3.2014 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{17}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.200691, 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} \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{15}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^17,x]

[Out]

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

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 \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{17}} \, 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 \frac{(a+b x)^6}{(d+e x)^{17}} \, 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}{e^6 (d+e x)^{17}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{16}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{15}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{14}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^{13}}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^{12}}+\frac{b^6}{e^6 (d+e x)^{11}}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x) (d+e x)^{16}}+\frac{2 b (b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{15}}-\frac{15 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac{20 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac{5 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{6 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}\\ \end{align*}

Mathematica [A]  time = 0.111198, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (55 a^2 b^4 e^2 \left (120 d^2 e^2 x^2+16 d^3 e x+d^4+560 d e^3 x^3+1820 e^4 x^4\right )+220 a^3 b^3 e^3 \left (16 d^2 e x+d^3+120 d e^2 x^2+560 e^3 x^3\right )+715 a^4 b^2 e^4 \left (d^2+16 d e x+120 e^2 x^2\right )+2002 a^5 b e^5 (d+16 e x)+5005 a^6 e^6+10 a b^5 e \left (120 d^3 e^2 x^2+560 d^2 e^3 x^3+16 d^4 e x+d^5+1820 d e^4 x^4+4368 e^5 x^5\right )+b^6 \left (120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+16 d^5 e x+d^6+4368 d e^5 x^5+8008 e^6 x^6\right )\right )}{80080 e^7 (a+b x) (d+e x)^{16}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^17,x]

[Out]

-(Sqrt[(a + b*x)^2]*(5005*a^6*e^6 + 2002*a^5*b*e^5*(d + 16*e*x) + 715*a^4*b^2*e^4*(d^2 + 16*d*e*x + 120*e^2*x^
2) + 220*a^3*b^3*e^3*(d^3 + 16*d^2*e*x + 120*d*e^2*x^2 + 560*e^3*x^3) + 55*a^2*b^4*e^2*(d^4 + 16*d^3*e*x + 120
*d^2*e^2*x^2 + 560*d*e^3*x^3 + 1820*e^4*x^4) + 10*a*b^5*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*d^2*e^3*x^
3 + 1820*d*e^4*x^4 + 4368*e^5*x^5) + b^6*(d^6 + 16*d^5*e*x + 120*d^4*e^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2*e^4*
x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)))/(80080*e^7*(a + b*x)*(d + e*x)^16)

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Maple [A]  time = 0.011, size = 392, normalized size = 1.1 \begin{align*} -{\frac{8008\,{x}^{6}{b}^{6}{e}^{6}+43680\,{x}^{5}a{b}^{5}{e}^{6}+4368\,{x}^{5}{b}^{6}d{e}^{5}+100100\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+18200\,{x}^{4}a{b}^{5}d{e}^{5}+1820\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+123200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+30800\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+5600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+560\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+85800\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+26400\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+6600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1200\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+120\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+32032\,x{a}^{5}b{e}^{6}+11440\,x{a}^{4}{b}^{2}d{e}^{5}+3520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+160\,xa{b}^{5}{d}^{4}{e}^{2}+16\,x{b}^{6}{d}^{5}e+5005\,{a}^{6}{e}^{6}+2002\,d{e}^{5}{a}^{5}b+715\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+220\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+55\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+10\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{80080\,{e}^{7} \left ( ex+d \right ) ^{16} \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^17,x)

[Out]

-1/80080/e^7*(8008*b^6*e^6*x^6+43680*a*b^5*e^6*x^5+4368*b^6*d*e^5*x^5+100100*a^2*b^4*e^6*x^4+18200*a*b^5*d*e^5
*x^4+1820*b^6*d^2*e^4*x^4+123200*a^3*b^3*e^6*x^3+30800*a^2*b^4*d*e^5*x^3+5600*a*b^5*d^2*e^4*x^3+560*b^6*d^3*e^
3*x^3+85800*a^4*b^2*e^6*x^2+26400*a^3*b^3*d*e^5*x^2+6600*a^2*b^4*d^2*e^4*x^2+1200*a*b^5*d^3*e^3*x^2+120*b^6*d^
4*e^2*x^2+32032*a^5*b*e^6*x+11440*a^4*b^2*d*e^5*x+3520*a^3*b^3*d^2*e^4*x+880*a^2*b^4*d^3*e^3*x+160*a*b^5*d^4*e
^2*x+16*b^6*d^5*e*x+5005*a^6*e^6+2002*a^5*b*d*e^5+715*a^4*b^2*d^2*e^4+220*a^3*b^3*d^3*e^3+55*a^2*b^4*d^4*e^2+1
0*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^16/(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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^17,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59117, size = 1183, normalized size = 3.27 \begin{align*} -\frac{8008 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 10 \, a b^{5} d^{5} e + 55 \, a^{2} b^{4} d^{4} e^{2} + 220 \, a^{3} b^{3} d^{3} e^{3} + 715 \, a^{4} b^{2} d^{2} e^{4} + 2002 \, a^{5} b d e^{5} + 5005 \, a^{6} e^{6} + 4368 \,{\left (b^{6} d e^{5} + 10 \, a b^{5} e^{6}\right )} x^{5} + 1820 \,{\left (b^{6} d^{2} e^{4} + 10 \, a b^{5} d e^{5} + 55 \, a^{2} b^{4} e^{6}\right )} x^{4} + 560 \,{\left (b^{6} d^{3} e^{3} + 10 \, a b^{5} d^{2} e^{4} + 55 \, a^{2} b^{4} d e^{5} + 220 \, a^{3} b^{3} e^{6}\right )} x^{3} + 120 \,{\left (b^{6} d^{4} e^{2} + 10 \, a b^{5} d^{3} e^{3} + 55 \, a^{2} b^{4} d^{2} e^{4} + 220 \, a^{3} b^{3} d e^{5} + 715 \, a^{4} b^{2} e^{6}\right )} x^{2} + 16 \,{\left (b^{6} d^{5} e + 10 \, a b^{5} d^{4} e^{2} + 55 \, a^{2} b^{4} d^{3} e^{3} + 220 \, a^{3} b^{3} d^{2} e^{4} + 715 \, a^{4} b^{2} d e^{5} + 2002 \, a^{5} b e^{6}\right )} x}{80080 \,{\left (e^{23} x^{16} + 16 \, d e^{22} x^{15} + 120 \, d^{2} e^{21} x^{14} + 560 \, d^{3} e^{20} x^{13} + 1820 \, d^{4} e^{19} x^{12} + 4368 \, d^{5} e^{18} x^{11} + 8008 \, d^{6} e^{17} x^{10} + 11440 \, d^{7} e^{16} x^{9} + 12870 \, d^{8} e^{15} x^{8} + 11440 \, d^{9} e^{14} x^{7} + 8008 \, d^{10} e^{13} x^{6} + 4368 \, d^{11} e^{12} x^{5} + 1820 \, d^{12} e^{11} x^{4} + 560 \, d^{13} e^{10} x^{3} + 120 \, d^{14} e^{9} x^{2} + 16 \, d^{15} e^{8} x + d^{16} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^17,x, algorithm="fricas")

[Out]

-1/80080*(8008*b^6*e^6*x^6 + b^6*d^6 + 10*a*b^5*d^5*e + 55*a^2*b^4*d^4*e^2 + 220*a^3*b^3*d^3*e^3 + 715*a^4*b^2
*d^2*e^4 + 2002*a^5*b*d*e^5 + 5005*a^6*e^6 + 4368*(b^6*d*e^5 + 10*a*b^5*e^6)*x^5 + 1820*(b^6*d^2*e^4 + 10*a*b^
5*d*e^5 + 55*a^2*b^4*e^6)*x^4 + 560*(b^6*d^3*e^3 + 10*a*b^5*d^2*e^4 + 55*a^2*b^4*d*e^5 + 220*a^3*b^3*e^6)*x^3
+ 120*(b^6*d^4*e^2 + 10*a*b^5*d^3*e^3 + 55*a^2*b^4*d^2*e^4 + 220*a^3*b^3*d*e^5 + 715*a^4*b^2*e^6)*x^2 + 16*(b^
6*d^5*e + 10*a*b^5*d^4*e^2 + 55*a^2*b^4*d^3*e^3 + 220*a^3*b^3*d^2*e^4 + 715*a^4*b^2*d*e^5 + 2002*a^5*b*e^6)*x)
/(e^23*x^16 + 16*d*e^22*x^15 + 120*d^2*e^21*x^14 + 560*d^3*e^20*x^13 + 1820*d^4*e^19*x^12 + 4368*d^5*e^18*x^11
+ 8008*d^6*e^17*x^10 + 11440*d^7*e^16*x^9 + 12870*d^8*e^15*x^8 + 11440*d^9*e^14*x^7 + 8008*d^10*e^13*x^6 + 43
68*d^11*e^12*x^5 + 1820*d^12*e^11*x^4 + 560*d^13*e^10*x^3 + 120*d^14*e^9*x^2 + 16*d^15*e^8*x + d^16*e^7)

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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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**17,x)

[Out]

Timed out

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Giac [A]  time = 1.17564, size = 702, normalized size = 1.94 \begin{align*} -\frac{{\left (8008 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 4368 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1820 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 560 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 120 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 16 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 43680 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 18200 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 5600 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 1200 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 160 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 100100 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 30800 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 6600 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 880 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 55 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 123200 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 26400 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 3520 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 220 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 85800 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 11440 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 715 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 32032 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 2002 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 5005 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{80080 \,{\left (x e + d\right )}^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^17,x, algorithm="giac")

[Out]

-1/80080*(8008*b^6*x^6*e^6*sgn(b*x + a) + 4368*b^6*d*x^5*e^5*sgn(b*x + a) + 1820*b^6*d^2*x^4*e^4*sgn(b*x + a)
+ 560*b^6*d^3*x^3*e^3*sgn(b*x + a) + 120*b^6*d^4*x^2*e^2*sgn(b*x + a) + 16*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*
sgn(b*x + a) + 43680*a*b^5*x^5*e^6*sgn(b*x + a) + 18200*a*b^5*d*x^4*e^5*sgn(b*x + a) + 5600*a*b^5*d^2*x^3*e^4*
sgn(b*x + a) + 1200*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 160*a*b^5*d^4*x*e^2*sgn(b*x + a) + 10*a*b^5*d^5*e*sgn(b*x
+ a) + 100100*a^2*b^4*x^4*e^6*sgn(b*x + a) + 30800*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 6600*a^2*b^4*d^2*x^2*e^4*
sgn(b*x + a) + 880*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 55*a^2*b^4*d^4*e^2*sgn(b*x + a) + 123200*a^3*b^3*x^3*e^6*s
gn(b*x + a) + 26400*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 3520*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 220*a^3*b^3*d^3*e^3
*sgn(b*x + a) + 85800*a^4*b^2*x^2*e^6*sgn(b*x + a) + 11440*a^4*b^2*d*x*e^5*sgn(b*x + a) + 715*a^4*b^2*d^2*e^4*
sgn(b*x + a) + 32032*a^5*b*x*e^6*sgn(b*x + a) + 2002*a^5*b*d*e^5*sgn(b*x + a) + 5005*a^6*e^6*sgn(b*x + a))*e^(
-7)/(x*e + d)^16