3.1990 \(\int (a+b x) (d+e x)^7 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=362 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14}}{14 e^7 (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)}{13 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^2}{4 e^7 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^3}{11 e^7 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^4}{2 e^7 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^5}{3 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^6}{8 e^7 (a+b x)} \]

[Out]

((b*d - a*e)^6*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)) - (2*b*(b*d - a*e)^5*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (3*b^2*(b*d - a*e)^4*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))
+ (5*b^4*(b*d - a*e)^2*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) - (6*b^5*(b*d - a*e)*(d +
e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (b^6*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(14*e^7*(a + b*x))

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Rubi [A]  time = 0.524812, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 43} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14}}{14 e^7 (a+b x)}-\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)}{13 e^7 (a+b x)}+\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^2}{4 e^7 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^3}{11 e^7 (a+b x)}+\frac{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^4}{2 e^7 (a+b x)}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^9 (b d-a e)^5}{3 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^8 (b d-a e)^6}{8 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*e^7*(a + b*x)) - (2*b*(b*d - a*e)^5*(d + e*x)^9*S
qrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (3*b^2*(b*d - a*e)^4*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(2*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x))
+ (5*b^4*(b*d - a*e)^2*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)) - (6*b^5*(b*d - a*e)*(d +
e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (b^6*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(14*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 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^7 \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 (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^7 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^7 \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^6 (d+e x)^7}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^8}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^9}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{10}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{11}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{12}}{e^6}+\frac{b^6 (d+e x)^{13}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^6 (d+e x)^8 \sqrt{a^2+2 a b x+b^2 x^2}}{8 e^7 (a+b x)}-\frac{2 b (b d-a e)^5 (d+e x)^9 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac{3 b^2 (b d-a e)^4 (d+e x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac{5 b^4 (b d-a e)^2 (d+e x)^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac{6 b^5 (b d-a e) (d+e x)^{13} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac{b^6 (d+e x)^{14} \sqrt{a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.193638, size = 602, normalized size = 1.66 \[ \frac{x \sqrt{(a+b x)^2} \left (1001 a^4 b^2 x^2 \left (1512 d^5 e^2 x^2+2100 d^4 e^3 x^3+1800 d^3 e^4 x^4+945 d^2 e^5 x^5+630 d^6 e x+120 d^7+280 d e^6 x^6+36 e^7 x^7\right )+364 a^3 b^3 x^3 \left (4620 d^5 e^2 x^2+6600 d^4 e^3 x^3+5775 d^3 e^4 x^4+3080 d^2 e^5 x^5+1848 d^6 e x+330 d^7+924 d e^6 x^6+120 e^7 x^7\right )+91 a^2 b^4 x^4 \left (11880 d^5 e^2 x^2+17325 d^4 e^3 x^3+15400 d^3 e^4 x^4+8316 d^2 e^5 x^5+4620 d^6 e x+792 d^7+2520 d e^6 x^6+330 e^7 x^7\right )+2002 a^5 b x \left (378 d^5 e^2 x^2+504 d^4 e^3 x^3+420 d^3 e^4 x^4+216 d^2 e^5 x^5+168 d^6 e x+36 d^7+63 d e^6 x^6+8 e^7 x^7\right )+3003 a^6 \left (56 d^5 e^2 x^2+70 d^4 e^3 x^3+56 d^3 e^4 x^4+28 d^2 e^5 x^5+28 d^6 e x+8 d^7+8 d e^6 x^6+e^7 x^7\right )+14 a b^5 x^5 \left (27027 d^5 e^2 x^2+40040 d^4 e^3 x^3+36036 d^3 e^4 x^4+19656 d^2 e^5 x^5+10296 d^6 e x+1716 d^7+6006 d e^6 x^6+792 e^7 x^7\right )+b^6 x^6 \left (56056 d^5 e^2 x^2+84084 d^4 e^3 x^3+76440 d^3 e^4 x^4+42042 d^2 e^5 x^5+21021 d^6 e x+3432 d^7+12936 d e^6 x^6+1716 e^7 x^7\right )\right )}{24024 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^7*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(3003*a^6*(8*d^7 + 28*d^6*e*x + 56*d^5*e^2*x^2 + 70*d^4*e^3*x^3 + 56*d^3*e^4*x^4 + 28*d^2
*e^5*x^5 + 8*d*e^6*x^6 + e^7*x^7) + 2002*a^5*b*x*(36*d^7 + 168*d^6*e*x + 378*d^5*e^2*x^2 + 504*d^4*e^3*x^3 + 4
20*d^3*e^4*x^4 + 216*d^2*e^5*x^5 + 63*d*e^6*x^6 + 8*e^7*x^7) + 1001*a^4*b^2*x^2*(120*d^7 + 630*d^6*e*x + 1512*
d^5*e^2*x^2 + 2100*d^4*e^3*x^3 + 1800*d^3*e^4*x^4 + 945*d^2*e^5*x^5 + 280*d*e^6*x^6 + 36*e^7*x^7) + 364*a^3*b^
3*x^3*(330*d^7 + 1848*d^6*e*x + 4620*d^5*e^2*x^2 + 6600*d^4*e^3*x^3 + 5775*d^3*e^4*x^4 + 3080*d^2*e^5*x^5 + 92
4*d*e^6*x^6 + 120*e^7*x^7) + 91*a^2*b^4*x^4*(792*d^7 + 4620*d^6*e*x + 11880*d^5*e^2*x^2 + 17325*d^4*e^3*x^3 +
15400*d^3*e^4*x^4 + 8316*d^2*e^5*x^5 + 2520*d*e^6*x^6 + 330*e^7*x^7) + 14*a*b^5*x^5*(1716*d^7 + 10296*d^6*e*x
+ 27027*d^5*e^2*x^2 + 40040*d^4*e^3*x^3 + 36036*d^3*e^4*x^4 + 19656*d^2*e^5*x^5 + 6006*d*e^6*x^6 + 792*e^7*x^7
) + b^6*x^6*(3432*d^7 + 21021*d^6*e*x + 56056*d^5*e^2*x^2 + 84084*d^4*e^3*x^3 + 76440*d^3*e^4*x^4 + 42042*d^2*
e^5*x^5 + 12936*d*e^6*x^6 + 1716*e^7*x^7)))/(24024*(a + b*x))

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Maple [B]  time = 0.008, size = 816, normalized size = 2.3 \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)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/24024*x*(1716*b^6*e^7*x^13+11088*a*b^5*e^7*x^12+12936*b^6*d*e^6*x^12+30030*a^2*b^4*e^7*x^11+84084*a*b^5*d*e^
6*x^11+42042*b^6*d^2*e^5*x^11+43680*a^3*b^3*e^7*x^10+229320*a^2*b^4*d*e^6*x^10+275184*a*b^5*d^2*e^5*x^10+76440
*b^6*d^3*e^4*x^10+36036*a^4*b^2*e^7*x^9+336336*a^3*b^3*d*e^6*x^9+756756*a^2*b^4*d^2*e^5*x^9+504504*a*b^5*d^3*e
^4*x^9+84084*b^6*d^4*e^3*x^9+16016*a^5*b*e^7*x^8+280280*a^4*b^2*d*e^6*x^8+1121120*a^3*b^3*d^2*e^5*x^8+1401400*
a^2*b^4*d^3*e^4*x^8+560560*a*b^5*d^4*e^3*x^8+56056*b^6*d^5*e^2*x^8+3003*a^6*e^7*x^7+126126*a^5*b*d*e^6*x^7+945
945*a^4*b^2*d^2*e^5*x^7+2102100*a^3*b^3*d^3*e^4*x^7+1576575*a^2*b^4*d^4*e^3*x^7+378378*a*b^5*d^5*e^2*x^7+21021
*b^6*d^6*e*x^7+24024*a^6*d*e^6*x^6+432432*a^5*b*d^2*e^5*x^6+1801800*a^4*b^2*d^3*e^4*x^6+2402400*a^3*b^3*d^4*e^
3*x^6+1081080*a^2*b^4*d^5*e^2*x^6+144144*a*b^5*d^6*e*x^6+3432*b^6*d^7*x^6+84084*a^6*d^2*e^5*x^5+840840*a^5*b*d
^3*e^4*x^5+2102100*a^4*b^2*d^4*e^3*x^5+1681680*a^3*b^3*d^5*e^2*x^5+420420*a^2*b^4*d^6*e*x^5+24024*a*b^5*d^7*x^
5+168168*a^6*d^3*e^4*x^4+1009008*a^5*b*d^4*e^3*x^4+1513512*a^4*b^2*d^5*e^2*x^4+672672*a^3*b^3*d^6*e*x^4+72072*
a^2*b^4*d^7*x^4+210210*a^6*d^4*e^3*x^3+756756*a^5*b*d^5*e^2*x^3+630630*a^4*b^2*d^6*e*x^3+120120*a^3*b^3*d^7*x^
3+168168*a^6*d^5*e^2*x^2+336336*a^5*b*d^6*e*x^2+120120*a^4*b^2*d^7*x^2+84084*a^6*d^6*e*x+72072*a^5*b*d^7*x+240
24*a^6*d^7)*((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)^7*(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 = 1.61133, size = 1500, normalized size = 4.14 \begin{align*} \frac{1}{14} \, b^{6} e^{7} x^{14} + a^{6} d^{7} x + \frac{1}{13} \,{\left (7 \, b^{6} d e^{6} + 6 \, a b^{5} e^{7}\right )} x^{13} + \frac{1}{4} \,{\left (7 \, b^{6} d^{2} e^{5} + 14 \, a b^{5} d e^{6} + 5 \, a^{2} b^{4} e^{7}\right )} x^{12} + \frac{1}{11} \,{\left (35 \, b^{6} d^{3} e^{4} + 126 \, a b^{5} d^{2} e^{5} + 105 \, a^{2} b^{4} d e^{6} + 20 \, a^{3} b^{3} e^{7}\right )} x^{11} + \frac{1}{2} \,{\left (7 \, b^{6} d^{4} e^{3} + 42 \, a b^{5} d^{3} e^{4} + 63 \, a^{2} b^{4} d^{2} e^{5} + 28 \, a^{3} b^{3} d e^{6} + 3 \, a^{4} b^{2} e^{7}\right )} x^{10} + \frac{1}{3} \,{\left (7 \, b^{6} d^{5} e^{2} + 70 \, a b^{5} d^{4} e^{3} + 175 \, a^{2} b^{4} d^{3} e^{4} + 140 \, a^{3} b^{3} d^{2} e^{5} + 35 \, a^{4} b^{2} d e^{6} + 2 \, a^{5} b e^{7}\right )} x^{9} + \frac{1}{8} \,{\left (7 \, b^{6} d^{6} e + 126 \, a b^{5} d^{5} e^{2} + 525 \, a^{2} b^{4} d^{4} e^{3} + 700 \, a^{3} b^{3} d^{3} e^{4} + 315 \, a^{4} b^{2} d^{2} e^{5} + 42 \, a^{5} b d e^{6} + a^{6} e^{7}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} d^{7} + 42 \, a b^{5} d^{6} e + 315 \, a^{2} b^{4} d^{5} e^{2} + 700 \, a^{3} b^{3} d^{4} e^{3} + 525 \, a^{4} b^{2} d^{3} e^{4} + 126 \, a^{5} b d^{2} e^{5} + 7 \, a^{6} d e^{6}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} d^{7} + 35 \, a^{2} b^{4} d^{6} e + 140 \, a^{3} b^{3} d^{5} e^{2} + 175 \, a^{4} b^{2} d^{4} e^{3} + 70 \, a^{5} b d^{3} e^{4} + 7 \, a^{6} d^{2} e^{5}\right )} x^{6} +{\left (3 \, a^{2} b^{4} d^{7} + 28 \, a^{3} b^{3} d^{6} e + 63 \, a^{4} b^{2} d^{5} e^{2} + 42 \, a^{5} b d^{4} e^{3} + 7 \, a^{6} d^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} d^{7} + 105 \, a^{4} b^{2} d^{6} e + 126 \, a^{5} b d^{5} e^{2} + 35 \, a^{6} d^{4} e^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} d^{7} + 14 \, a^{5} b d^{6} e + 7 \, a^{6} d^{5} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (6 \, a^{5} b d^{7} + 7 \, a^{6} d^{6} e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/14*b^6*e^7*x^14 + a^6*d^7*x + 1/13*(7*b^6*d*e^6 + 6*a*b^5*e^7)*x^13 + 1/4*(7*b^6*d^2*e^5 + 14*a*b^5*d*e^6 +
5*a^2*b^4*e^7)*x^12 + 1/11*(35*b^6*d^3*e^4 + 126*a*b^5*d^2*e^5 + 105*a^2*b^4*d*e^6 + 20*a^3*b^3*e^7)*x^11 + 1/
2*(7*b^6*d^4*e^3 + 42*a*b^5*d^3*e^4 + 63*a^2*b^4*d^2*e^5 + 28*a^3*b^3*d*e^6 + 3*a^4*b^2*e^7)*x^10 + 1/3*(7*b^6
*d^5*e^2 + 70*a*b^5*d^4*e^3 + 175*a^2*b^4*d^3*e^4 + 140*a^3*b^3*d^2*e^5 + 35*a^4*b^2*d*e^6 + 2*a^5*b*e^7)*x^9
+ 1/8*(7*b^6*d^6*e + 126*a*b^5*d^5*e^2 + 525*a^2*b^4*d^4*e^3 + 700*a^3*b^3*d^3*e^4 + 315*a^4*b^2*d^2*e^5 + 42*
a^5*b*d*e^6 + a^6*e^7)*x^8 + 1/7*(b^6*d^7 + 42*a*b^5*d^6*e + 315*a^2*b^4*d^5*e^2 + 700*a^3*b^3*d^4*e^3 + 525*a
^4*b^2*d^3*e^4 + 126*a^5*b*d^2*e^5 + 7*a^6*d*e^6)*x^7 + 1/2*(2*a*b^5*d^7 + 35*a^2*b^4*d^6*e + 140*a^3*b^3*d^5*
e^2 + 175*a^4*b^2*d^4*e^3 + 70*a^5*b*d^3*e^4 + 7*a^6*d^2*e^5)*x^6 + (3*a^2*b^4*d^7 + 28*a^3*b^3*d^6*e + 63*a^4
*b^2*d^5*e^2 + 42*a^5*b*d^4*e^3 + 7*a^6*d^3*e^4)*x^5 + 1/4*(20*a^3*b^3*d^7 + 105*a^4*b^2*d^6*e + 126*a^5*b*d^5
*e^2 + 35*a^6*d^4*e^3)*x^4 + (5*a^4*b^2*d^7 + 14*a^5*b*d^6*e + 7*a^6*d^5*e^2)*x^3 + 1/2*(6*a^5*b*d^7 + 7*a^6*d
^6*e)*x^2

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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)*(e*x+d)**7*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.19367, size = 1484, normalized size = 4.1 \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)^7*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/14*b^6*x^14*e^7*sgn(b*x + a) + 7/13*b^6*d*x^13*e^6*sgn(b*x + a) + 7/4*b^6*d^2*x^12*e^5*sgn(b*x + a) + 35/11*
b^6*d^3*x^11*e^4*sgn(b*x + a) + 7/2*b^6*d^4*x^10*e^3*sgn(b*x + a) + 7/3*b^6*d^5*x^9*e^2*sgn(b*x + a) + 7/8*b^6
*d^6*x^8*e*sgn(b*x + a) + 1/7*b^6*d^7*x^7*sgn(b*x + a) + 6/13*a*b^5*x^13*e^7*sgn(b*x + a) + 7/2*a*b^5*d*x^12*e
^6*sgn(b*x + a) + 126/11*a*b^5*d^2*x^11*e^5*sgn(b*x + a) + 21*a*b^5*d^3*x^10*e^4*sgn(b*x + a) + 70/3*a*b^5*d^4
*x^9*e^3*sgn(b*x + a) + 63/4*a*b^5*d^5*x^8*e^2*sgn(b*x + a) + 6*a*b^5*d^6*x^7*e*sgn(b*x + a) + a*b^5*d^7*x^6*s
gn(b*x + a) + 5/4*a^2*b^4*x^12*e^7*sgn(b*x + a) + 105/11*a^2*b^4*d*x^11*e^6*sgn(b*x + a) + 63/2*a^2*b^4*d^2*x^
10*e^5*sgn(b*x + a) + 175/3*a^2*b^4*d^3*x^9*e^4*sgn(b*x + a) + 525/8*a^2*b^4*d^4*x^8*e^3*sgn(b*x + a) + 45*a^2
*b^4*d^5*x^7*e^2*sgn(b*x + a) + 35/2*a^2*b^4*d^6*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^7*x^5*sgn(b*x + a) + 20/11*a
^3*b^3*x^11*e^7*sgn(b*x + a) + 14*a^3*b^3*d*x^10*e^6*sgn(b*x + a) + 140/3*a^3*b^3*d^2*x^9*e^5*sgn(b*x + a) + 1
75/2*a^3*b^3*d^3*x^8*e^4*sgn(b*x + a) + 100*a^3*b^3*d^4*x^7*e^3*sgn(b*x + a) + 70*a^3*b^3*d^5*x^6*e^2*sgn(b*x
+ a) + 28*a^3*b^3*d^6*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^7*x^4*sgn(b*x + a) + 3/2*a^4*b^2*x^10*e^7*sgn(b*x + a)
+ 35/3*a^4*b^2*d*x^9*e^6*sgn(b*x + a) + 315/8*a^4*b^2*d^2*x^8*e^5*sgn(b*x + a) + 75*a^4*b^2*d^3*x^7*e^4*sgn(b*
x + a) + 175/2*a^4*b^2*d^4*x^6*e^3*sgn(b*x + a) + 63*a^4*b^2*d^5*x^5*e^2*sgn(b*x + a) + 105/4*a^4*b^2*d^6*x^4*
e*sgn(b*x + a) + 5*a^4*b^2*d^7*x^3*sgn(b*x + a) + 2/3*a^5*b*x^9*e^7*sgn(b*x + a) + 21/4*a^5*b*d*x^8*e^6*sgn(b*
x + a) + 18*a^5*b*d^2*x^7*e^5*sgn(b*x + a) + 35*a^5*b*d^3*x^6*e^4*sgn(b*x + a) + 42*a^5*b*d^4*x^5*e^3*sgn(b*x
+ a) + 63/2*a^5*b*d^5*x^4*e^2*sgn(b*x + a) + 14*a^5*b*d^6*x^3*e*sgn(b*x + a) + 3*a^5*b*d^7*x^2*sgn(b*x + a) +
1/8*a^6*x^8*e^7*sgn(b*x + a) + a^6*d*x^7*e^6*sgn(b*x + a) + 7/2*a^6*d^2*x^6*e^5*sgn(b*x + a) + 7*a^6*d^3*x^5*e
^4*sgn(b*x + a) + 35/4*a^6*d^4*x^4*e^3*sgn(b*x + a) + 7*a^6*d^5*x^3*e^2*sgn(b*x + a) + 7/2*a^6*d^6*x^2*e*sgn(b
*x + a) + a^6*d^7*x*sgn(b*x + a)