3.1064 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=279 \[ -\frac{b^5 (d+e x)^2 (-6 a B e-A b e+7 b B d)}{2 e^8}+\frac{3 b^4 x (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^7}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8 (d+e x)}-\frac{5 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac{(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}+\frac{b^6 B (d+e x)^3}{3 e^8} \]

[Out]

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

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Rubi [A]  time = 0.413609, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{b^5 (d+e x)^2 (-6 a B e-A b e+7 b B d)}{2 e^8}+\frac{3 b^4 x (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^7}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8 (d+e x)}-\frac{5 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)}{e^8}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac{(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}+\frac{b^6 B (d+e x)^3}{3 e^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx &=\int \left (-\frac{3 b^4 (b d-a e) (-7 b B d+2 A b e+5 a B e)}{e^7}+\frac{(-b d+a e)^6 (-B d+A e)}{e^7 (d+e x)^5}+\frac{(-b d+a e)^5 (-7 b B d+6 A b e+a B e)}{e^7 (d+e x)^4}+\frac{3 b (b d-a e)^4 (-7 b B d+5 A b e+2 a B e)}{e^7 (d+e x)^3}-\frac{5 b^2 (b d-a e)^3 (-7 b B d+4 A b e+3 a B e)}{e^7 (d+e x)^2}+\frac{5 b^3 (b d-a e)^2 (-7 b B d+3 A b e+4 a B e)}{e^7 (d+e x)}+\frac{b^5 (-7 b B d+A b e+6 a B e) (d+e x)}{e^7}+\frac{b^6 B (d+e x)^2}{e^7}\right ) \, dx\\ &=\frac{3 b^4 (b d-a e) (7 b B d-2 A b e-5 a B e) x}{e^7}+\frac{(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}-\frac{(b d-a e)^5 (7 b B d-6 A b e-a B e)}{3 e^8 (d+e x)^3}+\frac{3 b (b d-a e)^4 (7 b B d-5 A b e-2 a B e)}{2 e^8 (d+e x)^2}-\frac{5 b^2 (b d-a e)^3 (7 b B d-4 A b e-3 a B e)}{e^8 (d+e x)}-\frac{b^5 (7 b B d-A b e-6 a B e) (d+e x)^2}{2 e^8}+\frac{b^6 B (d+e x)^3}{3 e^8}-\frac{5 b^3 (b d-a e)^2 (7 b B d-3 A b e-4 a B e) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.211996, size = 263, normalized size = 0.94 \[ \frac{-12 b^4 e x \left (-15 a^2 B e^2-6 a b e (A e-5 B d)+5 b^2 d (A e-3 B d)\right )+6 b^5 e^2 x^2 (6 a B e+A b e-5 b B d)-\frac{60 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{d+e x}-60 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)+\frac{18 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{(d+e x)^2}-\frac{4 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{(d+e x)^3}+\frac{3 (b d-a e)^6 (B d-A e)}{(d+e x)^4}+4 b^6 B e^3 x^3}{12 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]

[Out]

(-12*b^4*e*(-15*a^2*B*e^2 - 6*a*b*e*(-5*B*d + A*e) + 5*b^2*d*(-3*B*d + A*e))*x + 6*b^5*e^2*(-5*b*B*d + A*b*e +
6*a*B*e)*x^2 + 4*b^6*B*e^3*x^3 + (3*(b*d - a*e)^6*(B*d - A*e))/(d + e*x)^4 - (4*(b*d - a*e)^5*(7*b*B*d - 6*A*
b*e - a*B*e))/(d + e*x)^3 + (18*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(d + e*x)^2 - (60*b^2*(b*d - a*
e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e))/(d + e*x) - 60*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*Log[d + e*x
])/(12*e^8)

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Maple [B]  time = 0.018, size = 1177, normalized size = 4.2 \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)^6*(B*x+A)/(e*x+d)^5,x)

[Out]

-3*b/e^3/(e*x+d)^2*B*a^5+21/2*b^6/e^8/(e*x+d)^2*B*d^5-20*b^3/e^4/(e*x+d)*A*a^3+3*b^5/e^5*B*x^2*a-5/2*b^6/e^6*B
*x^2*d+6*b^5/e^5*A*a*x-5*b^6/e^6*A*d*x+15*b^4/e^5*a^2*B*x+15*b^6/e^7*B*d^2*x-2/e^2/(e*x+d)^3*A*a^5*b+2/e^7/(e*
x+d)^3*A*b^6*d^5-7/3/e^8/(e*x+d)^3*b^6*B*d^6+20*b^6/e^7/(e*x+d)*A*d^3-15*b^2/e^4/(e*x+d)*B*a^4-35*b^6/e^8/(e*x
+d)*B*d^4+15*b^4/e^5*ln(e*x+d)*A*a^2+15*b^6/e^7*ln(e*x+d)*A*d^2+20*b^3/e^5*ln(e*x+d)*B*a^3-35*b^6/e^8*ln(e*x+d
)*B*d^3-15/2*b^2/e^3/(e*x+d)^2*A*a^4-15/2*b^6/e^7/(e*x+d)^2*A*d^4-1/3/e^2/(e*x+d)^3*B*a^6-1/4/e/(e*x+d)^4*a^6*
A+1/3*b^6/e^5*B*x^3+1/2*b^6/e^5*A*x^2-15/e^4/(e*x+d)^3*B*a^4*b^2*d^2+80/3/e^5/(e*x+d)^3*B*a^3*b^3*d^3-25/e^6/(
e*x+d)^3*B*a^2*b^4*d^4+12/e^7/(e*x+d)^3*B*a*b^5*d^5+30*b^3/e^4/(e*x+d)^2*A*a^3*d-30*b^5/e^6*B*a*d*x+10/e^3/(e*
x+d)^3*A*a^4*b^2*d-20/e^4/(e*x+d)^3*A*a^3*b^3*d^2+20/e^5/(e*x+d)^3*A*a^2*b^4*d^3+15/4/e^6/(e*x+d)^4*B*a^2*b^4*
d^5-3/2/e^7/(e*x+d)^4*B*a*b^5*d^6-10/e^6/(e*x+d)^3*A*a*b^5*d^4+4/e^3/(e*x+d)^3*B*a^5*b*d-1/4/e^7/(e*x+d)^4*A*b
^6*d^6+1/4/e^2/(e*x+d)^4*B*d*a^6+1/4/e^8/(e*x+d)^4*b^6*B*d^7-45*b^5/e^7/(e*x+d)^2*B*a*d^4+60*b^4/e^5/(e*x+d)*A
*a^2*d-60*b^5/e^6/(e*x+d)*A*a*d^2+80*b^3/e^5/(e*x+d)*B*a^3*d-150*b^4/e^6/(e*x+d)*B*a^2*d^2+120*b^5/e^7/(e*x+d)
*B*a*d^3-30*b^5/e^6*ln(e*x+d)*A*a*d-75*b^4/e^6*ln(e*x+d)*B*a^2*d+90*b^5/e^7*ln(e*x+d)*B*d^2*a+3/2/e^2/(e*x+d)^
4*A*d*a^5*b-15/4/e^3/(e*x+d)^4*A*d^2*a^4*b^2+5/e^4/(e*x+d)^4*A*d^3*a^3*b^3-15/4/e^5/(e*x+d)^4*A*d^4*a^2*b^4+3/
2/e^6/(e*x+d)^4*A*a*b^5*d^5-3/2/e^3/(e*x+d)^4*B*d^2*a^5*b+15/4/e^4/(e*x+d)^4*B*d^3*a^4*b^2-5/e^5/(e*x+d)^4*B*d
^4*a^3*b^3-45*b^4/e^5/(e*x+d)^2*A*a^2*d^2+30*b^5/e^6/(e*x+d)^2*A*a*d^3+45/2*b^2/e^4/(e*x+d)^2*B*a^4*d-60*b^3/e
^5/(e*x+d)^2*B*a^3*d^2+75*b^4/e^6/(e*x+d)^2*B*a^2*d^3

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Maxima [B]  time = 1.39723, size = 1081, normalized size = 3.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(319*B*b^6*d^7 + 3*A*a^6*e^7 - 171*(6*B*a*b^5 + A*b^6)*d^6*e + 231*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 1
25*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*
d^2*e^5 + (B*a^6 + 6*A*a^5*b)*d*e^6 + 60*(7*B*b^6*d^4*e^3 - 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2
*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 18*(63*B*b^6*
d^5*e^2 - 35*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 50*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 30*(4*B*a^3*b^3 + 3*A*a^2*b^
4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + (2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 4*(259*B*b^6*d^6*e - 1
41*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 195*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 110*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e
^4 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x)/
(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*b^6*e^2*x^3 - 3*(5*B*b^6*d*e - (
6*B*a*b^5 + A*b^6)*e^2)*x^2 + 6*(15*B*b^6*d^2 - 5*(6*B*a*b^5 + A*b^6)*d*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*x
)/e^7 - 5*(7*B*b^6*d^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*b^3 + 3*A*
a^2*b^4)*e^3)*log(e*x + d)/e^8

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Fricas [B]  time = 2.01391, size = 2533, normalized size = 9.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(4*B*b^6*e^7*x^7 - 319*B*b^6*d^7 - 3*A*a^6*e^7 + 171*(6*B*a*b^5 + A*b^6)*d^6*e - 231*(5*B*a^2*b^4 + 2*A*a
*b^5)*d^5*e^2 + 125*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(2*B*a^5*
b + 5*A*a^4*b^2)*d^2*e^5 - (B*a^6 + 6*A*a^5*b)*d*e^6 - 2*(7*B*b^6*d*e^6 - 3*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 12*
(7*B*b^6*d^2*e^5 - 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 4*(139*B*b^6*d^3*e^4 -
51*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 36*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6)*x^4 + 4*(136*B*b^6*d^4*e^3 - 24*(6*B*a*b
^5 + A*b^6)*d^3*e^4 - 36*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 60*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 - 15*(3*B*a^
4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 6*(74*B*b^6*d^5*e^2 - 66*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 126*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^3*e^4 - 90*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b
+ 5*A*a^4*b^2)*e^7)*x^2 - 4*(214*B*b^6*d^6*e - 126*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 186*(5*B*a^2*b^4 + 2*A*a*b^5)
*d^4*e^3 - 110*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 3*(2*B*a^5*b + 5
*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e^7)*x - 60*(7*B*b^6*d^7 - 3*(6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^5*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (7*B*b^6*d^3*e^4 - 3*(6*B*a*b^5 + A*b^6)*d^2*e^
5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 4*(7*B*b^6*d^4*e^3 - 3*(6*B*a*b
^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6)*x^3 + 6*(7*B*b^
6*d^5*e^2 - 3*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*
d^2*e^5)*x^2 + 4*(7*B*b^6*d^6*e - 3*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - (4*B*a
^3*b^3 + 3*A*a^2*b^4)*d^3*e^4)*x)*log(e*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*
e^8)

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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)**6*(B*x+A)/(e*x+d)**5,x)

[Out]

Timed out

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Giac [B]  time = 3.08224, size = 1573, normalized size = 5.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^5,x, algorithm="giac")

[Out]

1/6*(2*B*b^6 - 3*(7*B*b^6*d*e - 6*B*a*b^5*e^2 - A*b^6*e^2)*e^(-1)/(x*e + d) + 18*(7*B*b^6*d^2*e^2 - 12*B*a*b^5
*d*e^3 - 2*A*b^6*d*e^3 + 5*B*a^2*b^4*e^4 + 2*A*a*b^5*e^4)*e^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-8) + 5*(7*B*b^6*
d^3 - 18*B*a*b^5*d^2*e - 3*A*b^6*d^2*e + 15*B*a^2*b^4*d*e^2 + 6*A*a*b^5*d*e^2 - 4*B*a^3*b^3*e^3 - 3*A*a^2*b^4*
e^3)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(420*B*b^6*d^4*e^36/(x*e + d) - 126*B*b^6*d^5*e^36/(x*
e + d)^2 + 28*B*b^6*d^6*e^36/(x*e + d)^3 - 3*B*b^6*d^7*e^36/(x*e + d)^4 - 1440*B*a*b^5*d^3*e^37/(x*e + d) - 24
0*A*b^6*d^3*e^37/(x*e + d) + 540*B*a*b^5*d^4*e^37/(x*e + d)^2 + 90*A*b^6*d^4*e^37/(x*e + d)^2 - 144*B*a*b^5*d^
5*e^37/(x*e + d)^3 - 24*A*b^6*d^5*e^37/(x*e + d)^3 + 18*B*a*b^5*d^6*e^37/(x*e + d)^4 + 3*A*b^6*d^6*e^37/(x*e +
d)^4 + 1800*B*a^2*b^4*d^2*e^38/(x*e + d) + 720*A*a*b^5*d^2*e^38/(x*e + d) - 900*B*a^2*b^4*d^3*e^38/(x*e + d)^
2 - 360*A*a*b^5*d^3*e^38/(x*e + d)^2 + 300*B*a^2*b^4*d^4*e^38/(x*e + d)^3 + 120*A*a*b^5*d^4*e^38/(x*e + d)^3 -
45*B*a^2*b^4*d^5*e^38/(x*e + d)^4 - 18*A*a*b^5*d^5*e^38/(x*e + d)^4 - 960*B*a^3*b^3*d*e^39/(x*e + d) - 720*A*
a^2*b^4*d*e^39/(x*e + d) + 720*B*a^3*b^3*d^2*e^39/(x*e + d)^2 + 540*A*a^2*b^4*d^2*e^39/(x*e + d)^2 - 320*B*a^3
*b^3*d^3*e^39/(x*e + d)^3 - 240*A*a^2*b^4*d^3*e^39/(x*e + d)^3 + 60*B*a^3*b^3*d^4*e^39/(x*e + d)^4 + 45*A*a^2*
b^4*d^4*e^39/(x*e + d)^4 + 180*B*a^4*b^2*e^40/(x*e + d) + 240*A*a^3*b^3*e^40/(x*e + d) - 270*B*a^4*b^2*d*e^40/
(x*e + d)^2 - 360*A*a^3*b^3*d*e^40/(x*e + d)^2 + 180*B*a^4*b^2*d^2*e^40/(x*e + d)^3 + 240*A*a^3*b^3*d^2*e^40/(
x*e + d)^3 - 45*B*a^4*b^2*d^3*e^40/(x*e + d)^4 - 60*A*a^3*b^3*d^3*e^40/(x*e + d)^4 + 36*B*a^5*b*e^41/(x*e + d)
^2 + 90*A*a^4*b^2*e^41/(x*e + d)^2 - 48*B*a^5*b*d*e^41/(x*e + d)^3 - 120*A*a^4*b^2*d*e^41/(x*e + d)^3 + 18*B*a
^5*b*d^2*e^41/(x*e + d)^4 + 45*A*a^4*b^2*d^2*e^41/(x*e + d)^4 + 4*B*a^6*e^42/(x*e + d)^3 + 24*A*a^5*b*e^42/(x*
e + d)^3 - 3*B*a^6*d*e^42/(x*e + d)^4 - 18*A*a^5*b*d*e^42/(x*e + d)^4 + 3*A*a^6*e^43/(x*e + d)^4)*e^(-44)