3.1746 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=423 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) (d+e x)}+\frac{5 b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^6 (a+b x)}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-5 a B e-A b e+6 b B d)}{4 e^7 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)} \]
[Out]
(5*b*(b*d - a*e)^3*(3*b*B*d - 2*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])/(e^7*(a + b*x)*(d + e*x)) - (5*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*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^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d +
e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) + (b^5*B*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5
*e^7*(a + b*x)) - ((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]*Log[d + e*x])/(e^7*
(a + b*x))
________________________________________________________________________________________
Rubi [A] time = 0.563251, antiderivative size = 423, 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} \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{e^7 (a+b x) (d+e x)}+\frac{5 b x \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{e^6 (a+b x)}-\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x)}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (-2 a B e-A b e+3 b B d)}{3 e^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-5 a B e-A b e+6 b B d)}{4 e^7 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)}{e^7 (a+b x)}+\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^7 (a+b x)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^2,x]
[Out]
(5*b*(b*d - a*e)^3*(3*b*B*d - 2*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])/(e^7*(a + b*x)*(d + e*x)) - (5*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*
e - a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*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^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d +
e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) + (b^5*B*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5
*e^7*(a + b*x)) - ((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]*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)^2} \, 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)^2} \, 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^6 (b d-a e)^3 (-3 b B d+2 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)^2}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e) (d+e x)}{e^6}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)^2}{e^6}+\frac{b^9 (-6 b B d+A b e+5 a B e) (d+e x)^3}{e^6}+\frac{b^{10} B (d+e x)^4}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{5 b (b d-a e)^3 (3 b B d-2 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}}{e^7 (a+b x) (d+e x)}-\frac{5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{5 b^3 (b d-a e) (3 b B d-A b e-2 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^4 (6 b B d-A b e-5 a B e) (d+e x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac{b^5 B (d+e x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\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} \log (d+e x)}{e^7 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.298301, size = 506, normalized size = 1.2 \[ \frac{\sqrt{(a+b x)^2} \left (100 a^2 b^3 e^2 \left (3 A e \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )+2 B \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )\right )+300 a^3 b^2 e^3 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (-4 d^2 e x+2 d^3-3 d e^2 x^2+e^3 x^3\right )\right )+300 a^4 b e^4 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+60 a^5 e^5 (B d-A e)+25 a b^4 e \left (4 A e \left (6 d^2 e^2 x^2+9 d^3 e x-3 d^4-2 d e^3 x^3+e^4 x^4\right )+B \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )\right )-60 (d+e x) (b d-a e)^4 \log (d+e x) (-a B e-5 A b e+6 b B d)+b^5 \left (5 A e \left (-30 d^3 e^2 x^2+10 d^2 e^3 x^3-48 d^4 e x+12 d^5-5 d e^4 x^4+3 e^5 x^5\right )-6 B \left (-30 d^4 e^2 x^2+10 d^3 e^3 x^3-5 d^2 e^4 x^4-50 d^5 e x+10 d^6+3 d e^5 x^5-2 e^6 x^6\right )\right )\right )}{60 e^7 (a+b x) (d+e x)} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^2,x]
[Out]
(Sqrt[(a + b*x)^2]*(60*a^5*e^5*(B*d - A*e) + 300*a^4*b*e^4*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 300*a^3*b^2*
e^3*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3)) + 100*a^2*b^3*e^2*(3*A*e*
(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)
) + 25*a*b^4*e*(4*A*e*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*e*x -
30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + b^5*(5*A*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2
+ 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) - 6*B*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*
d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)) - 60*(b*d - a*e)^4*(6*b*B*d - 5*A*b*e - a*B*e)*(d + e*x)*Log[d + e*x])
)/(60*e^7*(a + b*x)*(d + e*x))
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Maple [B] time = 0.021, size = 1084, normalized size = 2.6 \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)^2,x)
[Out]
1/60*((b*x+a)^2)^(5/2)*(1500*B*ln(e*x+d)*a*b^4*d^5*e-600*B*ln(e*x+d)*x*a^4*b*d*e^5+1800*B*ln(e*x+d)*x*a^3*b^2*
d^2*e^4-2400*B*ln(e*x+d)*x*a^2*b^3*d^3*e^3+1500*B*ln(e*x+d)*x*a*b^4*d^4*e^2-1200*A*ln(e*x+d)*x*a^3*b^2*d*e^5+1
800*A*ln(e*x+d)*x*a^2*b^3*d^2*e^4-1200*A*ln(e*x+d)*x*a*b^4*d^3*e^3+300*A*ln(e*x+d)*x*a^4*b*e^6+300*A*ln(e*x+d)
*x*b^5*d^4*e^2-360*B*ln(e*x+d)*x*b^5*d^5*e+600*B*a^3*b^2*d^3*e^3-600*B*a^2*b^3*d^4*e^2+300*B*a*b^4*d^5*e-300*A
*a*b^4*d^4*e^2-300*B*a^4*b*d^2*e^4-600*A*a^3*b^2*d^2*e^4+600*A*a^2*b^3*d^3*e^3-750*B*x^2*a*b^4*d^3*e^3+600*A*x
*a^3*b^2*d*e^5-1200*A*x*a^2*b^3*d^2*e^4+900*A*x*a*b^4*d^3*e^3+300*B*x*a^4*b*d*e^5-1200*B*x*a^3*b^2*d^2*e^4+120
0*B*x^2*a^2*b^3*d^2*e^4-900*B*x^2*a^3*b^2*d*e^5-1200*A*ln(e*x+d)*a^3*b^2*d^2*e^4+1800*A*ln(e*x+d)*a^2*b^3*d^3*
e^3+300*A*ln(e*x+d)*a^4*b*d*e^5+1800*B*x*a^2*b^3*d^3*e^3-1200*B*x*a*b^4*d^4*e^2-125*B*x^4*a*b^4*d*e^5-200*A*x^
3*a*b^4*d*e^5-400*B*x^3*a^2*b^3*d*e^5+250*B*x^3*a*b^4*d^2*e^4-900*A*x^2*a^2*b^3*d*e^5+600*A*x^2*a*b^4*d^2*e^4-
2400*B*ln(e*x+d)*a^2*b^3*d^4*e^2-1200*A*ln(e*x+d)*a*b^4*d^4*e^2-600*B*ln(e*x+d)*a^4*b*d^2*e^4+1800*B*ln(e*x+d)
*a^3*b^2*d^3*e^3-360*B*ln(e*x+d)*b^5*d^6+12*B*x^6*b^5*e^6+15*A*x^5*b^5*e^6+60*B*d*e^5*a^5+60*A*b^5*d^5*e+60*B*
ln(e*x+d)*x*a^5*e^6-60*A*a^5*e^6-60*B*b^5*d^6+30*B*x^4*b^5*d^2*e^4-60*B*x^3*b^5*d^3*e^3+600*A*x^2*a^3*b^2*e^6+
75*B*x^5*a*b^4*e^6-18*B*x^5*b^5*d*e^5+100*A*x^4*a*b^4*e^6-25*A*x^4*b^5*d*e^5+200*B*x^4*a^2*b^3*e^6+300*B*x*b^5
*d^5*e+300*A*x^3*a^2*b^3*e^6+50*A*x^3*b^5*d^2*e^4+300*B*x^3*a^3*b^2*e^6-240*A*x*b^5*d^4*e^2-150*A*x^2*b^5*d^3*
e^3+300*B*x^2*a^4*b*e^6+180*B*x^2*b^5*d^4*e^2+300*A*ln(e*x+d)*b^5*d^5*e+60*B*ln(e*x+d)*a^5*d*e^5+300*A*d*e^5*a
^4*b)/(b*x+a)^5/e^7/(e*x+d)
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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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.69203, size = 1661, normalized size = 3.93 \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)^2,x, algorithm="fricas")
[Out]
1/60*(12*B*b^5*e^6*x^6 - 60*B*b^5*d^6 - 60*A*a^5*e^6 + 60*(5*B*a*b^4 + A*b^5)*d^5*e - 300*(2*B*a^2*b^3 + A*a*b
^4)*d^4*e^2 + 600*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 - 300*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 60*(B*a^5 + 5*A*a^4*
b)*d*e^5 - 3*(6*B*b^5*d*e^5 - 5*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(6*B*b^5*d^2*e^4 - 5*(5*B*a*b^4 + A*b^5)*d*e^
5 + 20*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 10*(6*B*b^5*d^3*e^3 - 5*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 20*(2*B*a^2*b^
3 + A*a*b^4)*d*e^5 - 30*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 30*(6*B*b^5*d^4*e^2 - 5*(5*B*a*b^4 + A*b^5)*d^3*e^3
+ 20*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 30*(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
+ 60*(5*B*b^5*d^5*e - 4*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 15*(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 + 5*(B*a^4*b + 2*A*a^3*b^2)*d*e^5)*x - 60*(6*B*b^5*d^6 - 5*(5*B*a*b^4 + A*b^5)*d^5*e + 20*(2*
B*a^2*b^3 + A*a*b^4)*d^4*e^2 - 30*(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 + (6*B*b^5*d^5*e - 5*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 3
0*(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)*log(e*x + d
))/(e^8*x + d*e^7)
________________________________________________________________________________________
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)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.17192, size = 1222, normalized size = 2.89 \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)^2,x, algorithm="giac")
[Out]
-(6*B*b^5*d^5*sgn(b*x + a) - 25*B*a*b^4*d^4*e*sgn(b*x + a) - 5*A*b^5*d^4*e*sgn(b*x + a) + 40*B*a^2*b^3*d^3*e^2
*sgn(b*x + a) + 20*A*a*b^4*d^3*e^2*sgn(b*x + a) - 30*B*a^3*b^2*d^2*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^2*e^3*sgn
(b*x + a) + 10*B*a^4*b*d*e^4*sgn(b*x + a) + 20*A*a^3*b^2*d*e^4*sgn(b*x + a) - B*a^5*e^5*sgn(b*x + a) - 5*A*a^4
*b*e^5*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/60*(12*B*b^5*x^5*e^8*sgn(b*x + a) - 30*B*b^5*d*x^4*e^7*sgn(b
*x + a) + 60*B*b^5*d^2*x^3*e^6*sgn(b*x + a) - 120*B*b^5*d^3*x^2*e^5*sgn(b*x + a) + 300*B*b^5*d^4*x*e^4*sgn(b*x
+ a) + 75*B*a*b^4*x^4*e^8*sgn(b*x + a) + 15*A*b^5*x^4*e^8*sgn(b*x + a) - 200*B*a*b^4*d*x^3*e^7*sgn(b*x + a) -
40*A*b^5*d*x^3*e^7*sgn(b*x + a) + 450*B*a*b^4*d^2*x^2*e^6*sgn(b*x + a) + 90*A*b^5*d^2*x^2*e^6*sgn(b*x + a) -
1200*B*a*b^4*d^3*x*e^5*sgn(b*x + a) - 240*A*b^5*d^3*x*e^5*sgn(b*x + a) + 200*B*a^2*b^3*x^3*e^8*sgn(b*x + a) +
100*A*a*b^4*x^3*e^8*sgn(b*x + a) - 600*B*a^2*b^3*d*x^2*e^7*sgn(b*x + a) - 300*A*a*b^4*d*x^2*e^7*sgn(b*x + a) +
1800*B*a^2*b^3*d^2*x*e^6*sgn(b*x + a) + 900*A*a*b^4*d^2*x*e^6*sgn(b*x + a) + 300*B*a^3*b^2*x^2*e^8*sgn(b*x +
a) + 300*A*a^2*b^3*x^2*e^8*sgn(b*x + a) - 1200*B*a^3*b^2*d*x*e^7*sgn(b*x + a) - 1200*A*a^2*b^3*d*x*e^7*sgn(b*x
+ a) + 300*B*a^4*b*x*e^8*sgn(b*x + a) + 600*A*a^3*b^2*x*e^8*sgn(b*x + a))*e^(-10) - (B*b^5*d^6*sgn(b*x + a) -
5*B*a*b^4*d^5*e*sgn(b*x + a) - A*b^5*d^5*e*sgn(b*x + a) + 10*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 5*A*a*b^4*d^4*e
^2*sgn(b*x + a) - 10*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 10*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 5*B*a^4*b*d^2*e^4*sg
n(b*x + a) + 10*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))*e^(-7)/(x*e + d)