*x + d^5*e^6) + B*c^2*log(e*x + d)/e^6

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Fricas [A]  time = 1.46372, size = 1096, normalized size = 3.69 \begin{align*} \frac{137 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 60 \,{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 30 \,{\left (30 \, 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 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e^{2} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} e - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} - 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x + 60 \,{\left (B c^{2} e^{5} x^{5} + 5 \, B c^{2} d e^{4} x^{4} + 10 \, B c^{2} d^{2} e^{3} x^{3} + 10 \, B c^{2} d^{3} e^{2} x^{2} + 5 \, B c^{2} d^{4} e x + B c^{2} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} 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+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

1/60*(137*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B
*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 - 3*(B*a^2 + 2*A*a*b)*d*e^4 + 60*(5*B*c^2*d*e^4 - (2*B*b*c + A*c^2)*e^5)*x^4 +
30*(30*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 - (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^3 + 10*(110*B*c^2*d^3*e^2
- 12*(2*B*b*c + A*c^2)*d^2*e^3 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 +
5*(125*B*c^2*d^4*e - 12*(2*B*b*c + A*c^2)*d^3*e^2 - 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2
+ 2*A*a*c)*d*e^4 - 3*(B*a^2 + 2*A*a*b)*e^5)*x + 60*(B*c^2*e^5*x^5 + 5*B*c^2*d*e^4*x^4 + 10*B*c^2*d^2*e^3*x^3 +
10*B*c^2*d^3*e^2*x^2 + 5*B*c^2*d^4*e*x + B*c^2*d^5)*log(e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 +
10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6)

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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)*(c*x**2+b*x+a)**2/(e*x+d)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.13208, size = 575, normalized size = 1.94 \begin{align*} B c^{2} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (60 \,{\left (5 \, B c^{2} d e^{3} - 2 \, B b c e^{4} - A c^{2} e^{4}\right )} x^{4} + 30 \,{\left (30 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} - B b^{2} e^{4} - 2 \, B a c e^{4} - 2 \, A b c e^{4}\right )} x^{3} + 10 \,{\left (110 \, B c^{2} d^{3} e - 24 \, B b c d^{2} e^{2} - 12 \, A c^{2} d^{2} e^{2} - 3 \, B b^{2} d e^{3} - 6 \, B a c d e^{3} - 6 \, A b c d e^{3} - 4 \, B a b e^{4} - 2 \, A b^{2} e^{4} - 4 \, A a c e^{4}\right )} x^{2} + 5 \,{\left (125 \, B c^{2} d^{4} - 24 \, B b c d^{3} e - 12 \, A c^{2} d^{3} e - 3 \, B b^{2} d^{2} e^{2} - 6 \, B a c d^{2} e^{2} - 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} - 3 \, B a^{2} e^{4} - 6 \, A a b e^{4}\right )} x +{\left (137 \, B c^{2} d^{5} - 24 \, B b c d^{4} e - 12 \, A c^{2} d^{4} e - 3 \, B b^{2} d^{3} e^{2} - 6 \, B a c d^{3} e^{2} - 6 \, A b c d^{3} e^{2} - 4 \, B a b d^{2} e^{3} - 2 \, A b^{2} d^{2} e^{3} - 4 \, A a c d^{2} e^{3} - 3 \, B a^{2} d e^{4} - 6 \, A a b d e^{4} - 12 \, A a^{2} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

B*c^2*e^(-6)*log(abs(x*e + d)) + 1/60*(60*(5*B*c^2*d*e^3 - 2*B*b*c*e^4 - A*c^2*e^4)*x^4 + 30*(30*B*c^2*d^2*e^2
- 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 - B*b^2*e^4 - 2*B*a*c*e^4 - 2*A*b*c*e^4)*x^3 + 10*(110*B*c^2*d^3*e - 24*B*b*c
*d^2*e^2 - 12*A*c^2*d^2*e^2 - 3*B*b^2*d*e^3 - 6*B*a*c*d*e^3 - 6*A*b*c*d*e^3 - 4*B*a*b*e^4 - 2*A*b^2*e^4 - 4*A*
a*c*e^4)*x^2 + 5*(125*B*c^2*d^4 - 24*B*b*c*d^3*e - 12*A*c^2*d^3*e - 3*B*b^2*d^2*e^2 - 6*B*a*c*d^2*e^2 - 6*A*b*
c*d^2*e^2 - 4*B*a*b*d*e^3 - 2*A*b^2*d*e^3 - 4*A*a*c*d*e^3 - 3*B*a^2*e^4 - 6*A*a*b*e^4)*x + (137*B*c^2*d^5 - 24
*B*b*c*d^4*e - 12*A*c^2*d^4*e - 3*B*b^2*d^3*e^2 - 6*B*a*c*d^3*e^2 - 6*A*b*c*d^3*e^2 - 4*B*a*b*d^2*e^3 - 2*A*b^
2*d^2*e^3 - 4*A*a*c*d^2*e^3 - 3*B*a^2*d*e^4 - 6*A*a*b*d*e^4 - 12*A*a^2*e^5)*e^(-1))*e^(-5)/(x*e + d)^5