3.1748 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=425 \[ -\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^3}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7 (a+b x)}+\frac{5 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)} \]

[Out]

(5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(e^6*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e
^7*(a + b*x)*(d + e*x)^3) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) - (5*b*(b*d - a*e)^3*(3*b*B*
d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) -
(b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e
^7*(a + b*x)) + (b^5*B*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*
x)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi [A]  time = 1.13196, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^3}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7 (a+b x)}+\frac{5 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(e^6*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e
^7*(a + b*x)*(d + e*x)^3) + ((b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) - (5*b*(b*d - a*e)^3*(3*b*B*
d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) -
(b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e
^7*(a + b*x)) + (b^5*B*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*
x)) - (10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]*Log[d + e*x])/(e^7*(a + b*x))

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Rubi in Sympy [A]  time = 82.5051, size = 400, normalized size = 0.94 \[ \frac{10 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e + B a e - 2 B b d\right )}{3 e^{4} \left (a e - b d\right )} + \frac{5 b^{2} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right )}{3 e^{5}} + \frac{10 b^{2} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right )}{e^{6}} + \frac{10 b^{2} \left (a e - b d\right )^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (A b e + B a e - 2 B b d\right ) \log{\left (d + e x \right )}}{e^{7} \left (a + b x\right )} - \frac{5 b \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (A b e + B a e - 2 B b d\right )}{2 e^{3} \left (d + e x\right ) \left (a e - b d\right )} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{6 e \left (d + e x\right )^{3} \left (a e - b d\right )} - \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (A b e + B a e - 2 B b d\right )}{2 e^{2} \left (d + e x\right )^{2} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**4,x)

[Out]

10*b**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(A*b*e + B*a*e - 2*B*b*d)/(3*e**4*(a
*e - b*d)) + 5*b**2*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*b*e + B*a*
e - 2*B*b*d)/(3*e**5) + 10*b**2*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(A*
b*e + B*a*e - 2*B*b*d)/e**6 + 10*b**2*(a*e - b*d)**2*sqrt(a**2 + 2*a*b*x + b**2*
x**2)*(A*b*e + B*a*e - 2*B*b*d)*log(d + e*x)/(e**7*(a + b*x)) - 5*b*(a + b*x)*(a
**2 + 2*a*b*x + b**2*x**2)**(3/2)*(A*b*e + B*a*e - 2*B*b*d)/(2*e**3*(d + e*x)*(a
*e - b*d)) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(6*e*
(d + e*x)**3*(a*e - b*d)) - (a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(A*b*e + B*a*e -
2*B*b*d)/(2*e**2*(d + e*x)**2*(a*e - b*d))

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Mathematica [A]  time = 0.636179, size = 504, normalized size = 1.19 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 e^5 (2 A e+B (d+3 e x))+5 a^4 b e^4 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+3 d e x+3 e^2 x^2\right )-B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-10 a^2 b^3 e^2 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )-5 a b^4 e \left (2 A e \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+B \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )\right )+60 b^2 (d+e x)^3 (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)+b^5 \left (A e \left (-47 d^5-81 d^4 e x+9 d^3 e^2 x^2+63 d^2 e^3 x^3+15 d e^4 x^4-3 e^5 x^5\right )+2 B \left (37 d^6+51 d^5 e x-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+3 d e^5 x^5-e^6 x^6\right )\right )\right )}{6 e^7 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

-(Sqrt[(a + b*x)^2]*(a^5*e^5*(2*A*e + B*(d + 3*e*x)) + 5*a^4*b*e^4*(A*e*(d + 3*e
*x) + 2*B*(d^2 + 3*d*e*x + 3*e^2*x^2)) + 10*a^3*b^2*e^3*(2*A*e*(d^2 + 3*d*e*x +
3*e^2*x^2) - B*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) - 10*a^2*b^3*e^2*(A*d*e*(11*d
^2 + 27*d*e*x + 18*e^2*x^2) - 2*B*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3
*x^3 - 3*e^4*x^4)) - 5*a*b^4*e*(2*A*e*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*
d*e^3*x^3 + 3*e^4*x^4) + B*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3
- 15*d*e^4*x^4 + 3*e^5*x^5)) + b^5*(A*e*(-47*d^5 - 81*d^4*e*x + 9*d^3*e^2*x^2 +
63*d^2*e^3*x^3 + 15*d*e^4*x^4 - 3*e^5*x^5) + 2*B*(37*d^6 + 51*d^5*e*x - 39*d^4*
e^2*x^2 - 73*d^3*e^3*x^3 - 15*d^2*e^4*x^4 + 3*d*e^5*x^5 - e^6*x^6)) + 60*b^2*(b*
d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^3*Log[d + e*x]))/(6*e^7*(a + b*x)
*(d + e*x)^3)

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Maple [B]  time = 0.034, size = 1233, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x)

[Out]

1/6*((b*x+a)^2)^(5/2)*(47*A*b^5*d^5*e+180*B*x^2*a^3*b^2*d*e^5-20*A*a^3*b^2*d^2*e
^4+110*A*a^2*b^3*d^3*e^3-B*d*e^5*a^5-2*A*a^5*e^6-74*B*b^5*d^6-15*A*x^4*b^5*d*e^5
-720*B*ln(e*x+d)*x*a^2*b^3*d^3*e^3+60*B*x^4*a^2*b^3*e^6+15*B*x^5*a*b^4*e^6+235*B
*a*b^4*d^5*e+110*B*a^3*b^2*d^3*e^3+78*B*x^2*b^5*d^4*e^2-15*A*x*a^4*b*e^6+81*A*x*
b^5*d^4*e^2-720*B*ln(e*x+d)*x^2*a^2*b^3*d^2*e^4+900*B*ln(e*x+d)*x^2*a*b^4*d^3*e^
3+180*A*ln(e*x+d)*x^2*a^2*b^3*d*e^5-360*A*ln(e*x+d)*x^2*a*b^4*d^2*e^4+180*B*ln(e
*x+d)*x^2*a^3*b^2*d*e^5-360*B*ln(e*x+d)*x^2*b^5*d^4*e^2+60*A*ln(e*x+d)*x^3*a^2*b
^3*e^6+60*A*ln(e*x+d)*x^3*b^5*d^2*e^4+60*B*ln(e*x+d)*x^3*a^3*b^2*e^6-120*B*ln(e*
x+d)*x^3*b^5*d^3*e^3+180*A*ln(e*x+d)*x^2*b^5*d^3*e^3-120*A*ln(e*x+d)*x^3*a*b^4*d
*e^5-240*B*ln(e*x+d)*x^3*a^2*b^3*d*e^5+300*B*ln(e*x+d)*x^3*a*b^4*d^2*e^4-180*B*x
^2*a^2*b^3*d^2*e^4+180*A*x^2*a^2*b^3*d*e^5-90*A*x^2*a*b^4*d^2*e^4+90*A*x^3*a*b^4
*d*e^5+180*B*x^3*a^2*b^3*d*e^5-315*B*x^3*a*b^4*d^2*e^4-75*B*x^4*a*b^4*d*e^5+180*
A*ln(e*x+d)*x*b^5*d^4*e^2-360*B*ln(e*x+d)*x*b^5*d^5*e+60*B*ln(e*x+d)*a^3*b^2*d^3
*e^3-240*B*ln(e*x+d)*a^2*b^3*d^4*e^2+300*B*ln(e*x+d)*a*b^4*d^5*e+146*B*x^3*b^5*d
^3*e^3-60*A*x^2*a^3*b^2*e^6-9*A*x^2*b^5*d^3*e^3-30*B*x^2*a^4*b*e^6-5*A*d*e^5*a^4
*b+30*B*x^4*b^5*d^2*e^4-63*A*x^3*b^5*d^2*e^4+60*A*ln(e*x+d)*a^2*b^3*d^3*e^3-120*
A*ln(e*x+d)*a*b^4*d^4*e^2+270*A*x*a^2*b^3*d^2*e^4-270*A*x*a*b^4*d^3*e^3-30*B*x*a
^4*b*d*e^5+270*B*x*a^3*b^2*d^2*e^4-540*B*x*a^2*b^3*d^3*e^3+405*B*x*a*b^4*d^4*e^2
-45*B*x^2*a*b^4*d^3*e^3-60*A*x*a^3*b^2*d*e^5+2*B*x^6*b^5*e^6+3*A*x^5*b^5*e^6-3*B
*x*a^5*e^6-120*B*ln(e*x+d)*b^5*d^6+180*A*ln(e*x+d)*x*a^2*b^3*d^2*e^4-360*A*ln(e*
x+d)*x*a*b^4*d^3*e^3-260*B*a^2*b^3*d^4*e^2-130*A*a*b^4*d^4*e^2+180*B*ln(e*x+d)*x
*a^3*b^2*d^2*e^4+900*B*ln(e*x+d)*x*a*b^4*d^4*e^2-10*B*a^4*b*d^2*e^4-102*B*x*b^5*
d^5*e+60*A*ln(e*x+d)*b^5*d^5*e-6*B*x^5*b^5*d*e^5+30*A*x^4*a*b^4*e^6)/(b*x+a)^5/e
^7/(e*x+d)^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284333, size = 1214, normalized size = 2.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^4,x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*e^6*x^6 - 74*B*b^5*d^6 - 2*A*a^5*e^6 + 47*(5*B*a*b^4 + A*b^5)*d^5*e
- 130*(2*B*a^2*b^3 + A*a*b^4)*d^4*e^2 + 110*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 1
0*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - (B*a^5 + 5*A*a^4*b)*d*e^5 - 3*(2*B*b^5*d*e^5
- (5*B*a*b^4 + A*b^5)*e^6)*x^5 + 15*(2*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^
5 + 2*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + (146*B*b^5*d^3*e^3 - 63*(5*B*a*b^4 + A*
b^5)*d^2*e^4 + 90*(2*B*a^2*b^3 + A*a*b^4)*d*e^5)*x^3 + 3*(26*B*b^5*d^4*e^2 - 3*(
5*B*a*b^4 + A*b^5)*d^3*e^3 - 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 60*(B*a^3*b^2
+ A*a^2*b^3)*d*e^5 - 10*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 - 3*(34*B*b^5*d^5*e - 2
7*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 90*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 90*(B*a^3*b
^2 + A*a^2*b^3)*d^2*e^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + (B*a^5 + 5*A*a^4*b)
*e^6)*x - 60*(2*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + 2*(2*B*a^2*b^3 + A*a*b^4
)*d^4*e^2 - (B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + (2*B*b^5*d^3*e^3 - (5*B*a*b^4 + A*
b^5)*d^2*e^4 + 2*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 - (B*a^3*b^2 + A*a^2*b^3)*e^6)*x^
3 + 3*(2*B*b^5*d^4*e^2 - (5*B*a*b^4 + A*b^5)*d^3*e^3 + 2*(2*B*a^2*b^3 + A*a*b^4)
*d^2*e^4 - (B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 + 3*(2*B*b^5*d^5*e - (5*B*a*b^4 +
A*b^5)*d^4*e^2 + 2*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - (B*a^3*b^2 + A*a^2*b^3)*d^2
*e^4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.305115, size = 1180, normalized size = 2.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/(e*x + d)^4,x, algorithm="giac")

[Out]

-10*(2*B*b^5*d^3*sign(b*x + a) - 5*B*a*b^4*d^2*e*sign(b*x + a) - A*b^5*d^2*e*sig
n(b*x + a) + 4*B*a^2*b^3*d*e^2*sign(b*x + a) + 2*A*a*b^4*d*e^2*sign(b*x + a) - B
*a^3*b^2*e^3*sign(b*x + a) - A*a^2*b^3*e^3*sign(b*x + a))*e^(-7)*ln(abs(x*e + d)
) + 1/6*(2*B*b^5*x^3*e^8*sign(b*x + a) - 12*B*b^5*d*x^2*e^7*sign(b*x + a) + 60*B
*b^5*d^2*x*e^6*sign(b*x + a) + 15*B*a*b^4*x^2*e^8*sign(b*x + a) + 3*A*b^5*x^2*e^
8*sign(b*x + a) - 120*B*a*b^4*d*x*e^7*sign(b*x + a) - 24*A*b^5*d*x*e^7*sign(b*x
+ a) + 60*B*a^2*b^3*x*e^8*sign(b*x + a) + 30*A*a*b^4*x*e^8*sign(b*x + a))*e^(-12
) - 1/6*(74*B*b^5*d^6*sign(b*x + a) - 235*B*a*b^4*d^5*e*sign(b*x + a) - 47*A*b^5
*d^5*e*sign(b*x + a) + 260*B*a^2*b^3*d^4*e^2*sign(b*x + a) + 130*A*a*b^4*d^4*e^2
*sign(b*x + a) - 110*B*a^3*b^2*d^3*e^3*sign(b*x + a) - 110*A*a^2*b^3*d^3*e^3*sig
n(b*x + a) + 10*B*a^4*b*d^2*e^4*sign(b*x + a) + 20*A*a^3*b^2*d^2*e^4*sign(b*x +
a) + B*a^5*d*e^5*sign(b*x + a) + 5*A*a^4*b*d*e^5*sign(b*x + a) + 2*A*a^5*e^6*sig
n(b*x + a) + 30*(3*B*b^5*d^4*e^2*sign(b*x + a) - 10*B*a*b^4*d^3*e^3*sign(b*x + a
) - 2*A*b^5*d^3*e^3*sign(b*x + a) + 12*B*a^2*b^3*d^2*e^4*sign(b*x + a) + 6*A*a*b
^4*d^2*e^4*sign(b*x + a) - 6*B*a^3*b^2*d*e^5*sign(b*x + a) - 6*A*a^2*b^3*d*e^5*s
ign(b*x + a) + B*a^4*b*e^6*sign(b*x + a) + 2*A*a^3*b^2*e^6*sign(b*x + a))*x^2 +
3*(54*B*b^5*d^5*e*sign(b*x + a) - 175*B*a*b^4*d^4*e^2*sign(b*x + a) - 35*A*b^5*d
^4*e^2*sign(b*x + a) + 200*B*a^2*b^3*d^3*e^3*sign(b*x + a) + 100*A*a*b^4*d^3*e^3
*sign(b*x + a) - 90*B*a^3*b^2*d^2*e^4*sign(b*x + a) - 90*A*a^2*b^3*d^2*e^4*sign(
b*x + a) + 10*B*a^4*b*d*e^5*sign(b*x + a) + 20*A*a^3*b^2*d*e^5*sign(b*x + a) + B
*a^5*e^6*sign(b*x + a) + 5*A*a^4*b*e^6*sign(b*x + a))*x)*e^(-7)/(x*e + d)^3