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

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

[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 = 0.429413, 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, Rules used = {770, 77} \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (-5 a B e-A b e+6 b B d)}{2 e^7 (a+b x)}+\frac{5 b^3 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{e^6 (a+b x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^7 (a+b x) (d+e x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{2 e^7 (a+b x) (d+e x)^2}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{3 e^7 (a+b x) (d+e x)^3}-\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3}{3 e^7 (a+b x)} \]

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))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

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) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^4} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6}-\frac{b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^4}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^3}-\frac{5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^2}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)}{e^6}+\frac{b^{10} B (d+e x)^2}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{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)}-\frac{(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}+\frac{(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}-\frac{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)}-\frac{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)}+\frac{b^5 B (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{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)}\\ \end{align*}

Mathematica [A]  time = 0.355237, size = 504, normalized size = 1.19 \[ -\frac{\sqrt{(a+b x)^2} \left (-10 a^2 b^3 e^2 \left (A d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 B \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )\right )+10 a^3 b^2 e^3 \left (2 A e \left (d^2+3 d e x+3 e^2 x^2\right )-B d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )+5 a^4 b e^4 \left (A e (d+3 e x)+2 B \left (d^2+3 d e x+3 e^2 x^2\right )\right )+a^5 e^5 (2 A e+B (d+3 e x))-5 a b^4 e \left (2 A e \left (-9 d^2 e^2 x^2-27 d^3 e x-13 d^4+9 d e^3 x^3+3 e^4 x^4\right )+B \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )\right )+60 b^2 (d+e x)^3 (b d-a e)^2 \log (d+e x) (-a B e-A b e+2 b B d)+b^5 \left (A e \left (9 d^3 e^2 x^2+63 d^2 e^3 x^3-81 d^4 e x-47 d^5+15 d e^4 x^4-3 e^5 x^5\right )+2 B \left (-39 d^4 e^2 x^2-73 d^3 e^3 x^3-15 d^2 e^4 x^4+51 d^5 e x+37 d^6+3 d e^5 x^5-e^6 x^6\right )\right )\right )}{6 e^7 (a+b x) (d+e x)^3} \]

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*L
og[d + e*x]))/(6*e^7*(a + b*x)*(d + e*x)^3)

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Maple [B]  time = 0.023, size = 1233, normalized size = 2.9 \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)*(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)*(300*B*ln(e*x+d)*a*b^4*d^5*e+180*B*ln(e*x+d)*x*a^3*b^2*d^2*e^4-720*B*ln(e*x+d)*x*a^2*b^3
*d^3*e^3+900*B*ln(e*x+d)*x*a*b^4*d^4*e^2+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+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+110*B*a^3*b^2*d^3*e^3-260*B*a^2*b^3*d^4*e^2+235*B*a*b^4
*d^5*e-130*A*a*b^4*d^4*e^2-10*B*a^4*b*d^2*e^4-20*A*a^3*b^2*d^2*e^4+110*A*a^2*b^3*d^3*e^3+180*A*ln(e*x+d)*x^2*b
^5*d^3*e^3-360*B*ln(e*x+d)*x^2*b^5*d^4*e^2+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+60*A*ln(e*x+d)*x^3*a^2*b^3*e^6-45*B*x^2*a*b^4*d^3*e^3-60*A*x*a^3*b^2*d*e^5+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^4-180*B*x^2*a^2*b^3*d^2*e^4+180
*B*x^2*a^3*b^2*d*e^5+60*A*ln(e*x+d)*a^2*b^3*d^3*e^3-540*B*x*a^2*b^3*d^3*e^3+405*B*x*a*b^4*d^4*e^2-75*B*x^4*a*b
^4*d*e^5+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+180*A*x^2*a^2*b^3*d*e^5-90*A*x^2
*a*b^4*d^2*e^4-240*B*ln(e*x+d)*a^2*b^3*d^4*e^2-120*A*ln(e*x+d)*a*b^4*d^4*e^2+60*B*ln(e*x+d)*a^3*b^2*d^3*e^3-12
0*B*ln(e*x+d)*b^5*d^6+2*B*x^6*b^5*e^6+3*A*x^5*b^5*e^6-3*B*x*a^5*e^6-B*d*e^5*a^5+47*A*b^5*d^5*e-2*A*a^5*e^6-74*
B*b^5*d^6+30*B*x^4*b^5*d^2*e^4+146*B*x^3*b^5*d^3*e^3-60*A*x^2*a^3*b^2*e^6+15*B*x^5*a*b^4*e^6-6*B*x^5*b^5*d*e^5
+30*A*x^4*a*b^4*e^6-15*A*x^4*b^5*d*e^5+60*B*x^4*a^2*b^3*e^6-102*B*x*b^5*d^5*e-63*A*x^3*b^5*d^2*e^4-15*A*x*a^4*
b*e^6+81*A*x*b^5*d^4*e^2-9*A*x^2*b^5*d^3*e^3-30*B*x^2*a^4*b*e^6+78*B*x^2*b^5*d^4*e^2+60*A*ln(e*x+d)*b^5*d^5*e-
5*A*d*e^5*a^4*b+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^5-720*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+300*B*ln(e*x+d)*x^3*a*b^4*d^2*e^
4-120*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)/(b*x+a)^5/e^7/(e*x+d)^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.66038, size = 1827, normalized size = 4.3 \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^2*x^2+2*a*b*x+a^2)^(5/2)/(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 - 10*(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 - 27*(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. \begin{align*} \text{Timed out} \end{align*}

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 [B]  time = 1.16698, size = 1180, normalized size = 2.78 \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^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

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