3.1741 \(\int (A+B x) (d+e x)^3 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=259 \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]

[Out]

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

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Rubi [A]  time = 0.460646, antiderivative size = 259, 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} \[ \frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^8 (-4 a B e+A b e+3 b B d)}{9 b^5}+\frac{3 e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e) (-2 a B e+A b e+b B d)}{8 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{7 b^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^3}{6 b^5}+\frac{B e^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^9}{10 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.211097, size = 478, normalized size = 1.85 \[ \frac{x \sqrt{(a+b x)^2} \left (60 a^3 b^2 x^2 \left (7 A \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )+3 B x \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right )+90 a^2 b^3 x^3 \left (2 A \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )+B x \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )\right )+210 a^4 b x \left (3 A \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )+B x \left (45 d^2 e x+20 d^3+36 d e^2 x^2+10 e^3 x^3\right )\right )+126 a^5 \left (5 A \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+B x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right )+5 a b^4 x^4 \left (9 A \left (140 d^2 e x+56 d^3+120 d e^2 x^2+35 e^3 x^3\right )+5 B x \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )\right )+b^5 x^5 \left (5 A \left (216 d^2 e x+84 d^3+189 d e^2 x^2+56 e^3 x^3\right )+3 B x \left (315 d^2 e x+120 d^3+280 d e^2 x^2+84 e^3 x^3\right )\right )\right )}{2520 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(126*a^5*(5*A*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + B*x*(10*d^3 + 20*d^2*e*x + 15
*d*e^2*x^2 + 4*e^3*x^3)) + 210*a^4*b*x*(3*A*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + B*x*(20*d^3 + 4
5*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3)) + 60*a^3*b^2*x^2*(7*A*(20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3
) + 3*B*x*(35*d^3 + 84*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)) + 90*a^2*b^3*x^3*(2*A*(35*d^3 + 84*d^2*e*x + 70*d
*e^2*x^2 + 20*e^3*x^3) + B*x*(56*d^3 + 140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3)) + 5*a*b^4*x^4*(9*A*(56*d^3 +
140*d^2*e*x + 120*d*e^2*x^2 + 35*e^3*x^3) + 5*B*x*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3)) + b^5*
x^5*(5*A*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 3*B*x*(120*d^3 + 315*d^2*e*x + 280*d*e^2*x^2 +
84*e^3*x^3))))/(2520*(a + b*x))

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Maple [B]  time = 0.009, size = 676, normalized size = 2.6 \begin{align*}{\frac{x \left ( 252\,B{e}^{3}{b}^{5}{x}^{9}+280\,{x}^{8}A{b}^{5}{e}^{3}+1400\,{x}^{8}B{e}^{3}a{b}^{4}+840\,{x}^{8}B{b}^{5}d{e}^{2}+1575\,{x}^{7}Aa{b}^{4}{e}^{3}+945\,{x}^{7}A{b}^{5}d{e}^{2}+3150\,{x}^{7}B{e}^{3}{a}^{2}{b}^{3}+4725\,{x}^{7}Ba{b}^{4}d{e}^{2}+945\,{x}^{7}B{b}^{5}{d}^{2}e+3600\,{x}^{6}A{a}^{2}{b}^{3}{e}^{3}+5400\,{x}^{6}Aa{b}^{4}d{e}^{2}+1080\,{x}^{6}A{b}^{5}{d}^{2}e+3600\,{x}^{6}B{e}^{3}{a}^{3}{b}^{2}+10800\,{x}^{6}B{a}^{2}{b}^{3}d{e}^{2}+5400\,{x}^{6}Ba{b}^{4}{d}^{2}e+360\,{x}^{6}B{b}^{5}{d}^{3}+4200\,{x}^{5}A{a}^{3}{b}^{2}{e}^{3}+12600\,{x}^{5}A{a}^{2}{b}^{3}d{e}^{2}+6300\,{x}^{5}Aa{b}^{4}{d}^{2}e+420\,{x}^{5}A{d}^{3}{b}^{5}+2100\,{x}^{5}B{e}^{3}{a}^{4}b+12600\,{x}^{5}B{a}^{3}{b}^{2}d{e}^{2}+12600\,{x}^{5}B{a}^{2}{b}^{3}{d}^{2}e+2100\,{x}^{5}Ba{b}^{4}{d}^{3}+2520\,{x}^{4}A{a}^{4}b{e}^{3}+15120\,{x}^{4}A{a}^{3}{b}^{2}d{e}^{2}+15120\,{x}^{4}A{a}^{2}{b}^{3}{d}^{2}e+2520\,{x}^{4}A{d}^{3}a{b}^{4}+504\,{x}^{4}B{e}^{3}{a}^{5}+7560\,{x}^{4}B{a}^{4}bd{e}^{2}+15120\,{x}^{4}B{a}^{3}{b}^{2}{d}^{2}e+5040\,{x}^{4}B{a}^{2}{b}^{3}{d}^{3}+630\,{x}^{3}A{a}^{5}{e}^{3}+9450\,{x}^{3}A{a}^{4}bd{e}^{2}+18900\,{x}^{3}A{a}^{3}{b}^{2}{d}^{2}e+6300\,{x}^{3}A{d}^{3}{a}^{2}{b}^{3}+1890\,{x}^{3}B{a}^{5}d{e}^{2}+9450\,{x}^{3}B{a}^{4}b{d}^{2}e+6300\,{x}^{3}B{a}^{3}{b}^{2}{d}^{3}+2520\,{x}^{2}A{a}^{5}d{e}^{2}+12600\,{x}^{2}A{a}^{4}b{d}^{2}e+8400\,{x}^{2}A{d}^{3}{a}^{3}{b}^{2}+2520\,{x}^{2}B{a}^{5}{d}^{2}e+4200\,{x}^{2}B{a}^{4}b{d}^{3}+3780\,xA{a}^{5}{d}^{2}e+6300\,xA{d}^{3}{a}^{4}b+1260\,xB{a}^{5}{d}^{3}+2520\,A{d}^{3}{a}^{5} \right ) }{2520\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*x*(252*B*b^5*e^3*x^9+280*A*b^5*e^3*x^8+1400*B*a*b^4*e^3*x^8+840*B*b^5*d*e^2*x^8+1575*A*a*b^4*e^3*x^7+94
5*A*b^5*d*e^2*x^7+3150*B*a^2*b^3*e^3*x^7+4725*B*a*b^4*d*e^2*x^7+945*B*b^5*d^2*e*x^7+3600*A*a^2*b^3*e^3*x^6+540
0*A*a*b^4*d*e^2*x^6+1080*A*b^5*d^2*e*x^6+3600*B*a^3*b^2*e^3*x^6+10800*B*a^2*b^3*d*e^2*x^6+5400*B*a*b^4*d^2*e*x
^6+360*B*b^5*d^3*x^6+4200*A*a^3*b^2*e^3*x^5+12600*A*a^2*b^3*d*e^2*x^5+6300*A*a*b^4*d^2*e*x^5+420*A*b^5*d^3*x^5
+2100*B*a^4*b*e^3*x^5+12600*B*a^3*b^2*d*e^2*x^5+12600*B*a^2*b^3*d^2*e*x^5+2100*B*a*b^4*d^3*x^5+2520*A*a^4*b*e^
3*x^4+15120*A*a^3*b^2*d*e^2*x^4+15120*A*a^2*b^3*d^2*e*x^4+2520*A*a*b^4*d^3*x^4+504*B*a^5*e^3*x^4+7560*B*a^4*b*
d*e^2*x^4+15120*B*a^3*b^2*d^2*e*x^4+5040*B*a^2*b^3*d^3*x^4+630*A*a^5*e^3*x^3+9450*A*a^4*b*d*e^2*x^3+18900*A*a^
3*b^2*d^2*e*x^3+6300*A*a^2*b^3*d^3*x^3+1890*B*a^5*d*e^2*x^3+9450*B*a^4*b*d^2*e*x^3+6300*B*a^3*b^2*d^3*x^3+2520
*A*a^5*d*e^2*x^2+12600*A*a^4*b*d^2*e*x^2+8400*A*a^3*b^2*d^3*x^2+2520*B*a^5*d^2*e*x^2+4200*B*a^4*b*d^3*x^2+3780
*A*a^5*d^2*e*x+6300*A*a^4*b*d^3*x+1260*B*a^5*d^3*x+2520*A*a^5*d^3)*((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)^3*(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 = 1.58746, size = 1122, normalized size = 4.33 \begin{align*} \frac{1}{10} \, B b^{5} e^{3} x^{10} + A a^{5} d^{3} x + \frac{1}{9} \,{\left (3 \, B b^{5} d e^{2} +{\left (5 \, B a b^{4} + A b^{5}\right )} e^{3}\right )} x^{9} + \frac{1}{8} \,{\left (3 \, B b^{5} d^{2} e + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d e^{2} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{3}\right )} x^{8} + \frac{1}{7} \,{\left (B b^{5} d^{3} + 3 \,{\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e + 15 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{2} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{6} \,{\left ({\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} + 15 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{2} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} + 30 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e + 15 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{2} +{\left (B a^{5} + 5 \, A a^{4} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{5} e^{3} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} + 15 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{5} d e^{2} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{5} d^{2} e +{\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/10*B*b^5*e^3*x^10 + A*a^5*d^3*x + 1/9*(3*B*b^5*d*e^2 + (5*B*a*b^4 + A*b^5)*e^3)*x^9 + 1/8*(3*B*b^5*d^2*e + 3
*(5*B*a*b^4 + A*b^5)*d*e^2 + 5*(2*B*a^2*b^3 + A*a*b^4)*e^3)*x^8 + 1/7*(B*b^5*d^3 + 3*(5*B*a*b^4 + A*b^5)*d^2*e
+ 15*(2*B*a^2*b^3 + A*a*b^4)*d*e^2 + 10*(B*a^3*b^2 + A*a^2*b^3)*e^3)*x^7 + 1/6*((5*B*a*b^4 + A*b^5)*d^3 + 15*
(2*B*a^2*b^3 + A*a*b^4)*d^2*e + 30*(B*a^3*b^2 + A*a^2*b^3)*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*e^3)*x^6 + 1/5*(5
*(2*B*a^2*b^3 + A*a*b^4)*d^3 + 30*(B*a^3*b^2 + A*a^2*b^3)*d^2*e + 15*(B*a^4*b + 2*A*a^3*b^2)*d*e^2 + (B*a^5 +
5*A*a^4*b)*e^3)*x^5 + 1/4*(A*a^5*e^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*d^3 + 15*(B*a^4*b + 2*A*a^3*b^2)*d^2*e + 3*(
B*a^5 + 5*A*a^4*b)*d*e^2)*x^4 + 1/3*(3*A*a^5*d*e^2 + 5*(B*a^4*b + 2*A*a^3*b^2)*d^3 + 3*(B*a^5 + 5*A*a^4*b)*d^2
*e)*x^3 + 1/2*(3*A*a^5*d^2*e + (B*a^5 + 5*A*a^4*b)*d^3)*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 )^{3} \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)**3*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**3*((a + b*x)**2)**(5/2), x)

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Giac [B]  time = 1.17202, size = 1261, normalized size = 4.87 \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)^3*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/10*B*b^5*x^10*e^3*sgn(b*x + a) + 1/3*B*b^5*d*x^9*e^2*sgn(b*x + a) + 3/8*B*b^5*d^2*x^8*e*sgn(b*x + a) + 1/7*B
*b^5*d^3*x^7*sgn(b*x + a) + 5/9*B*a*b^4*x^9*e^3*sgn(b*x + a) + 1/9*A*b^5*x^9*e^3*sgn(b*x + a) + 15/8*B*a*b^4*d
*x^8*e^2*sgn(b*x + a) + 3/8*A*b^5*d*x^8*e^2*sgn(b*x + a) + 15/7*B*a*b^4*d^2*x^7*e*sgn(b*x + a) + 3/7*A*b^5*d^2
*x^7*e*sgn(b*x + a) + 5/6*B*a*b^4*d^3*x^6*sgn(b*x + a) + 1/6*A*b^5*d^3*x^6*sgn(b*x + a) + 5/4*B*a^2*b^3*x^8*e^
3*sgn(b*x + a) + 5/8*A*a*b^4*x^8*e^3*sgn(b*x + a) + 30/7*B*a^2*b^3*d*x^7*e^2*sgn(b*x + a) + 15/7*A*a*b^4*d*x^7
*e^2*sgn(b*x + a) + 5*B*a^2*b^3*d^2*x^6*e*sgn(b*x + a) + 5/2*A*a*b^4*d^2*x^6*e*sgn(b*x + a) + 2*B*a^2*b^3*d^3*
x^5*sgn(b*x + a) + A*a*b^4*d^3*x^5*sgn(b*x + a) + 10/7*B*a^3*b^2*x^7*e^3*sgn(b*x + a) + 10/7*A*a^2*b^3*x^7*e^3
*sgn(b*x + a) + 5*B*a^3*b^2*d*x^6*e^2*sgn(b*x + a) + 5*A*a^2*b^3*d*x^6*e^2*sgn(b*x + a) + 6*B*a^3*b^2*d^2*x^5*
e*sgn(b*x + a) + 6*A*a^2*b^3*d^2*x^5*e*sgn(b*x + a) + 5/2*B*a^3*b^2*d^3*x^4*sgn(b*x + a) + 5/2*A*a^2*b^3*d^3*x
^4*sgn(b*x + a) + 5/6*B*a^4*b*x^6*e^3*sgn(b*x + a) + 5/3*A*a^3*b^2*x^6*e^3*sgn(b*x + a) + 3*B*a^4*b*d*x^5*e^2*
sgn(b*x + a) + 6*A*a^3*b^2*d*x^5*e^2*sgn(b*x + a) + 15/4*B*a^4*b*d^2*x^4*e*sgn(b*x + a) + 15/2*A*a^3*b^2*d^2*x
^4*e*sgn(b*x + a) + 5/3*B*a^4*b*d^3*x^3*sgn(b*x + a) + 10/3*A*a^3*b^2*d^3*x^3*sgn(b*x + a) + 1/5*B*a^5*x^5*e^3
*sgn(b*x + a) + A*a^4*b*x^5*e^3*sgn(b*x + a) + 3/4*B*a^5*d*x^4*e^2*sgn(b*x + a) + 15/4*A*a^4*b*d*x^4*e^2*sgn(b
*x + a) + B*a^5*d^2*x^3*e*sgn(b*x + a) + 5*A*a^4*b*d^2*x^3*e*sgn(b*x + a) + 1/2*B*a^5*d^3*x^2*sgn(b*x + a) + 5
/2*A*a^4*b*d^3*x^2*sgn(b*x + a) + 1/4*A*a^5*x^4*e^3*sgn(b*x + a) + A*a^5*d*x^3*e^2*sgn(b*x + a) + 3/2*A*a^5*d^
2*x^2*e*sgn(b*x + a) + A*a^5*d^3*x*sgn(b*x + a)