3.1121 \(\int \frac{A+B x}{(a+b x) (d+e x)^5} \, dx\)

Optimal. Leaf size=178 \[ \frac{b^2 (A b-a B)}{(d+e x) (b d-a e)^4}+\frac{b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac{b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5}+\frac{b (A b-a B)}{2 (d+e x)^2 (b d-a e)^3}+\frac{A b-a B}{3 (d+e x)^3 (b d-a e)^2}-\frac{B d-A e}{4 e (d+e x)^4 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.159693, antiderivative size = 178, 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} \[ \frac{b^2 (A b-a B)}{(d+e x) (b d-a e)^4}+\frac{b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac{b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5}+\frac{b (A b-a B)}{2 (d+e x)^2 (b d-a e)^3}+\frac{A b-a B}{3 (d+e x)^3 (b d-a e)^2}-\frac{B d-A e}{4 e (d+e x)^4 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.11093, size = 183, normalized size = 1.03 \[ \frac{12 b^2 e (d+e x)^3 (A b-a B) (b d-a e)+12 b^3 e (d+e x)^4 (A b-a B) \log (a+b x)-12 b^3 e (d+e x)^4 (A b-a B) \log (d+e x)+4 e (d+e x) (A b-a B) (b d-a e)^3+6 b e (d+e x)^2 (A b-a B) (b d-a e)^2-3 (b d-a e)^4 (B d-A e)}{12 e (d+e x)^4 (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 267, normalized size = 1.5 \begin{align*} -{\frac{A}{ \left ( 4\,ae-4\,bd \right ) \left ( ex+d \right ) ^{4}}}+{\frac{Bd}{ \left ( 4\,ae-4\,bd \right ) e \left ( ex+d \right ) ^{4}}}-{\frac{A{b}^{2}}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bba}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{4}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{5}}}+{\frac{Ab}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-{\frac{Ba}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}A}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-{\frac{{b}^{2}Ba}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{5}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)/(e*x+d)^5,x)

[Out]

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

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Maxima [B]  time = 1.45891, size = 927, normalized size = 5.21 \begin{align*} -\frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} + 12 \,{\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} +{\left (13 \, B a b^{2} - 25 \, A b^{3}\right )} d^{3} e -{\left (5 \, B a^{2} b - 23 \, A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} - 13 \, A a^{2} b\right )} d e^{3} + 6 \,{\left (7 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} -{\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \,{\left (13 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 5 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{3} +{\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x}{12 \,{\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5} +{\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*a*b^3 - A*b^4)*log(b*x + a)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d
*e^4 - a^5*e^5) + (B*a*b^3 - A*b^4)*log(e*x + d)/(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^
2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5) - 1/12*(3*B*b^3*d^4 + 3*A*a^3*e^4 + 12*(B*a*b^2 - A*b^3)*e^4*x^3 + (13*B*a*b^
2 - 25*A*b^3)*d^3*e - (5*B*a^2*b - 23*A*a*b^2)*d^2*e^2 + (B*a^3 - 13*A*a^2*b)*d*e^3 + 6*(7*(B*a*b^2 - A*b^3)*d
*e^3 - (B*a^2*b - A*a*b^2)*e^4)*x^2 + 4*(13*(B*a*b^2 - A*b^3)*d^2*e^2 - 5*(B*a^2*b - A*a*b^2)*d*e^3 + (B*a^3 -
A*a^2*b)*e^4)*x)/(b^4*d^8*e - 4*a*b^3*d^7*e^2 + 6*a^2*b^2*d^6*e^3 - 4*a^3*b*d^5*e^4 + a^4*d^4*e^5 + (b^4*d^4*
e^5 - 4*a*b^3*d^3*e^6 + 6*a^2*b^2*d^2*e^7 - 4*a^3*b*d*e^8 + a^4*e^9)*x^4 + 4*(b^4*d^5*e^4 - 4*a*b^3*d^4*e^5 +
6*a^2*b^2*d^3*e^6 - 4*a^3*b*d^2*e^7 + a^4*d*e^8)*x^3 + 6*(b^4*d^6*e^3 - 4*a*b^3*d^5*e^4 + 6*a^2*b^2*d^4*e^5 -
4*a^3*b*d^3*e^6 + a^4*d^2*e^7)*x^2 + 4*(b^4*d^7*e^2 - 4*a*b^3*d^6*e^3 + 6*a^2*b^2*d^5*e^4 - 4*a^3*b*d^4*e^5 +
a^4*d^3*e^6)*x)

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Fricas [B]  time = 1.69835, size = 1820, normalized size = 10.22 \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)/(b*x+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(3*B*b^4*d^5 - 3*A*a^4*e^5 + 5*(2*B*a*b^3 - 5*A*b^4)*d^4*e - 6*(3*B*a^2*b^2 - 8*A*a*b^3)*d^3*e^2 + 6*(B*
a^3*b - 6*A*a^2*b^2)*d^2*e^3 - (B*a^4 - 16*A*a^3*b)*d*e^4 + 12*((B*a*b^3 - A*b^4)*d*e^4 - (B*a^2*b^2 - A*a*b^3
)*e^5)*x^3 + 6*(7*(B*a*b^3 - A*b^4)*d^2*e^3 - 8*(B*a^2*b^2 - A*a*b^3)*d*e^4 + (B*a^3*b - A*a^2*b^2)*e^5)*x^2 +
4*(13*(B*a*b^3 - A*b^4)*d^3*e^2 - 18*(B*a^2*b^2 - A*a*b^3)*d^2*e^3 + 6*(B*a^3*b - A*a^2*b^2)*d*e^4 - (B*a^4 -
A*a^3*b)*e^5)*x + 12*((B*a*b^3 - A*b^4)*e^5*x^4 + 4*(B*a*b^3 - A*b^4)*d*e^4*x^3 + 6*(B*a*b^3 - A*b^4)*d^2*e^3
*x^2 + 4*(B*a*b^3 - A*b^4)*d^3*e^2*x + (B*a*b^3 - A*b^4)*d^4*e)*log(b*x + a) - 12*((B*a*b^3 - A*b^4)*e^5*x^4 +
4*(B*a*b^3 - A*b^4)*d*e^4*x^3 + 6*(B*a*b^3 - A*b^4)*d^2*e^3*x^2 + 4*(B*a*b^3 - A*b^4)*d^3*e^2*x + (B*a*b^3 -
A*b^4)*d^4*e)*log(e*x + d))/(b^5*d^9*e - 5*a*b^4*d^8*e^2 + 10*a^2*b^3*d^7*e^3 - 10*a^3*b^2*d^6*e^4 + 5*a^4*b*d
^5*e^5 - a^5*d^4*e^6 + (b^5*d^5*e^5 - 5*a*b^4*d^4*e^6 + 10*a^2*b^3*d^3*e^7 - 10*a^3*b^2*d^2*e^8 + 5*a^4*b*d*e^
9 - a^5*e^10)*x^4 + 4*(b^5*d^6*e^4 - 5*a*b^4*d^5*e^5 + 10*a^2*b^3*d^4*e^6 - 10*a^3*b^2*d^3*e^7 + 5*a^4*b*d^2*e
^8 - a^5*d*e^9)*x^3 + 6*(b^5*d^7*e^3 - 5*a*b^4*d^6*e^4 + 10*a^2*b^3*d^5*e^5 - 10*a^3*b^2*d^4*e^6 + 5*a^4*b*d^3
*e^7 - a^5*d^2*e^8)*x^2 + 4*(b^5*d^8*e^2 - 5*a*b^4*d^7*e^3 + 10*a^2*b^3*d^6*e^4 - 10*a^3*b^2*d^5*e^5 + 5*a^4*b
*d^4*e^6 - a^5*d^3*e^7)*x)

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Sympy [B]  time = 5.05795, size = 1132, normalized size = 6.36 \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)/(b*x+a)/(e*x+d)**5,x)

[Out]

-b**3*(-A*b + B*a)*log(x + (-A*a*b**4*e - A*b**5*d + B*a**2*b**3*e + B*a*b**4*d - a**6*b**3*e**6*(-A*b + B*a)/
(a*e - b*d)**5 + 6*a**5*b**4*d*e**5*(-A*b + B*a)/(a*e - b*d)**5 - 15*a**4*b**5*d**2*e**4*(-A*b + B*a)/(a*e - b
*d)**5 + 20*a**3*b**6*d**3*e**3*(-A*b + B*a)/(a*e - b*d)**5 - 15*a**2*b**7*d**4*e**2*(-A*b + B*a)/(a*e - b*d)*
*5 + 6*a*b**8*d**5*e*(-A*b + B*a)/(a*e - b*d)**5 - b**9*d**6*(-A*b + B*a)/(a*e - b*d)**5)/(-2*A*b**5*e + 2*B*a
*b**4*e))/(a*e - b*d)**5 + b**3*(-A*b + B*a)*log(x + (-A*a*b**4*e - A*b**5*d + B*a**2*b**3*e + B*a*b**4*d + a*
*6*b**3*e**6*(-A*b + B*a)/(a*e - b*d)**5 - 6*a**5*b**4*d*e**5*(-A*b + B*a)/(a*e - b*d)**5 + 15*a**4*b**5*d**2*
e**4*(-A*b + B*a)/(a*e - b*d)**5 - 20*a**3*b**6*d**3*e**3*(-A*b + B*a)/(a*e - b*d)**5 + 15*a**2*b**7*d**4*e**2
*(-A*b + B*a)/(a*e - b*d)**5 - 6*a*b**8*d**5*e*(-A*b + B*a)/(a*e - b*d)**5 + b**9*d**6*(-A*b + B*a)/(a*e - b*d
)**5)/(-2*A*b**5*e + 2*B*a*b**4*e))/(a*e - b*d)**5 - (3*A*a**3*e**4 - 13*A*a**2*b*d*e**3 + 23*A*a*b**2*d**2*e*
*2 - 25*A*b**3*d**3*e + B*a**3*d*e**3 - 5*B*a**2*b*d**2*e**2 + 13*B*a*b**2*d**3*e + 3*B*b**3*d**4 + x**3*(-12*
A*b**3*e**4 + 12*B*a*b**2*e**4) + x**2*(6*A*a*b**2*e**4 - 42*A*b**3*d*e**3 - 6*B*a**2*b*e**4 + 42*B*a*b**2*d*e
**3) + x*(-4*A*a**2*b*e**4 + 20*A*a*b**2*d*e**3 - 52*A*b**3*d**2*e**2 + 4*B*a**3*e**4 - 20*B*a**2*b*d*e**3 + 5
2*B*a*b**2*d**2*e**2))/(12*a**4*d**4*e**5 - 48*a**3*b*d**5*e**4 + 72*a**2*b**2*d**6*e**3 - 48*a*b**3*d**7*e**2
+ 12*b**4*d**8*e + x**4*(12*a**4*e**9 - 48*a**3*b*d*e**8 + 72*a**2*b**2*d**2*e**7 - 48*a*b**3*d**3*e**6 + 12*
b**4*d**4*e**5) + x**3*(48*a**4*d*e**8 - 192*a**3*b*d**2*e**7 + 288*a**2*b**2*d**3*e**6 - 192*a*b**3*d**4*e**5
+ 48*b**4*d**5*e**4) + x**2*(72*a**4*d**2*e**7 - 288*a**3*b*d**3*e**6 + 432*a**2*b**2*d**4*e**5 - 288*a*b**3*
d**5*e**4 + 72*b**4*d**6*e**3) + x*(48*a**4*d**3*e**6 - 192*a**3*b*d**4*e**5 + 288*a**2*b**2*d**5*e**4 - 192*a
*b**3*d**6*e**3 + 48*b**4*d**7*e**2))

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Giac [B]  time = 3.36288, size = 713, normalized size = 4.01 \begin{align*} -\frac{{\left (B a b^{3} e - A b^{4} e\right )} \log \left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{\frac{3 \, B b^{3} d^{4} e^{3}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B a b^{2} e^{4}}{x e + d} - \frac{12 \, A b^{3} e^{4}}{x e + d} + \frac{6 \, B a b^{2} d e^{4}}{{\left (x e + d\right )}^{2}} - \frac{6 \, A b^{3} d e^{4}}{{\left (x e + d\right )}^{2}} + \frac{4 \, B a b^{2} d^{2} e^{4}}{{\left (x e + d\right )}^{3}} - \frac{4 \, A b^{3} d^{2} e^{4}}{{\left (x e + d\right )}^{3}} - \frac{9 \, B a b^{2} d^{3} e^{4}}{{\left (x e + d\right )}^{4}} - \frac{3 \, A b^{3} d^{3} e^{4}}{{\left (x e + d\right )}^{4}} - \frac{6 \, B a^{2} b e^{5}}{{\left (x e + d\right )}^{2}} + \frac{6 \, A a b^{2} e^{5}}{{\left (x e + d\right )}^{2}} - \frac{8 \, B a^{2} b d e^{5}}{{\left (x e + d\right )}^{3}} + \frac{8 \, A a b^{2} d e^{5}}{{\left (x e + d\right )}^{3}} + \frac{9 \, B a^{2} b d^{2} e^{5}}{{\left (x e + d\right )}^{4}} + \frac{9 \, A a b^{2} d^{2} e^{5}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{3} e^{6}}{{\left (x e + d\right )}^{3}} - \frac{4 \, A a^{2} b e^{6}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{3} d e^{6}}{{\left (x e + d\right )}^{4}} - \frac{9 \, A a^{2} b d e^{6}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{3} e^{7}}{{\left (x e + d\right )}^{4}}}{12 \,{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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