3.1104 \(\int \frac{A+B x}{(a+b x) (d+e x)^5} \, dx\)

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

[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 = 0.349995, 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 \[ \frac{b^3 (A b-a B) \log (a+b x)}{(b d-a e)^5}-\frac{b^3 (A b-a B) \log (d+e x)}{(b d-a e)^5}+\frac{b^2 (A b-a B)}{(d+e x) (b d-a e)^4}+\frac{b (A b-a B)}{2 (d+e x)^2 (b d-a e)^3}+\frac{A b-a B}{3 (d+e x)^3 (b d-a e)^2}-\frac{B d-A e}{4 e (d+e x)^4 (b d-a e)} \]

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

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Rubi in Sympy [A]  time = 60.2527, size = 146, normalized size = 0.82 \[ - \frac{b^{3} \left (A b - B a\right ) \log{\left (a + b x \right )}}{\left (a e - b d\right )^{5}} + \frac{b^{3} \left (A b - B a\right ) \log{\left (d + e x \right )}}{\left (a e - b d\right )^{5}} + \frac{b^{2} \left (A b - B a\right )}{\left (d + e x\right ) \left (a e - b d\right )^{4}} - \frac{b \left (A b - B a\right )}{2 \left (d + e x\right )^{2} \left (a e - b d\right )^{3}} + \frac{A b - B a}{3 \left (d + e x\right )^{3} \left (a e - b d\right )^{2}} - \frac{A e - B d}{4 e \left (d + e x\right )^{4} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.188343, size = 183, normalized size = 1.03 \[ \frac{12 b^3 e (d+e x)^4 (A b-a B) \log (a+b x)-12 b^3 e (d+e x)^4 (A b-a B) \log (d+e x)+12 b^2 e (d+e x)^3 (A b-a B) (b d-a e)+4 e (d+e x) (A b-a B) (b d-a e)^3+6 b e (d+e x)^2 (A b-a B) (b d-a e)^2-3 (b d-a e)^4 (B d-A e)}{12 e (d+e x)^4 (b d-a e)^5} \]

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 = 0.02, size = 267, normalized size = 1.5 \[ -{\frac{A}{ \left ( 4\,ae-4\,bd \right ) \left ( ex+d \right ) ^{4}}}+{\frac{Bd}{ \left ( 4\,ae-4\,bd \right ) e \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2}A}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+{\frac{Bba}{2\, \left ( ae-bd \right ) ^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{4}\ln \left ( ex+d \right ) A}{ \left ( ae-bd \right ) ^{5}}}-{\frac{{b}^{3}\ln \left ( ex+d \right ) Ba}{ \left ( ae-bd \right ) ^{5}}}+{\frac{Ab}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}-{\frac{Ba}{3\, \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) ^{3}}}+{\frac{{b}^{3}A}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-{\frac{{b}^{2}Ba}{ \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) }}-{\frac{{b}^{4}\ln \left ( bx+a \right ) A}{ \left ( ae-bd \right ) ^{5}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) Ba}{ \left ( ae-bd \right ) ^{5}}} \]

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 [A]  time = 1.41961, size = 927, normalized size = 5.21 \[ -\frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac{3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} + 12 \,{\left (B a b^{2} - A b^{3}\right )} e^{4} x^{3} +{\left (13 \, B a b^{2} - 25 \, A b^{3}\right )} d^{3} e -{\left (5 \, B a^{2} b - 23 \, A a b^{2}\right )} d^{2} e^{2} +{\left (B a^{3} - 13 \, A a^{2} b\right )} d e^{3} + 6 \,{\left (7 \,{\left (B a b^{2} - A b^{3}\right )} d e^{3} -{\left (B a^{2} b - A a b^{2}\right )} e^{4}\right )} x^{2} + 4 \,{\left (13 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e^{2} - 5 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{3} +{\left (B a^{3} - A a^{2} b\right )} e^{4}\right )} x}{12 \,{\left (b^{4} d^{8} e - 4 \, a b^{3} d^{7} e^{2} + 6 \, a^{2} b^{2} d^{6} e^{3} - 4 \, a^{3} b d^{5} e^{4} + a^{4} d^{4} e^{5} +{\left (b^{4} d^{4} e^{5} - 4 \, a b^{3} d^{3} e^{6} + 6 \, a^{2} b^{2} d^{2} e^{7} - 4 \, a^{3} b d e^{8} + a^{4} e^{9}\right )} x^{4} + 4 \,{\left (b^{4} d^{5} e^{4} - 4 \, a b^{3} d^{4} e^{5} + 6 \, a^{2} b^{2} d^{3} e^{6} - 4 \, a^{3} b d^{2} e^{7} + a^{4} d e^{8}\right )} x^{3} + 6 \,{\left (b^{4} d^{6} e^{3} - 4 \, a b^{3} d^{5} e^{4} + 6 \, a^{2} b^{2} d^{4} e^{5} - 4 \, a^{3} b d^{3} e^{6} + a^{4} d^{2} e^{7}\right )} x^{2} + 4 \,{\left (b^{4} d^{7} e^{2} - 4 \, a b^{3} d^{6} e^{3} + 6 \, a^{2} b^{2} d^{5} e^{4} - 4 \, a^{3} b d^{4} e^{5} + a^{4} d^{3} e^{6}\right )} x\right )}} \]

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 [A]  time = 0.22304, size = 1245, normalized size = 6.99 \[ \text{result too large to display} \]

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 [A]  time = 18.0924, size = 1132, normalized size = 6.36 \[ \text{result too large to display} \]

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 + 52*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 + 2
88*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/XCAS [A]  time = 0.220201, size = 713, normalized size = 4.01 \[ -\frac{{\left (B a b^{3} e - A b^{4} e\right )}{\rm ln}\left ({\left | -b + \frac{b d}{x e + d} - \frac{a e}{x e + d} \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac{\frac{3 \, B b^{3} d^{4} e^{3}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B a b^{2} e^{4}}{x e + d} - \frac{12 \, A b^{3} e^{4}}{x e + d} + \frac{6 \, B a b^{2} d e^{4}}{{\left (x e + d\right )}^{2}} - \frac{6 \, A b^{3} d e^{4}}{{\left (x e + d\right )}^{2}} + \frac{4 \, B a b^{2} d^{2} e^{4}}{{\left (x e + d\right )}^{3}} - \frac{4 \, A b^{3} d^{2} e^{4}}{{\left (x e + d\right )}^{3}} - \frac{9 \, B a b^{2} d^{3} e^{4}}{{\left (x e + d\right )}^{4}} - \frac{3 \, A b^{3} d^{3} e^{4}}{{\left (x e + d\right )}^{4}} - \frac{6 \, B a^{2} b e^{5}}{{\left (x e + d\right )}^{2}} + \frac{6 \, A a b^{2} e^{5}}{{\left (x e + d\right )}^{2}} - \frac{8 \, B a^{2} b d e^{5}}{{\left (x e + d\right )}^{3}} + \frac{8 \, A a b^{2} d e^{5}}{{\left (x e + d\right )}^{3}} + \frac{9 \, B a^{2} b d^{2} e^{5}}{{\left (x e + d\right )}^{4}} + \frac{9 \, A a b^{2} d^{2} e^{5}}{{\left (x e + d\right )}^{4}} + \frac{4 \, B a^{3} e^{6}}{{\left (x e + d\right )}^{3}} - \frac{4 \, A a^{2} b e^{6}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B a^{3} d e^{6}}{{\left (x e + d\right )}^{4}} - \frac{9 \, A a^{2} b d e^{6}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A a^{3} e^{7}}{{\left (x e + d\right )}^{4}}}{12 \,{\left (b^{4} d^{4} e^{4} - 4 \, a b^{3} d^{3} e^{5} + 6 \, a^{2} b^{2} d^{2} e^{6} - 4 \, a^{3} b d e^{7} + a^{4} e^{8}\right )}} \]

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)*ln(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)