3.1720 \(\int (A+B x) (d+e x)^5 (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=298 \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.508242, antiderivative size = 298, 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} \[ -\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (-3 a B e-A b e+4 b B d)}{9 e^5 (a+b x)}+\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e) (-a B e-A b e+2 b B d)}{8 e^5 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^7 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6 (b d-a e)^3 (B d-A e)}{6 e^5 (a+b x)}+\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10}}{10 e^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.223912, size = 496, normalized size = 1.66 \[ \frac{x \sqrt{(a+b x)^2} \left (45 a^2 b x \left (4 A \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )+B x \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )\right )+60 a^3 \left (7 A \left (20 d^3 e^2 x^2+15 d^2 e^3 x^3+15 d^4 e x+6 d^5+6 d e^4 x^4+e^5 x^5\right )+B x \left (105 d^3 e^2 x^2+84 d^2 e^3 x^3+70 d^4 e x+21 d^5+35 d e^4 x^4+6 e^5 x^5\right )\right )+15 a b^2 x^2 \left (3 A \left (336 d^3 e^2 x^2+280 d^2 e^3 x^3+210 d^4 e x+56 d^5+120 d e^4 x^4+21 e^5 x^5\right )+B x \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )\right )+b^3 x^3 \left (5 A \left (840 d^3 e^2 x^2+720 d^2 e^3 x^3+504 d^4 e x+126 d^5+315 d e^4 x^4+56 e^5 x^5\right )+2 B x \left (1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+1050 d^4 e x+252 d^5+700 d e^4 x^4+126 e^5 x^5\right )\right )\right )}{2520 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(60*a^3*(7*A*(6*d^5 + 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^
5) + B*x*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5)) + 45*a^2*b*x*(4*
A*(21*d^5 + 70*d^4*e*x + 105*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 35*d*e^4*x^4 + 6*e^5*x^5) + B*x*(56*d^5 + 210*d^4*
e*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5)) + 15*a*b^2*x^2*(3*A*(56*d^5 + 210*d^4*e
*x + 336*d^3*e^2*x^2 + 280*d^2*e^3*x^3 + 120*d*e^4*x^4 + 21*e^5*x^5) + B*x*(126*d^5 + 504*d^4*e*x + 840*d^3*e^
2*x^2 + 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5)) + b^3*x^3*(5*A*(126*d^5 + 504*d^4*e*x + 840*d^3*e^2*x^2
+ 720*d^2*e^3*x^3 + 315*d*e^4*x^4 + 56*e^5*x^5) + 2*B*x*(252*d^5 + 1050*d^4*e*x + 1800*d^3*e^2*x^2 + 1575*d^2
*e^3*x^3 + 700*d*e^4*x^4 + 126*e^5*x^5))))/(2520*(a + b*x))

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Maple [B]  time = 0.007, size = 676, normalized size = 2.3 \begin{align*}{\frac{x \left ( 252\,{b}^{3}B{e}^{5}{x}^{9}+280\,{x}^{8}A{b}^{3}{e}^{5}+840\,{x}^{8}Ba{b}^{2}{e}^{5}+1400\,{x}^{8}{b}^{3}Bd{e}^{4}+945\,{x}^{7}Aa{b}^{2}{e}^{5}+1575\,{x}^{7}A{b}^{3}d{e}^{4}+945\,{x}^{7}B{a}^{2}b{e}^{5}+4725\,{x}^{7}Ba{b}^{2}d{e}^{4}+3150\,{x}^{7}{b}^{3}B{d}^{2}{e}^{3}+1080\,{x}^{6}A{a}^{2}b{e}^{5}+5400\,{x}^{6}Aa{b}^{2}d{e}^{4}+3600\,{x}^{6}A{b}^{3}{d}^{2}{e}^{3}+360\,{x}^{6}B{e}^{5}{a}^{3}+5400\,{x}^{6}B{a}^{2}bd{e}^{4}+10800\,{x}^{6}Ba{b}^{2}{d}^{2}{e}^{3}+3600\,{x}^{6}{b}^{3}B{d}^{3}{e}^{2}+420\,{x}^{5}A{a}^{3}{e}^{5}+6300\,{x}^{5}A{a}^{2}bd{e}^{4}+12600\,{x}^{5}Aa{b}^{2}{d}^{2}{e}^{3}+4200\,{x}^{5}A{b}^{3}{d}^{3}{e}^{2}+2100\,{x}^{5}B{a}^{3}d{e}^{4}+12600\,{x}^{5}B{a}^{2}b{d}^{2}{e}^{3}+12600\,{x}^{5}Ba{b}^{2}{d}^{3}{e}^{2}+2100\,{x}^{5}{b}^{3}B{d}^{4}e+2520\,{x}^{4}A{a}^{3}d{e}^{4}+15120\,{x}^{4}A{a}^{2}b{d}^{2}{e}^{3}+15120\,{x}^{4}Aa{b}^{2}{d}^{3}{e}^{2}+2520\,{x}^{4}A{b}^{3}{d}^{4}e+5040\,{x}^{4}B{a}^{3}{d}^{2}{e}^{3}+15120\,{x}^{4}B{a}^{2}b{d}^{3}{e}^{2}+7560\,{x}^{4}Ba{b}^{2}{d}^{4}e+504\,{x}^{4}{b}^{3}B{d}^{5}+6300\,{x}^{3}A{a}^{3}{d}^{2}{e}^{3}+18900\,{x}^{3}A{a}^{2}b{d}^{3}{e}^{2}+9450\,{x}^{3}Aa{b}^{2}{d}^{4}e+630\,{x}^{3}A{b}^{3}{d}^{5}+6300\,{x}^{3}B{a}^{3}{d}^{3}{e}^{2}+9450\,{x}^{3}B{a}^{2}b{d}^{4}e+1890\,{x}^{3}Ba{b}^{2}{d}^{5}+8400\,{x}^{2}A{a}^{3}{d}^{3}{e}^{2}+12600\,{x}^{2}A{a}^{2}b{d}^{4}e+2520\,{x}^{2}Aa{b}^{2}{d}^{5}+4200\,{x}^{2}B{a}^{3}{d}^{4}e+2520\,{x}^{2}B{a}^{2}b{d}^{5}+6300\,xA{a}^{3}{d}^{4}e+3780\,xA{a}^{2}b{d}^{5}+1260\,xB{a}^{3}{d}^{5}+2520\,A{a}^{3}{d}^{5} \right ) }{2520\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*x*(252*B*b^3*e^5*x^9+280*A*b^3*e^5*x^8+840*B*a*b^2*e^5*x^8+1400*B*b^3*d*e^4*x^8+945*A*a*b^2*e^5*x^7+157
5*A*b^3*d*e^4*x^7+945*B*a^2*b*e^5*x^7+4725*B*a*b^2*d*e^4*x^7+3150*B*b^3*d^2*e^3*x^7+1080*A*a^2*b*e^5*x^6+5400*
A*a*b^2*d*e^4*x^6+3600*A*b^3*d^2*e^3*x^6+360*B*a^3*e^5*x^6+5400*B*a^2*b*d*e^4*x^6+10800*B*a*b^2*d^2*e^3*x^6+36
00*B*b^3*d^3*e^2*x^6+420*A*a^3*e^5*x^5+6300*A*a^2*b*d*e^4*x^5+12600*A*a*b^2*d^2*e^3*x^5+4200*A*b^3*d^3*e^2*x^5
+2100*B*a^3*d*e^4*x^5+12600*B*a^2*b*d^2*e^3*x^5+12600*B*a*b^2*d^3*e^2*x^5+2100*B*b^3*d^4*e*x^5+2520*A*a^3*d*e^
4*x^4+15120*A*a^2*b*d^2*e^3*x^4+15120*A*a*b^2*d^3*e^2*x^4+2520*A*b^3*d^4*e*x^4+5040*B*a^3*d^2*e^3*x^4+15120*B*
a^2*b*d^3*e^2*x^4+7560*B*a*b^2*d^4*e*x^4+504*B*b^3*d^5*x^4+6300*A*a^3*d^2*e^3*x^3+18900*A*a^2*b*d^3*e^2*x^3+94
50*A*a*b^2*d^4*e*x^3+630*A*b^3*d^5*x^3+6300*B*a^3*d^3*e^2*x^3+9450*B*a^2*b*d^4*e*x^3+1890*B*a*b^2*d^5*x^3+8400
*A*a^3*d^3*e^2*x^2+12600*A*a^2*b*d^4*e*x^2+2520*A*a*b^2*d^5*x^2+4200*B*a^3*d^4*e*x^2+2520*B*a^2*b*d^5*x^2+6300
*A*a^3*d^4*e*x+3780*A*a^2*b*d^5*x+1260*B*a^3*d^5*x+2520*A*a^3*d^5)*((b*x+a)^2)^(3/2)/(b*x+a)^3

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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)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.61404, size = 1106, normalized size = 3.71 \begin{align*} \frac{1}{10} \, B b^{3} e^{5} x^{10} + A a^{3} d^{5} x + \frac{1}{9} \,{\left (5 \, B b^{3} d e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{9} + \frac{1}{8} \,{\left (10 \, B b^{3} d^{2} e^{3} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{8} + \frac{1}{7} \,{\left (10 \, B b^{3} d^{3} e^{2} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{4} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{7} + \frac{1}{6} \,{\left (5 \, B b^{3} d^{4} e + A a^{3} e^{5} + 10 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{5} + 5 \, A a^{3} d e^{4} + 5 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 30 \,{\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (10 \, A a^{3} d^{2} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} + 15 \,{\left (B a^{2} b + A a b^{2}\right )} d^{4} e + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, A a^{3} d^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} d^{5} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} e\right )} x^{3} + \frac{1}{2} \,{\left (5 \, A a^{3} d^{4} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17523, size = 1245, normalized size = 4.18 \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)^5*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

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