3.2011 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{14}} \, dx\)

Optimal. Leaf size=360 \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) + (b*(b*d - a*e)^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e
^7*(a + b*x)*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^10)
- (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^9) + (3*b^5*(b*d - a*e)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^8) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
*(d + e*x)^7)

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Rubi [A]  time = 0.192599, antiderivative size = 360, 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} \[ -\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac{3 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^7 (a+b x) (d+e x)^8}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{3 e^7 (a+b x) (d+e x)^9}+\frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{e^7 (a+b x) (d+e x)^{10}}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{11 e^7 (a+b x) (d+e x)^{11}}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{2 e^7 (a+b x) (d+e x)^{12}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{13 e^7 (a+b x) (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)*(d + e*x)^13) + (b*(b*d - a*e)^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e
^7*(a + b*x)*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)^10)
- (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^9) + (3*b^5*(b*d - a*e)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^8) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)
*(d + e*x)^7)

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

Mathematica [A]  time = 0.119403, size = 295, normalized size = 0.82 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^2 b^4 e^2 \left (78 d^2 e^2 x^2+13 d^3 e x+d^4+286 d e^3 x^3+715 e^4 x^4\right )+84 a^3 b^3 e^3 \left (13 d^2 e x+d^3+78 d e^2 x^2+286 e^3 x^3\right )+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+462 a^5 b e^5 (d+13 e x)+924 a^6 e^6+7 a b^5 e \left (78 d^3 e^2 x^2+286 d^2 e^3 x^3+13 d^4 e x+d^5+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+13 d^5 e x+d^6+1287 d e^5 x^5+1716 e^6 x^6\right )\right )}{12012 e^7 (a+b x) (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^14,x]

[Out]

-(Sqrt[(a + b*x)^2]*(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 210*a^4*b^2*e^4*(d^2 + 13*d*e*x + 78*e^2*x^2)
+ 84*a^3*b^3*e^3*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78*d^2*e
^2*x^2 + 286*d*e^3*x^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d
*e^4*x^4 + 1287*e^5*x^5) + b^6*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d
*e^5*x^5 + 1716*e^6*x^6)))/(12012*e^7*(a + b*x)*(d + e*x)^13)

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Maple [A]  time = 0.01, size = 392, normalized size = 1.1 \begin{align*} -{\frac{1716\,{x}^{6}{b}^{6}{e}^{6}+9009\,{x}^{5}a{b}^{5}{e}^{6}+1287\,{x}^{5}{b}^{6}d{e}^{5}+20020\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+5005\,{x}^{4}a{b}^{5}d{e}^{5}+715\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+24024\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+8008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+2002\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+286\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+16380\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+6552\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+2184\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+546\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+78\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+6006\,x{a}^{5}b{e}^{6}+2730\,x{a}^{4}{b}^{2}d{e}^{5}+1092\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+364\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+91\,xa{b}^{5}{d}^{4}{e}^{2}+13\,x{b}^{6}{d}^{5}e+924\,{a}^{6}{e}^{6}+462\,d{e}^{5}{a}^{5}b+210\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+84\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+28\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+7\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{12012\,{e}^{7} \left ( ex+d \right ) ^{13} \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)^14,x)

[Out]

-1/12012/e^7*(1716*b^6*e^6*x^6+9009*a*b^5*e^6*x^5+1287*b^6*d*e^5*x^5+20020*a^2*b^4*e^6*x^4+5005*a*b^5*d*e^5*x^
4+715*b^6*d^2*e^4*x^4+24024*a^3*b^3*e^6*x^3+8008*a^2*b^4*d*e^5*x^3+2002*a*b^5*d^2*e^4*x^3+286*b^6*d^3*e^3*x^3+
16380*a^4*b^2*e^6*x^2+6552*a^3*b^3*d*e^5*x^2+2184*a^2*b^4*d^2*e^4*x^2+546*a*b^5*d^3*e^3*x^2+78*b^6*d^4*e^2*x^2
+6006*a^5*b*e^6*x+2730*a^4*b^2*d*e^5*x+1092*a^3*b^3*d^2*e^4*x+364*a^2*b^4*d^3*e^3*x+91*a*b^5*d^4*e^2*x+13*b^6*
d^5*e*x+924*a^6*e^6+462*a^5*b*d*e^5+210*a^4*b^2*d^2*e^4+84*a^3*b^3*d^3*e^3+28*a^2*b^4*d^4*e^2+7*a*b^5*d^5*e+b^
6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^13/(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)^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60925, size = 1064, normalized size = 2.96 \begin{align*} -\frac{1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \,{\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \,{\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \,{\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \,{\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \,{\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \,{\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} 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)^14,x, algorithm="fricas")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d
^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5
+ 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6
*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*
a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 1
3*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^
7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e
^8*x + d^13*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)**14,x)

[Out]

Timed out

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Giac [A]  time = 1.16744, size = 702, normalized size = 1.95 \begin{align*} -\frac{{\left (1716 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 1287 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 715 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 286 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 78 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 13 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 9009 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 5005 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 2002 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 546 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 91 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 20020 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 8008 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 364 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 28 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 24024 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 6552 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 1092 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 16380 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 2730 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 6006 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 462 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 924 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{12012 \,{\left (x e + d\right )}^{13}} \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)^14,x, algorithm="giac")

[Out]

-1/12012*(1716*b^6*x^6*e^6*sgn(b*x + a) + 1287*b^6*d*x^5*e^5*sgn(b*x + a) + 715*b^6*d^2*x^4*e^4*sgn(b*x + a) +
286*b^6*d^3*x^3*e^3*sgn(b*x + a) + 78*b^6*d^4*x^2*e^2*sgn(b*x + a) + 13*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sg
n(b*x + a) + 9009*a*b^5*x^5*e^6*sgn(b*x + a) + 5005*a*b^5*d*x^4*e^5*sgn(b*x + a) + 2002*a*b^5*d^2*x^3*e^4*sgn(
b*x + a) + 546*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 91*a*b^5*d^4*x*e^2*sgn(b*x + a) + 7*a*b^5*d^5*e*sgn(b*x + a) +
20020*a^2*b^4*x^4*e^6*sgn(b*x + a) + 8008*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 2184*a^2*b^4*d^2*x^2*e^4*sgn(b*x +
a) + 364*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 28*a^2*b^4*d^4*e^2*sgn(b*x + a) + 24024*a^3*b^3*x^3*e^6*sgn(b*x + a
) + 6552*a^3*b^3*d*x^2*e^5*sgn(b*x + a) + 1092*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 84*a^3*b^3*d^3*e^3*sgn(b*x + a
) + 16380*a^4*b^2*x^2*e^6*sgn(b*x + a) + 2730*a^4*b^2*d*x*e^5*sgn(b*x + a) + 210*a^4*b^2*d^2*e^4*sgn(b*x + a)
+ 6006*a^5*b*x*e^6*sgn(b*x + a) + 462*a^5*b*d*e^5*sgn(b*x + a) + 924*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^13