3.1747 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=424 \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.479467, antiderivative size = 424, 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} \[ -\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}-\frac{10 b^2 x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.320854, size = 501, normalized size = 1.18 \[ \frac{\sqrt{(a+b x)^2} \left (60 a^2 b^3 e^2 \left (A e \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )+B \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )\right )+60 a^3 b^2 e^3 \left (A d e (3 d+4 e x)+B \left (-4 d^2 e x-5 d^3+4 d e^2 x^2+2 e^3 x^3\right )\right )-30 a^4 b e^4 (A e (d+2 e x)-B d (3 d+4 e x))-6 a^5 e^5 (A e+B (d+2 e x))+10 a b^4 e \left (3 A e \left (-11 d^2 e^2 x^2+2 d^3 e x+7 d^4-4 d e^3 x^3+e^4 x^4\right )+B \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )\right )+60 b (d+e x)^2 (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)+b^5 \left (2 A e \left (63 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x-27 d^5-5 d e^4 x^4+2 e^5 x^5\right )+3 B \left (-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-16 d^5 e x+22 d^6-2 d e^5 x^5+e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(Sqrt[(a + b*x)^2]*(-6*a^5*e^5*(A*e + B*(d + 2*e*x)) - 30*a^4*b*e^4*(A*e*(d + 2*e*x) - B*d*(3*d + 4*e*x)) + 60
*a^3*b^2*e^3*(A*d*e*(3*d + 4*e*x) + B*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 60*a^2*b^3*e^2*(A*e*(-
5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + B*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4))
+ 10*a*b^4*e*(3*A*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + B*(-27*d^5 + 6*d^4*e*x + 6
3*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5)) + b^5*(2*A*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2
+ 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + 3*B*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d
^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6)) + 60*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2*Log[d + e*x]
))/(12*e^7*(a + b*x)*(d + e*x)^2)

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Maple [B]  time = 0.023, size = 1205, normalized size = 2.8 \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)^3,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(-600*B*ln(e*x+d)*a*b^4*d^5*e+120*B*ln(e*x+d)*x*a^4*b*d*e^5-720*B*ln(e*x+d)*x*a^3*b^2*d
^2*e^4+1440*B*ln(e*x+d)*x*a^2*b^3*d^3*e^3-1200*B*ln(e*x+d)*x*a*b^4*d^4*e^2+240*A*ln(e*x+d)*x*a^3*b^2*d*e^5-720
*A*ln(e*x+d)*x*a^2*b^3*d^2*e^4+720*A*ln(e*x+d)*x*a*b^4*d^3*e^3-240*A*ln(e*x+d)*x*b^5*d^4*e^2+360*B*ln(e*x+d)*x
*b^5*d^5*e-300*B*a^3*b^2*d^3*e^3+420*B*a^2*b^3*d^4*e^2-270*B*a*b^4*d^5*e+210*A*a*b^4*d^4*e^2+90*B*a^4*b*d^2*e^
4+180*A*a^3*b^2*d^2*e^4-300*A*a^2*b^3*d^3*e^3+60*B*ln(e*x+d)*x^2*a^4*b*e^6+120*A*ln(e*x+d)*x^2*a^3*b^2*e^6-120
*A*ln(e*x+d)*x^2*b^5*d^3*e^3+180*B*ln(e*x+d)*x^2*b^5*d^4*e^2+630*B*x^2*a*b^4*d^3*e^3+240*A*x*a^3*b^2*d*e^5-240
*A*x*a^2*b^3*d^2*e^4+60*A*x*a*b^4*d^3*e^3+120*B*x*a^4*b*d*e^5-240*B*x*a^3*b^2*d^2*e^4-660*B*x^2*a^2*b^3*d^2*e^
4+240*B*x^2*a^3*b^2*d*e^5+120*A*ln(e*x+d)*a^3*b^2*d^2*e^4-360*A*ln(e*x+d)*a^2*b^3*d^3*e^3+120*B*x*a^2*b^3*d^3*
e^3+60*B*x*a*b^4*d^4*e^2-50*B*x^4*a*b^4*d*e^5-120*A*x^3*a*b^4*d*e^5-240*B*x^3*a^2*b^3*d*e^5+200*B*x^3*a*b^4*d^
2*e^4+240*A*x^2*a^2*b^3*d*e^5-330*A*x^2*a*b^4*d^2*e^4+720*B*ln(e*x+d)*a^2*b^3*d^4*e^2+360*A*ln(e*x+d)*a*b^4*d^
4*e^2+60*B*ln(e*x+d)*a^4*b*d^2*e^4-360*B*ln(e*x+d)*a^3*b^2*d^3*e^3+180*B*ln(e*x+d)*b^5*d^6+3*B*x^6*b^5*e^6+4*A
*x^5*b^5*e^6-12*B*x*a^5*e^6-6*B*d*e^5*a^5-54*A*b^5*d^5*e-6*A*a^5*e^6+66*B*b^5*d^6+15*B*x^4*b^5*d^2*e^4-60*B*x^
3*b^5*d^3*e^3+20*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^6-10*A*x^4*b^5*d*e^5+60*B*x^4*a^2*b^3*e^6-
48*B*x*b^5*d^5*e+120*A*x^3*a^2*b^3*e^6+40*A*x^3*b^5*d^2*e^4+120*B*x^3*a^3*b^2*e^6-60*A*x*a^4*b*e^6+12*A*x*b^5*
d^4*e^2+126*A*x^2*b^5*d^3*e^3-204*B*x^2*b^5*d^4*e^2-120*A*ln(e*x+d)*b^5*d^5*e-30*A*d*e^5*a^4*b-360*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^4-360*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^4-600*B*ln(e*x+d)*x^2*a*b^4*d^3*e^3)/(b*x+a)^5/e^7/(e*x+d)^2

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.71, size = 1789, normalized size = 4.22 \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^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*e^6*x^6 + 66*B*b^5*d^6 - 6*A*a^5*e^6 - 54*(5*B*a*b^4 + A*b^5)*d^5*e + 210*(2*B*a^2*b^3 + A*a*b^4
)*d^4*e^2 - 300*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 90*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 6*(B*a^5 + 5*A*a^4*b)*d
*e^5 - 2*(3*B*b^5*d*e^5 - 2*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(3*B*b^5*d^2*e^4 - 2*(5*B*a*b^4 + A*b^5)*d*e^5 +
6*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 20*(3*B*b^5*d^3*e^3 - 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 6*(2*B*a^2*b^3 + A*
a*b^4)*d*e^5 - 6*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 - 6*(34*B*b^5*d^4*e^2 - 21*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 55*
(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 - 12*(4*B*b^5*d^5*e - (5*B*a*b^4 + A*b
^5)*d^4*e^2 - 5*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 20*(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*(3*B*b^5*d^6 - 2*(5*B*a*b^4 + A*b^5)*d^5*e + 6*(2*B*a^2*b^3 + A*a*
b^4)*d^4*e^2 - 6*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + (B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + (3*B*b^5*d^4*e^2 - 2*(5*B
*a*b^4 + A*b^5)*d^3*e^3 + 6*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 6*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*a^4*b + 2*A
*a^3*b^2)*e^6)*x^2 + 2*(3*B*b^5*d^5*e - 2*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 6*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 6*
(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + (B*a^4*b + 2*A*a^3*b^2)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.21466, size = 1197, normalized size = 2.82 \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)^3,x, algorithm="giac")

[Out]

5*(3*B*b^5*d^4*sgn(b*x + a) - 10*B*a*b^4*d^3*e*sgn(b*x + a) - 2*A*b^5*d^3*e*sgn(b*x + a) + 12*B*a^2*b^3*d^2*e^
2*sgn(b*x + a) + 6*A*a*b^4*d^2*e^2*sgn(b*x + a) - 6*B*a^3*b^2*d*e^3*sgn(b*x + a) - 6*A*a^2*b^3*d*e^3*sgn(b*x +
a) + B*a^4*b*e^4*sgn(b*x + a) + 2*A*a^3*b^2*e^4*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/12*(3*B*b^5*x^4*e^
9*sgn(b*x + a) - 12*B*b^5*d*x^3*e^8*sgn(b*x + a) + 36*B*b^5*d^2*x^2*e^7*sgn(b*x + a) - 120*B*b^5*d^3*x*e^6*sgn
(b*x + a) + 20*B*a*b^4*x^3*e^9*sgn(b*x + a) + 4*A*b^5*x^3*e^9*sgn(b*x + a) - 90*B*a*b^4*d*x^2*e^8*sgn(b*x + a)
- 18*A*b^5*d*x^2*e^8*sgn(b*x + a) + 360*B*a*b^4*d^2*x*e^7*sgn(b*x + a) + 72*A*b^5*d^2*x*e^7*sgn(b*x + a) + 60
*B*a^2*b^3*x^2*e^9*sgn(b*x + a) + 30*A*a*b^4*x^2*e^9*sgn(b*x + a) - 360*B*a^2*b^3*d*x*e^8*sgn(b*x + a) - 180*A
*a*b^4*d*x*e^8*sgn(b*x + a) + 120*B*a^3*b^2*x*e^9*sgn(b*x + a) + 120*A*a^2*b^3*x*e^9*sgn(b*x + a))*e^(-12) + 1
/2*(11*B*b^5*d^6*sgn(b*x + a) - 45*B*a*b^4*d^5*e*sgn(b*x + a) - 9*A*b^5*d^5*e*sgn(b*x + a) + 70*B*a^2*b^3*d^4*
e^2*sgn(b*x + a) + 35*A*a*b^4*d^4*e^2*sgn(b*x + a) - 50*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 50*A*a^2*b^3*d^3*e^3*
sgn(b*x + a) + 15*B*a^4*b*d^2*e^4*sgn(b*x + a) + 30*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) - A*a^5*e^6*sgn(b*x + a) + 2*(6*B*b^5*d^5*e*sgn(b*x + a) - 25*B*a*b^4*d^4*e^2*s
gn(b*x + a) - 5*A*b^5*d^4*e^2*sgn(b*x + a) + 40*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 20*A*a*b^4*d^3*e^3*sgn(b*x +
a) - 30*B*a^3*b^2*d^2*e^4*sgn(b*x + a) - 30*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) + 2
0*A*a^3*b^2*d*e^5*sgn(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)^2