3.1749 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^5} \, dx\)
Optimal. Leaf size=421 \[ -\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac{b^5 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]
[Out]
-((b^4*(5*b*B*d - A*b*e - 5*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^5*B*x^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b
*x)*(d + e*x)^4) + ((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])/(3*e^7*(a + b*x)*
(d + e*x)^3) - (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])/(2*e^7*(a + b*x)*
(d + e*x)^2) + (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])/(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)*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 = 0.392602, antiderivative size = 421, 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} \[ -\frac{b^4 x \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+5 b B d)}{e^6 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^7 (a+b x) (d+e x)}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{2 e^7 (a+b x) (d+e x)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{3 e^7 (a+b x) (d+e x)^3}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{4 e^7 (a+b x) (d+e x)^4}+\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)}{e^7 (a+b x)}+\frac{b^5 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^5,x]
[Out]
-((b^4*(5*b*B*d - A*b*e - 5*a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x))) + (b^5*B*x^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*e^5*(a + b*x)) - ((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b
*x)*(d + e*x)^4) + ((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])/(3*e^7*(a + b*x)*
(d + e*x)^3) - (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])/(2*e^7*(a + b*x)*
(d + e*x)^2) + (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])/(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)*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)^5} \, 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)^5} \, 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{b^9 (-5 b B d+A b e+5 a B e)}{e^6}+\frac{b^{10} B x}{e^5}-\frac{b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^5}+\frac{b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^4}-\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)^3}+\frac{10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^2}-\frac{5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{b^4 (5 b B d-A b e-5 a B e) x \sqrt{a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}+\frac{b^5 B x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^5 (a+b x)}-\frac{(b d-a e)^5 (B d-A e) \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\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}}{3 e^7 (a+b x) (d+e x)^3}-\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}}{2 e^7 (a+b x) (d+e x)^2}+\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}}{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) \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 = 0.338375, size = 497, normalized size = 1.18 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 b^3 e^2 \left (3 A e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )-B d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )+10 a^3 b^2 e^3 \left (A e \left (d^2+4 d e x+6 e^2 x^2\right )+3 B \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+5 a^4 b e^4 \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )+a^5 e^5 (3 A e+B (d+4 e x))-5 a b^4 e \left (A d e \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )-B \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )\right )-60 b^3 (d+e x)^4 (b d-a e) \log (d+e x) (-2 a B e-A b e+3 b B d)+b^5 \left (A e \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )-3 B \left (132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4+168 d^5 e x+57 d^6-12 d e^5 x^5+2 e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^5,x]
[Out]
-(Sqrt[(a + b*x)^2]*(a^5*e^5*(3*A*e + B*(d + 4*e*x)) + 5*a^4*b*e^4*(A*e*(d + 4*e*x) + B*(d^2 + 4*d*e*x + 6*e^2
*x^2)) + 10*a^3*b^2*e^3*(A*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + 3*B*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) +
10*a^2*b^3*e^2*(3*A*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - B*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 +
48*e^3*x^3)) - 5*a*b^4*e*(A*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) - B*(77*d^5 + 248*d^4*e*x
+ 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + b^5*(A*e*(77*d^5 + 248*d^4*e*x + 252*d^3*e^
2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - 3*B*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3
*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6)) - 60*b^3*(b*d - a*e)*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^
4*Log[d + e*x]))/(12*e^7*(a + b*x)*(d + e*x)^4)
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Maple [B] time = 0.02, size = 1163, 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)^5,x)
[Out]
1/12*((b*x+a)^2)^(5/2)*(-300*B*ln(e*x+d)*a*b^4*d^5*e+480*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*b^4*d^3*e^3-240*A*ln(e*x+d)*x*b^5*d^4*e^2+720*B*ln(e*x+d)*x*b^5*d^5*e-30*B*a^3*b
^2*d^3*e^3+250*B*a^2*b^3*d^4*e^2-385*B*a*b^4*d^5*e+125*A*a*b^4*d^4*e^2-5*B*a^4*b*d^2*e^4-10*A*a^3*b^2*d^2*e^4-
30*A*a^2*b^3*d^3*e^3-360*A*ln(e*x+d)*x^2*b^5*d^3*e^3+1080*B*ln(e*x+d)*x^2*b^5*d^4*e^2-240*A*ln(e*x+d)*x^3*b^5*
d^2*e^4+720*B*ln(e*x+d)*x^3*b^5*d^3*e^3-60*A*ln(e*x+d)*x^4*b^5*d*e^5+120*B*ln(e*x+d)*x^4*a^2*b^3*e^6+180*B*ln(
e*x+d)*x^4*b^5*d^2*e^4+60*A*ln(e*x+d)*x^4*a*b^4*e^6-1260*B*x^2*a*b^4*d^3*e^3-40*A*x*a^3*b^2*d*e^5-120*A*x*a^2*
b^3*d^2*e^4+440*A*x*a*b^4*d^3*e^3-20*B*x*a^4*b*d*e^5-120*B*x*a^3*b^2*d^2*e^4+1080*B*x^2*a^2*b^3*d^2*e^4-180*B*
x^2*a^3*b^2*d*e^5+880*B*x*a^2*b^3*d^3*e^3-1240*B*x*a*b^4*d^4*e^2+240*B*x^4*a*b^4*d*e^5+240*A*x^3*a*b^4*d*e^5+4
80*B*x^3*a^2*b^3*d*e^5-240*B*x^3*a*b^4*d^2*e^4-180*A*x^2*a^2*b^3*d*e^5+540*A*x^2*a*b^4*d^2*e^4+120*B*ln(e*x+d)
*a^2*b^3*d^4*e^2+60*A*ln(e*x+d)*a*b^4*d^4*e^2+180*B*ln(e*x+d)*b^5*d^6+6*B*x^6*b^5*e^6+12*A*x^5*b^5*e^6-4*B*x*a
^5*e^6-B*d*e^5*a^5-77*A*b^5*d^5*e-3*A*a^5*e^6+171*B*b^5*d^6-204*B*x^4*b^5*d^2*e^4-96*B*x^3*b^5*d^3*e^3-60*A*x^
2*a^3*b^2*e^6+60*B*x^5*a*b^4*e^6-36*B*x^5*b^5*d*e^5+48*A*x^4*b^5*d*e^5+504*B*x*b^5*d^5*e-120*A*x^3*a^2*b^3*e^6
-48*A*x^3*b^5*d^2*e^4-120*B*x^3*a^3*b^2*e^6-20*A*x*a^4*b*e^6-248*A*x*b^5*d^4*e^2-252*A*x^2*b^5*d^3*e^3-30*B*x^
2*a^4*b*e^6+396*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-300*B*ln(e*x+d)*x^4*a*b^4*d*e^5+360
*A*ln(e*x+d)*x^2*a*b^4*d^2*e^4+720*B*ln(e*x+d)*x^2*a^2*b^3*d^2*e^4-1800*B*ln(e*x+d)*x^2*a*b^4*d^3*e^3-1200*B*l
n(e*x+d)*x^3*a*b^4*d^2*e^4+240*A*ln(e*x+d)*x^3*a*b^4*d*e^5+480*B*ln(e*x+d)*x^3*a^2*b^3*d*e^5)/(b*x+a)^5/e^7/(e
*x+d)^4
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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)^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.68251, size = 1783, normalized size = 4.24 \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)^5,x, algorithm="fricas")
[Out]
1/12*(6*B*b^5*e^6*x^6 + 171*B*b^5*d^6 - 3*A*a^5*e^6 - 77*(5*B*a*b^4 + A*b^5)*d^5*e + 125*(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 - 5*(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 - 12*(3*B*b^5*d*e^5 - (5*B*a*b^4 + A*b^5)*e^6)*x^5 - 12*(17*B*b^5*d^2*e^4 - 4*(5*B*a*b^4 + A*b^5)*d*e^5)*x^4
- 24*(4*B*b^5*d^3*e^3 + 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 - 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 5*(B*a^3*b^2 + A*a
^2*b^3)*e^6)*x^3 + 6*(66*B*b^5*d^4*e^2 - 42*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 90*(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 - 5*(B*a^4*b + 2*A*a^3*b^2)*e^6)*x^2 + 4*(126*B*b^5*d^5*e - 62*(5*B*a*b^4 +
A*b^5)*d^4*e^2 + 110*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 30*(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 - (B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(3*B*b^5*d^6 - (5*B*a*b^4 + A*b^5)*d^5*e + (2*B*a^2*b^3 + A*a*
b^4)*d^4*e^2 + (3*B*b^5*d^2*e^4 - (5*B*a*b^4 + A*b^5)*d*e^5 + (2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 4*(3*B*b^5*d^
3*e^3 - (5*B*a*b^4 + A*b^5)*d^2*e^4 + (2*B*a^2*b^3 + A*a*b^4)*d*e^5)*x^3 + 6*(3*B*b^5*d^4*e^2 - (5*B*a*b^4 + A
*b^5)*d^3*e^3 + (2*B*a^2*b^3 + A*a*b^4)*d^2*e^4)*x^2 + 4*(3*B*b^5*d^5*e - (5*B*a*b^4 + A*b^5)*d^4*e^2 + (2*B*a
^2*b^3 + A*a*b^4)*d^3*e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*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)**5,x)
[Out]
Timed out
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Giac [B] time = 1.19383, size = 1175, normalized size = 2.79 \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)^5,x, algorithm="giac")
[Out]
5*(3*B*b^5*d^2*sgn(b*x + a) - 5*B*a*b^4*d*e*sgn(b*x + a) - A*b^5*d*e*sgn(b*x + a) + 2*B*a^2*b^3*e^2*sgn(b*x +
a) + A*a*b^4*e^2*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/2*(B*b^5*x^2*e^5*sgn(b*x + a) - 10*B*b^5*d*x*e^4*s
gn(b*x + a) + 10*B*a*b^4*x*e^5*sgn(b*x + a) + 2*A*b^5*x*e^5*sgn(b*x + a))*e^(-10) + 1/12*(171*B*b^5*d^6*sgn(b*
x + a) - 385*B*a*b^4*d^5*e*sgn(b*x + a) - 77*A*b^5*d^5*e*sgn(b*x + a) + 250*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 1
25*A*a*b^4*d^4*e^2*sgn(b*x + a) - 30*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 30*A*a^2*b^3*d^3*e^3*sgn(b*x + a) - 5*B*
a^4*b*d^2*e^4*sgn(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*sg
n(b*x + a) - 3*A*a^5*e^6*sgn(b*x + a) + 120*(2*B*b^5*d^3*e^3*sgn(b*x + a) - 5*B*a*b^4*d^2*e^4*sgn(b*x + a) - A
*b^5*d^2*e^4*sgn(b*x + a) + 4*B*a^2*b^3*d*e^5*sgn(b*x + a) + 2*A*a*b^4*d*e^5*sgn(b*x + a) - B*a^3*b^2*e^6*sgn(
b*x + a) - A*a^2*b^3*e^6*sgn(b*x + a))*x^3 + 30*(21*B*b^5*d^4*e^2*sgn(b*x + a) - 50*B*a*b^4*d^3*e^3*sgn(b*x +
a) - 10*A*b^5*d^3*e^3*sgn(b*x + a) + 36*B*a^2*b^3*d^2*e^4*sgn(b*x + a) + 18*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 + 4*(141*B*b^5*d^5*e*sgn(b*x + a) - 325*B*a*b^4*d^4*e^2*sgn(b*x + a) - 65*A*b^5*d^4*e^2*sgn(b*x
+ a) + 220*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 110*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) - 5*B*a^4*b*d*e^5*sgn(b*x + a) - 10*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)^4