3.1740 \(\int (A+B x) (d+e x)^4 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=324 \[ \frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (-5 a B e+A b e+4 b B d)}{10 b^6}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{9 b^6}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{4 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^4}{6 b^6}+\frac{B e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*
e - 5*a*B*e)*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^6) + (e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*
e)*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^6) + (2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a +
b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^9*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(10*b^6) + (B*e^4*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*b^6)

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Rubi [A]  time = 0.624054, antiderivative size = 324, 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} \[ \frac{e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9 (-5 a B e+A b e+4 b B d)}{10 b^6}+\frac{2 e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{9 b^6}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{4 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^3 (-5 a B e+4 A b e+b B d)}{7 b^6}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^4}{6 b^6}+\frac{B e^4 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^{10}}{11 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(b*d - a*e)^4*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^6) + ((b*d - a*e)^3*(b*B*d + 4*A*b*
e - 5*a*B*e)*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*b^6) + (e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*
e)*(a + b*x)^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^6) + (2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e)*(a +
b*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*b^6) + (e^3*(4*b*B*d + A*b*e - 5*a*B*e)*(a + b*x)^9*Sqrt[a^2 + 2*a*b*
x + b^2*x^2])/(10*b^6) + (B*e^4*(a + b*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*b^6)

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 (A+B x) (d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \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{(A b-a B) (b d-a e)^4 \left (a b+b^2 x\right )^5}{b^5}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e) \left (a b+b^2 x\right )^6}{b^6}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) \left (a b+b^2 x\right )^7}{b^7}+\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) \left (a b+b^2 x\right )^8}{b^8}+\frac{e^3 (4 b B d+A b e-5 a B e) \left (a b+b^2 x\right )^9}{b^9}+\frac{B e^4 \left (a b+b^2 x\right )^{10}}{b^{10}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{(A b-a B) (b d-a e)^4 (a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^6}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e) (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac{e (b d-a e)^2 (2 b B d+3 A b e-5 a B e) (a+b x)^7 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^6}+\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e) (a+b x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac{e^3 (4 b B d+A b e-5 a B e) (a+b x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{10 b^6}+\frac{B e^4 (a+b x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{11 b^6}\\ \end{align*}

Mathematica [A]  time = 0.256272, size = 611, normalized size = 1.89 \[ \frac{x \sqrt{(a+b x)^2} \left (165 a^3 b^2 x^2 \left (8 A \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )+3 B x \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )\right )+55 a^2 b^3 x^3 \left (9 A \left (280 d^2 e^2 x^2+224 d^3 e x+70 d^4+160 d e^3 x^3+35 e^4 x^4\right )+4 B x \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )\right )+330 a^4 b x \left (7 A \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )+2 B x \left (126 d^2 e^2 x^2+105 d^3 e x+35 d^4+70 d e^3 x^3+15 e^4 x^4\right )\right )+462 a^5 \left (6 A \left (10 d^2 e^2 x^2+10 d^3 e x+5 d^4+5 d e^3 x^3+e^4 x^4\right )+B x \left (45 d^2 e^2 x^2+40 d^3 e x+15 d^4+24 d e^3 x^3+5 e^4 x^4\right )\right )+55 a b^4 x^4 \left (2 A \left (540 d^2 e^2 x^2+420 d^3 e x+126 d^4+315 d e^3 x^3+70 e^4 x^4\right )+B x \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )\right )+b^5 x^5 \left (11 A \left (945 d^2 e^2 x^2+720 d^3 e x+210 d^4+560 d e^3 x^3+126 e^4 x^4\right )+6 B x \left (1540 d^2 e^2 x^2+1155 d^3 e x+330 d^4+924 d e^3 x^3+210 e^4 x^4\right )\right )\right )}{13860 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(462*a^5*(6*A*(5*d^4 + 10*d^3*e*x + 10*d^2*e^2*x^2 + 5*d*e^3*x^3 + e^4*x^4) + B*x*(15*d^4
+ 40*d^3*e*x + 45*d^2*e^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4)) + 330*a^4*b*x*(7*A*(15*d^4 + 40*d^3*e*x + 45*d^2*e
^2*x^2 + 24*d*e^3*x^3 + 5*e^4*x^4) + 2*B*x*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4
)) + 165*a^3*b^2*x^2*(8*A*(35*d^4 + 105*d^3*e*x + 126*d^2*e^2*x^2 + 70*d*e^3*x^3 + 15*e^4*x^4) + 3*B*x*(70*d^4
+ 224*d^3*e*x + 280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4)) + 55*a^2*b^3*x^3*(9*A*(70*d^4 + 224*d^3*e*x +
280*d^2*e^2*x^2 + 160*d*e^3*x^3 + 35*e^4*x^4) + 4*B*x*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3
+ 70*e^4*x^4)) + 55*a*b^4*x^4*(2*A*(126*d^4 + 420*d^3*e*x + 540*d^2*e^2*x^2 + 315*d*e^3*x^3 + 70*e^4*x^4) + B
*x*(210*d^4 + 720*d^3*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4)) + b^5*x^5*(11*A*(210*d^4 + 720*d^3
*e*x + 945*d^2*e^2*x^2 + 560*d*e^3*x^3 + 126*e^4*x^4) + 6*B*x*(330*d^4 + 1155*d^3*e*x + 1540*d^2*e^2*x^2 + 924
*d*e^3*x^3 + 210*e^4*x^4))))/(13860*(a + b*x))

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Maple [B]  time = 0.009, size = 872, normalized size = 2.7 \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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/13860*x*(1260*B*b^5*e^4*x^10+1386*A*b^5*e^4*x^9+6930*B*a*b^4*e^4*x^9+5544*B*b^5*d*e^3*x^9+7700*A*a*b^4*e^4*x
^8+6160*A*b^5*d*e^3*x^8+15400*B*a^2*b^3*e^4*x^8+30800*B*a*b^4*d*e^3*x^8+9240*B*b^5*d^2*e^2*x^8+17325*A*a^2*b^3
*e^4*x^7+34650*A*a*b^4*d*e^3*x^7+10395*A*b^5*d^2*e^2*x^7+17325*B*a^3*b^2*e^4*x^7+69300*B*a^2*b^3*d*e^3*x^7+519
75*B*a*b^4*d^2*e^2*x^7+6930*B*b^5*d^3*e*x^7+19800*A*a^3*b^2*e^4*x^6+79200*A*a^2*b^3*d*e^3*x^6+59400*A*a*b^4*d^
2*e^2*x^6+7920*A*b^5*d^3*e*x^6+9900*B*a^4*b*e^4*x^6+79200*B*a^3*b^2*d*e^3*x^6+118800*B*a^2*b^3*d^2*e^2*x^6+396
00*B*a*b^4*d^3*e*x^6+1980*B*b^5*d^4*x^6+11550*A*a^4*b*e^4*x^5+92400*A*a^3*b^2*d*e^3*x^5+138600*A*a^2*b^3*d^2*e
^2*x^5+46200*A*a*b^4*d^3*e*x^5+2310*A*b^5*d^4*x^5+2310*B*a^5*e^4*x^5+46200*B*a^4*b*d*e^3*x^5+138600*B*a^3*b^2*
d^2*e^2*x^5+92400*B*a^2*b^3*d^3*e*x^5+11550*B*a*b^4*d^4*x^5+2772*A*a^5*e^4*x^4+55440*A*a^4*b*d*e^3*x^4+166320*
A*a^3*b^2*d^2*e^2*x^4+110880*A*a^2*b^3*d^3*e*x^4+13860*A*a*b^4*d^4*x^4+11088*B*a^5*d*e^3*x^4+83160*B*a^4*b*d^2
*e^2*x^4+110880*B*a^3*b^2*d^3*e*x^4+27720*B*a^2*b^3*d^4*x^4+13860*A*a^5*d*e^3*x^3+103950*A*a^4*b*d^2*e^2*x^3+1
38600*A*a^3*b^2*d^3*e*x^3+34650*A*a^2*b^3*d^4*x^3+20790*B*a^5*d^2*e^2*x^3+69300*B*a^4*b*d^3*e*x^3+34650*B*a^3*
b^2*d^4*x^3+27720*A*a^5*d^2*e^2*x^2+92400*A*a^4*b*d^3*e*x^2+46200*A*a^3*b^2*d^4*x^2+18480*B*a^5*d^3*e*x^2+2310
0*B*a^4*b*d^4*x^2+27720*A*a^5*d^3*e*x+34650*A*a^4*b*d^4*x+6930*B*a^5*d^4*x+13860*A*a^5*d^4)*((b*x+a)^2)^(5/2)/
(b*x+a)^5

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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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.61878, size = 1420, normalized size = 4.38 \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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/11*B*b^5*e^4*x^11 + A*a^5*d^4*x + 1/10*(4*B*b^5*d*e^3 + (5*B*a*b^4 + A*b^5)*e^4)*x^10 + 1/9*(6*B*b^5*d^2*e^2
+ 4*(5*B*a*b^4 + A*b^5)*d*e^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^4)*x^9 + 1/4*(2*B*b^5*d^3*e + 3*(5*B*a*b^4 + A*b^
5)*d^2*e^2 + 10*(2*B*a^2*b^3 + A*a*b^4)*d*e^3 + 5*(B*a^3*b^2 + A*a^2*b^3)*e^4)*x^8 + 1/7*(B*b^5*d^4 + 4*(5*B*a
*b^4 + A*b^5)*d^3*e + 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^2 + 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^3 + 5*(B*a^4*b + 2*A
*a^3*b^2)*e^4)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^4 + 20*(2*B*a^2*b^3 + A*a*b^4)*d^3*e + 60*(B*a^3*b^2 + A*a^2*b
^3)*d^2*e^2 + 20*(B*a^4*b + 2*A*a^3*b^2)*d*e^3 + (B*a^5 + 5*A*a^4*b)*e^4)*x^6 + 1/5*(A*a^5*e^4 + 5*(2*B*a^2*b^
3 + A*a*b^4)*d^4 + 40*(B*a^3*b^2 + A*a^2*b^3)*d^3*e + 30*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^2 + 4*(B*a^5 + 5*A*a^4*
b)*d*e^3)*x^5 + 1/2*(2*A*a^5*d*e^3 + 5*(B*a^3*b^2 + A*a^2*b^3)*d^4 + 10*(B*a^4*b + 2*A*a^3*b^2)*d^3*e + 3*(B*a
^5 + 5*A*a^4*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^5*d^2*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^4 + 4*(B*a^5 + 5*A*a^4*b)*d^
3*e)*x^3 + 1/2*(4*A*a^5*d^3*e + (B*a^5 + 5*A*a^4*b)*d^4)*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B x\right ) \left (d + e x\right )^{4} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)*(d + e*x)**4*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.17609, size = 1609, normalized size = 4.97 \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)*(e*x+d)^4*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/11*B*b^5*x^11*e^4*sgn(b*x + a) + 2/5*B*b^5*d*x^10*e^3*sgn(b*x + a) + 2/3*B*b^5*d^2*x^9*e^2*sgn(b*x + a) + 1/
2*B*b^5*d^3*x^8*e*sgn(b*x + a) + 1/7*B*b^5*d^4*x^7*sgn(b*x + a) + 1/2*B*a*b^4*x^10*e^4*sgn(b*x + a) + 1/10*A*b
^5*x^10*e^4*sgn(b*x + a) + 20/9*B*a*b^4*d*x^9*e^3*sgn(b*x + a) + 4/9*A*b^5*d*x^9*e^3*sgn(b*x + a) + 15/4*B*a*b
^4*d^2*x^8*e^2*sgn(b*x + a) + 3/4*A*b^5*d^2*x^8*e^2*sgn(b*x + a) + 20/7*B*a*b^4*d^3*x^7*e*sgn(b*x + a) + 4/7*A
*b^5*d^3*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^4*x^6*sgn(b*x + a) + 1/6*A*b^5*d^4*x^6*sgn(b*x + a) + 10/9*B*a^2*b
^3*x^9*e^4*sgn(b*x + a) + 5/9*A*a*b^4*x^9*e^4*sgn(b*x + a) + 5*B*a^2*b^3*d*x^8*e^3*sgn(b*x + a) + 5/2*A*a*b^4*
d*x^8*e^3*sgn(b*x + a) + 60/7*B*a^2*b^3*d^2*x^7*e^2*sgn(b*x + a) + 30/7*A*a*b^4*d^2*x^7*e^2*sgn(b*x + a) + 20/
3*B*a^2*b^3*d^3*x^6*e*sgn(b*x + a) + 10/3*A*a*b^4*d^3*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^4*x^5*sgn(b*x + a) +
A*a*b^4*d^4*x^5*sgn(b*x + a) + 5/4*B*a^3*b^2*x^8*e^4*sgn(b*x + a) + 5/4*A*a^2*b^3*x^8*e^4*sgn(b*x + a) + 40/7*
B*a^3*b^2*d*x^7*e^3*sgn(b*x + a) + 40/7*A*a^2*b^3*d*x^7*e^3*sgn(b*x + a) + 10*B*a^3*b^2*d^2*x^6*e^2*sgn(b*x +
a) + 10*A*a^2*b^3*d^2*x^6*e^2*sgn(b*x + a) + 8*B*a^3*b^2*d^3*x^5*e*sgn(b*x + a) + 8*A*a^2*b^3*d^3*x^5*e*sgn(b*
x + a) + 5/2*B*a^3*b^2*d^4*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^4*x^4*sgn(b*x + a) + 5/7*B*a^4*b*x^7*e^4*sgn(b*x
+ a) + 10/7*A*a^3*b^2*x^7*e^4*sgn(b*x + a) + 10/3*B*a^4*b*d*x^6*e^3*sgn(b*x + a) + 20/3*A*a^3*b^2*d*x^6*e^3*s
gn(b*x + a) + 6*B*a^4*b*d^2*x^5*e^2*sgn(b*x + a) + 12*A*a^3*b^2*d^2*x^5*e^2*sgn(b*x + a) + 5*B*a^4*b*d^3*x^4*e
*sgn(b*x + a) + 10*A*a^3*b^2*d^3*x^4*e*sgn(b*x + a) + 5/3*B*a^4*b*d^4*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^4*x^
3*sgn(b*x + a) + 1/6*B*a^5*x^6*e^4*sgn(b*x + a) + 5/6*A*a^4*b*x^6*e^4*sgn(b*x + a) + 4/5*B*a^5*d*x^5*e^3*sgn(b
*x + a) + 4*A*a^4*b*d*x^5*e^3*sgn(b*x + a) + 3/2*B*a^5*d^2*x^4*e^2*sgn(b*x + a) + 15/2*A*a^4*b*d^2*x^4*e^2*sgn
(b*x + a) + 4/3*B*a^5*d^3*x^3*e*sgn(b*x + a) + 20/3*A*a^4*b*d^3*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^4*x^2*sgn(b*x
+ a) + 5/2*A*a^4*b*d^4*x^2*sgn(b*x + a) + 1/5*A*a^5*x^5*e^4*sgn(b*x + a) + A*a^5*d*x^4*e^3*sgn(b*x + a) + 2*A
*a^5*d^2*x^3*e^2*sgn(b*x + a) + 2*A*a^5*d^3*x^2*e*sgn(b*x + a) + A*a^5*d^4*x*sgn(b*x + a)