3.1988 \(\int (a+b x) (d+e x)^9 (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=362 \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{16}}{16 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15} (b d-a e)}{5 e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)^2}{14 e^7 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^3}{13 e^7 (a+b x)}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^4}{4 e^7 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^5}{11 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^6}{10 e^7 (a+b x)} \]

[Out]

((b*d - a*e)^6*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)) - (6*b*(b*d - a*e)^5*(d + e*x)^1
1*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (5*b^2*(b*d - a*e)^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(4*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*
x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)) - (2*b^5*(b*d - a*e
)*(d + e*x)^15*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (b^6*(d + e*x)^16*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(16*e^7*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.687499, 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} \[ \frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{16}}{16 e^7 (a+b x)}-\frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15} (b d-a e)}{5 e^7 (a+b x)}+\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{14} (b d-a e)^2}{14 e^7 (a+b x)}-\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13} (b d-a e)^3}{13 e^7 (a+b x)}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{12} (b d-a e)^4}{4 e^7 (a+b x)}-\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11} (b d-a e)^5}{11 e^7 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{10} (b d-a e)^6}{10 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((b*d - a*e)^6*(d + e*x)^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)) - (6*b*(b*d - a*e)^5*(d + e*x)^1
1*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) + (5*b^2*(b*d - a*e)^4*(d + e*x)^12*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(4*e^7*(a + b*x)) - (20*b^3*(b*d - a*e)^3*(d + e*x)^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*
x)) + (15*b^4*(b*d - a*e)^2*(d + e*x)^14*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)) - (2*b^5*(b*d - a*e
)*(d + e*x)^15*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (b^6*(d + e*x)^16*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(16*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)^9 \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)^9 \, 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)^9 \, 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)^9}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{10}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{11}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{12}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{13}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{14}}{e^6}+\frac{b^6 (d+e x)^{15}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{(b d-a e)^6 (d+e x)^{10} \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x)}-\frac{6 b (b d-a e)^5 (d+e x)^{11} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac{5 b^2 (b d-a e)^4 (d+e x)^{12} \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{13} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{14} \sqrt{a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x)}-\frac{2 b^5 (b d-a e) (d+e x)^{15} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac{b^6 (d+e x)^{16} \sqrt{a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x)}\\ \end{align*}

Mathematica [B]  time = 0.239326, size = 756, normalized size = 2.09 \[ \frac{x \sqrt{(a+b x)^2} \left (1820 a^4 b^2 x^2 \left (4752 d^7 e^2 x^2+9240 d^6 e^3 x^3+11880 d^5 e^4 x^4+10395 d^4 e^5 x^5+6160 d^3 e^6 x^6+2376 d^2 e^7 x^7+1485 d^8 e x+220 d^9+540 d e^8 x^8+55 e^9 x^9\right )+560 a^3 b^3 x^3 \left (17160 d^7 e^2 x^2+34320 d^6 e^3 x^3+45045 d^5 e^4 x^4+40040 d^4 e^5 x^5+24024 d^3 e^6 x^6+9360 d^2 e^7 x^7+5148 d^8 e x+715 d^9+2145 d e^8 x^8+220 e^9 x^9\right )+120 a^2 b^4 x^4 \left (51480 d^7 e^2 x^2+105105 d^6 e^3 x^3+140140 d^5 e^4 x^4+126126 d^4 e^5 x^5+76440 d^3 e^6 x^6+30030 d^2 e^7 x^7+15015 d^8 e x+2002 d^9+6930 d e^8 x^8+715 e^9 x^9\right )+4368 a^5 b x \left (990 d^7 e^2 x^2+1848 d^6 e^3 x^3+2310 d^5 e^4 x^4+1980 d^4 e^5 x^5+1155 d^3 e^6 x^6+440 d^2 e^7 x^7+330 d^8 e x+55 d^9+99 d e^8 x^8+10 e^9 x^9\right )+8008 a^6 \left (120 d^7 e^2 x^2+210 d^6 e^3 x^3+252 d^5 e^4 x^4+210 d^4 e^5 x^5+120 d^3 e^6 x^6+45 d^2 e^7 x^7+45 d^8 e x+10 d^9+10 d e^8 x^8+e^9 x^9\right )+16 a b^5 x^5 \left (135135 d^7 e^2 x^2+280280 d^6 e^3 x^3+378378 d^5 e^4 x^4+343980 d^4 e^5 x^5+210210 d^3 e^6 x^6+83160 d^2 e^7 x^7+38610 d^8 e x+5005 d^9+19305 d e^8 x^8+2002 e^9 x^9\right )+b^6 x^6 \left (320320 d^7 e^2 x^2+672672 d^6 e^3 x^3+917280 d^5 e^4 x^4+840840 d^4 e^5 x^5+517440 d^3 e^6 x^6+205920 d^2 e^7 x^7+90090 d^8 e x+11440 d^9+48048 d e^8 x^8+5005 e^9 x^9\right )\right )}{80080 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^9*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(x*Sqrt[(a + b*x)^2]*(8008*a^6*(10*d^9 + 45*d^8*e*x + 120*d^7*e^2*x^2 + 210*d^6*e^3*x^3 + 252*d^5*e^4*x^4 + 21
0*d^4*e^5*x^5 + 120*d^3*e^6*x^6 + 45*d^2*e^7*x^7 + 10*d*e^8*x^8 + e^9*x^9) + 4368*a^5*b*x*(55*d^9 + 330*d^8*e*
x + 990*d^7*e^2*x^2 + 1848*d^6*e^3*x^3 + 2310*d^5*e^4*x^4 + 1980*d^4*e^5*x^5 + 1155*d^3*e^6*x^6 + 440*d^2*e^7*
x^7 + 99*d*e^8*x^8 + 10*e^9*x^9) + 1820*a^4*b^2*x^2*(220*d^9 + 1485*d^8*e*x + 4752*d^7*e^2*x^2 + 9240*d^6*e^3*
x^3 + 11880*d^5*e^4*x^4 + 10395*d^4*e^5*x^5 + 6160*d^3*e^6*x^6 + 2376*d^2*e^7*x^7 + 540*d*e^8*x^8 + 55*e^9*x^9
) + 560*a^3*b^3*x^3*(715*d^9 + 5148*d^8*e*x + 17160*d^7*e^2*x^2 + 34320*d^6*e^3*x^3 + 45045*d^5*e^4*x^4 + 4004
0*d^4*e^5*x^5 + 24024*d^3*e^6*x^6 + 9360*d^2*e^7*x^7 + 2145*d*e^8*x^8 + 220*e^9*x^9) + 120*a^2*b^4*x^4*(2002*d
^9 + 15015*d^8*e*x + 51480*d^7*e^2*x^2 + 105105*d^6*e^3*x^3 + 140140*d^5*e^4*x^4 + 126126*d^4*e^5*x^5 + 76440*
d^3*e^6*x^6 + 30030*d^2*e^7*x^7 + 6930*d*e^8*x^8 + 715*e^9*x^9) + 16*a*b^5*x^5*(5005*d^9 + 38610*d^8*e*x + 135
135*d^7*e^2*x^2 + 280280*d^6*e^3*x^3 + 378378*d^5*e^4*x^4 + 343980*d^4*e^5*x^5 + 210210*d^3*e^6*x^6 + 83160*d^
2*e^7*x^7 + 19305*d*e^8*x^8 + 2002*e^9*x^9) + b^6*x^6*(11440*d^9 + 90090*d^8*e*x + 320320*d^7*e^2*x^2 + 672672
*d^6*e^3*x^3 + 917280*d^5*e^4*x^4 + 840840*d^4*e^5*x^5 + 517440*d^3*e^6*x^6 + 205920*d^2*e^7*x^7 + 48048*d*e^8
*x^8 + 5005*e^9*x^9)))/(80080*(a + b*x))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 1034, normalized size = 2.9 \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)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/80080*x*(5005*b^6*e^9*x^15+32032*a*b^5*e^9*x^14+48048*b^6*d*e^8*x^14+85800*a^2*b^4*e^9*x^13+308880*a*b^5*d*e
^8*x^13+205920*b^6*d^2*e^7*x^13+123200*a^3*b^3*e^9*x^12+831600*a^2*b^4*d*e^8*x^12+1330560*a*b^5*d^2*e^7*x^12+5
17440*b^6*d^3*e^6*x^12+100100*a^4*b^2*e^9*x^11+1201200*a^3*b^3*d*e^8*x^11+3603600*a^2*b^4*d^2*e^7*x^11+3363360
*a*b^5*d^3*e^6*x^11+840840*b^6*d^4*e^5*x^11+43680*a^5*b*e^9*x^10+982800*a^4*b^2*d*e^8*x^10+5241600*a^3*b^3*d^2
*e^7*x^10+9172800*a^2*b^4*d^3*e^6*x^10+5503680*a*b^5*d^4*e^5*x^10+917280*b^6*d^5*e^4*x^10+8008*a^6*e^9*x^9+432
432*a^5*b*d*e^8*x^9+4324320*a^4*b^2*d^2*e^7*x^9+13453440*a^3*b^3*d^3*e^6*x^9+15135120*a^2*b^4*d^4*e^5*x^9+6054
048*a*b^5*d^5*e^4*x^9+672672*b^6*d^6*e^3*x^9+80080*a^6*d*e^8*x^8+1921920*a^5*b*d^2*e^7*x^8+11211200*a^4*b^2*d^
3*e^6*x^8+22422400*a^3*b^3*d^4*e^5*x^8+16816800*a^2*b^4*d^5*e^4*x^8+4484480*a*b^5*d^6*e^3*x^8+320320*b^6*d^7*e
^2*x^8+360360*a^6*d^2*e^7*x^7+5045040*a^5*b*d^3*e^6*x^7+18918900*a^4*b^2*d^4*e^5*x^7+25225200*a^3*b^3*d^5*e^4*
x^7+12612600*a^2*b^4*d^6*e^3*x^7+2162160*a*b^5*d^7*e^2*x^7+90090*b^6*d^8*e*x^7+960960*a^6*d^3*e^6*x^6+8648640*
a^5*b*d^4*e^5*x^6+21621600*a^4*b^2*d^5*e^4*x^6+19219200*a^3*b^3*d^6*e^3*x^6+6177600*a^2*b^4*d^7*e^2*x^6+617760
*a*b^5*d^8*e*x^6+11440*b^6*d^9*x^6+1681680*a^6*d^4*e^5*x^5+10090080*a^5*b*d^5*e^4*x^5+16816800*a^4*b^2*d^6*e^3
*x^5+9609600*a^3*b^3*d^7*e^2*x^5+1801800*a^2*b^4*d^8*e*x^5+80080*a*b^5*d^9*x^5+2018016*a^6*d^5*e^4*x^4+8072064
*a^5*b*d^6*e^3*x^4+8648640*a^4*b^2*d^7*e^2*x^4+2882880*a^3*b^3*d^8*e*x^4+240240*a^2*b^4*d^9*x^4+1681680*a^6*d^
6*e^3*x^3+4324320*a^5*b*d^7*e^2*x^3+2702700*a^4*b^2*d^8*e*x^3+400400*a^3*b^3*d^9*x^3+960960*a^6*d^7*e^2*x^2+14
41440*a^5*b*d^8*e*x^2+400400*a^4*b^2*d^9*x^2+360360*a^6*d^8*e*x+240240*a^5*b*d^9*x+80080*a^6*d^9)*((b*x+a)^2)^
(5/2)/(b*x+a)^5

________________________________________________________________________________________

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)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.63046, size = 1918, normalized size = 5.3 \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)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/16*b^6*e^9*x^16 + a^6*d^9*x + 1/5*(3*b^6*d*e^8 + 2*a*b^5*e^9)*x^15 + 3/14*(12*b^6*d^2*e^7 + 18*a*b^5*d*e^8 +
5*a^2*b^4*e^9)*x^14 + 1/13*(84*b^6*d^3*e^6 + 216*a*b^5*d^2*e^7 + 135*a^2*b^4*d*e^8 + 20*a^3*b^3*e^9)*x^13 + 1
/4*(42*b^6*d^4*e^5 + 168*a*b^5*d^3*e^6 + 180*a^2*b^4*d^2*e^7 + 60*a^3*b^3*d*e^8 + 5*a^4*b^2*e^9)*x^12 + 3/11*(
42*b^6*d^5*e^4 + 252*a*b^5*d^4*e^5 + 420*a^2*b^4*d^3*e^6 + 240*a^3*b^3*d^2*e^7 + 45*a^4*b^2*d*e^8 + 2*a^5*b*e^
9)*x^11 + 1/10*(84*b^6*d^6*e^3 + 756*a*b^5*d^5*e^4 + 1890*a^2*b^4*d^4*e^5 + 1680*a^3*b^3*d^3*e^6 + 540*a^4*b^2
*d^2*e^7 + 54*a^5*b*d*e^8 + a^6*e^9)*x^10 + (4*b^6*d^7*e^2 + 56*a*b^5*d^6*e^3 + 210*a^2*b^4*d^5*e^4 + 280*a^3*
b^3*d^4*e^5 + 140*a^4*b^2*d^3*e^6 + 24*a^5*b*d^2*e^7 + a^6*d*e^8)*x^9 + 9/8*(b^6*d^8*e + 24*a*b^5*d^7*e^2 + 14
0*a^2*b^4*d^6*e^3 + 280*a^3*b^3*d^5*e^4 + 210*a^4*b^2*d^4*e^5 + 56*a^5*b*d^3*e^6 + 4*a^6*d^2*e^7)*x^8 + 1/7*(b
^6*d^9 + 54*a*b^5*d^8*e + 540*a^2*b^4*d^7*e^2 + 1680*a^3*b^3*d^6*e^3 + 1890*a^4*b^2*d^5*e^4 + 756*a^5*b*d^4*e^
5 + 84*a^6*d^3*e^6)*x^7 + 1/2*(2*a*b^5*d^9 + 45*a^2*b^4*d^8*e + 240*a^3*b^3*d^7*e^2 + 420*a^4*b^2*d^6*e^3 + 25
2*a^5*b*d^5*e^4 + 42*a^6*d^4*e^5)*x^6 + 3/5*(5*a^2*b^4*d^9 + 60*a^3*b^3*d^8*e + 180*a^4*b^2*d^7*e^2 + 168*a^5*
b*d^6*e^3 + 42*a^6*d^5*e^4)*x^5 + 1/4*(20*a^3*b^3*d^9 + 135*a^4*b^2*d^8*e + 216*a^5*b*d^7*e^2 + 84*a^6*d^6*e^3
)*x^4 + (5*a^4*b^2*d^9 + 18*a^5*b*d^8*e + 12*a^6*d^7*e^2)*x^3 + 3/2*(2*a^5*b*d^9 + 3*a^6*d^8*e)*x^2

________________________________________________________________________________________

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)**9*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.20838, size = 1872, normalized size = 5.17 \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)^9*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/16*b^6*x^16*e^9*sgn(b*x + a) + 3/5*b^6*d*x^15*e^8*sgn(b*x + a) + 18/7*b^6*d^2*x^14*e^7*sgn(b*x + a) + 84/13*
b^6*d^3*x^13*e^6*sgn(b*x + a) + 21/2*b^6*d^4*x^12*e^5*sgn(b*x + a) + 126/11*b^6*d^5*x^11*e^4*sgn(b*x + a) + 42
/5*b^6*d^6*x^10*e^3*sgn(b*x + a) + 4*b^6*d^7*x^9*e^2*sgn(b*x + a) + 9/8*b^6*d^8*x^8*e*sgn(b*x + a) + 1/7*b^6*d
^9*x^7*sgn(b*x + a) + 2/5*a*b^5*x^15*e^9*sgn(b*x + a) + 27/7*a*b^5*d*x^14*e^8*sgn(b*x + a) + 216/13*a*b^5*d^2*
x^13*e^7*sgn(b*x + a) + 42*a*b^5*d^3*x^12*e^6*sgn(b*x + a) + 756/11*a*b^5*d^4*x^11*e^5*sgn(b*x + a) + 378/5*a*
b^5*d^5*x^10*e^4*sgn(b*x + a) + 56*a*b^5*d^6*x^9*e^3*sgn(b*x + a) + 27*a*b^5*d^7*x^8*e^2*sgn(b*x + a) + 54/7*a
*b^5*d^8*x^7*e*sgn(b*x + a) + a*b^5*d^9*x^6*sgn(b*x + a) + 15/14*a^2*b^4*x^14*e^9*sgn(b*x + a) + 135/13*a^2*b^
4*d*x^13*e^8*sgn(b*x + a) + 45*a^2*b^4*d^2*x^12*e^7*sgn(b*x + a) + 1260/11*a^2*b^4*d^3*x^11*e^6*sgn(b*x + a) +
189*a^2*b^4*d^4*x^10*e^5*sgn(b*x + a) + 210*a^2*b^4*d^5*x^9*e^4*sgn(b*x + a) + 315/2*a^2*b^4*d^6*x^8*e^3*sgn(
b*x + a) + 540/7*a^2*b^4*d^7*x^7*e^2*sgn(b*x + a) + 45/2*a^2*b^4*d^8*x^6*e*sgn(b*x + a) + 3*a^2*b^4*d^9*x^5*sg
n(b*x + a) + 20/13*a^3*b^3*x^13*e^9*sgn(b*x + a) + 15*a^3*b^3*d*x^12*e^8*sgn(b*x + a) + 720/11*a^3*b^3*d^2*x^1
1*e^7*sgn(b*x + a) + 168*a^3*b^3*d^3*x^10*e^6*sgn(b*x + a) + 280*a^3*b^3*d^4*x^9*e^5*sgn(b*x + a) + 315*a^3*b^
3*d^5*x^8*e^4*sgn(b*x + a) + 240*a^3*b^3*d^6*x^7*e^3*sgn(b*x + a) + 120*a^3*b^3*d^7*x^6*e^2*sgn(b*x + a) + 36*
a^3*b^3*d^8*x^5*e*sgn(b*x + a) + 5*a^3*b^3*d^9*x^4*sgn(b*x + a) + 5/4*a^4*b^2*x^12*e^9*sgn(b*x + a) + 135/11*a
^4*b^2*d*x^11*e^8*sgn(b*x + a) + 54*a^4*b^2*d^2*x^10*e^7*sgn(b*x + a) + 140*a^4*b^2*d^3*x^9*e^6*sgn(b*x + a) +
945/4*a^4*b^2*d^4*x^8*e^5*sgn(b*x + a) + 270*a^4*b^2*d^5*x^7*e^4*sgn(b*x + a) + 210*a^4*b^2*d^6*x^6*e^3*sgn(b
*x + a) + 108*a^4*b^2*d^7*x^5*e^2*sgn(b*x + a) + 135/4*a^4*b^2*d^8*x^4*e*sgn(b*x + a) + 5*a^4*b^2*d^9*x^3*sgn(
b*x + a) + 6/11*a^5*b*x^11*e^9*sgn(b*x + a) + 27/5*a^5*b*d*x^10*e^8*sgn(b*x + a) + 24*a^5*b*d^2*x^9*e^7*sgn(b*
x + a) + 63*a^5*b*d^3*x^8*e^6*sgn(b*x + a) + 108*a^5*b*d^4*x^7*e^5*sgn(b*x + a) + 126*a^5*b*d^5*x^6*e^4*sgn(b*
x + a) + 504/5*a^5*b*d^6*x^5*e^3*sgn(b*x + a) + 54*a^5*b*d^7*x^4*e^2*sgn(b*x + a) + 18*a^5*b*d^8*x^3*e*sgn(b*x
+ a) + 3*a^5*b*d^9*x^2*sgn(b*x + a) + 1/10*a^6*x^10*e^9*sgn(b*x + a) + a^6*d*x^9*e^8*sgn(b*x + a) + 9/2*a^6*d
^2*x^8*e^7*sgn(b*x + a) + 12*a^6*d^3*x^7*e^6*sgn(b*x + a) + 21*a^6*d^4*x^6*e^5*sgn(b*x + a) + 126/5*a^6*d^5*x^
5*e^4*sgn(b*x + a) + 21*a^6*d^6*x^4*e^3*sgn(b*x + a) + 12*a^6*d^7*x^3*e^2*sgn(b*x + a) + 9/2*a^6*d^8*x^2*e*sgn
(b*x + a) + a^6*d^9*x*sgn(b*x + a)