3.2008 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=200 \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{120 (d+e x)^8 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{30 (d+e x)^9 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{10 (d+e x)^{10} (b d-a e)} \]

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(120*(b*d - a*e)^
3*(d + e*x)^8) + (b^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(840*(b*d - a*e)^4*(d + e*x)^7)

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Rubi [A]  time = 0.085155, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {770, 21, 45, 37} \[ \frac{b^3 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{120 (d+e x)^8 (b d-a e)^3}+\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{30 (d+e x)^9 (b d-a e)^2}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^6}{10 (d+e x)^{10} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

((a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(120*(b*d - a*e)^
3*(d + e*x)^8) + (b^3*(a + b*x)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(840*(b*d - a*e)^4*(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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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)^{11}} \, 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)^{11}} \, 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)^{11}} \, dx}{a b+b^2 x}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac{\left (3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e) \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac{\left (b^3 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac{b^2 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac{\left (b^4 \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3 \left (a b+b^2 x\right )}\\ &=\frac{(a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{30 (b d-a e)^2 (d+e x)^9}+\frac{b^2 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{120 (b d-a e)^3 (d+e x)^8}+\frac{b^3 (a+b x)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{840 (b d-a e)^4 (d+e x)^7}\\ \end{align*}

Mathematica [A]  time = 0.116416, size = 295, normalized size = 1.48 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^2 b^4 e^2 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )+20 a^3 b^3 e^3 \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+56 a^5 b e^5 (d+10 e x)+84 a^6 e^6+4 a b^5 e \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+10 d^5 e x+d^6+252 d e^5 x^5+210 e^6 x^6\right )\right )}{840 e^7 (a+b x) (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^11,x]

[Out]

-(Sqrt[(a + b*x)^2]*(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 2
0*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*
x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^
4*x^4 + 252*e^5*x^5) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*
x^5 + 210*e^6*x^6)))/(840*e^7*(a + b*x)*(d + e*x)^10)

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Maple [B]  time = 0.01, size = 392, normalized size = 2. \begin{align*} -{\frac{210\,{x}^{6}{b}^{6}{e}^{6}+1008\,{x}^{5}a{b}^{5}{e}^{6}+252\,{x}^{5}{b}^{6}d{e}^{5}+2100\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+840\,{x}^{4}a{b}^{5}d{e}^{5}+210\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1200\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+480\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+120\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+1575\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+900\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+450\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+180\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+45\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+560\,x{a}^{5}b{e}^{6}+350\,x{a}^{4}{b}^{2}d{e}^{5}+200\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+100\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+40\,xa{b}^{5}{d}^{4}{e}^{2}+10\,x{b}^{6}{d}^{5}e+84\,{a}^{6}{e}^{6}+56\,d{e}^{5}{a}^{5}b+35\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+10\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+4\,a{b}^{5}{d}^{5}e+{b}^{6}{d}^{6}}{840\,{e}^{7} \left ( ex+d \right ) ^{10} \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)^11,x)

[Out]

-1/840/e^7*(210*b^6*e^6*x^6+1008*a*b^5*e^6*x^5+252*b^6*d*e^5*x^5+2100*a^2*b^4*e^6*x^4+840*a*b^5*d*e^5*x^4+210*
b^6*d^2*e^4*x^4+2400*a^3*b^3*e^6*x^3+1200*a^2*b^4*d*e^5*x^3+480*a*b^5*d^2*e^4*x^3+120*b^6*d^3*e^3*x^3+1575*a^4
*b^2*e^6*x^2+900*a^3*b^3*d*e^5*x^2+450*a^2*b^4*d^2*e^4*x^2+180*a*b^5*d^3*e^3*x^2+45*b^6*d^4*e^2*x^2+560*a^5*b*
e^6*x+350*a^4*b^2*d*e^5*x+200*a^3*b^3*d^2*e^4*x+100*a^2*b^4*d^3*e^3*x+40*a*b^5*d^4*e^2*x+10*b^6*d^5*e*x+84*a^6
*e^6+56*a^5*b*d*e^5+35*a^4*b^2*d^2*e^4+20*a^3*b^3*d^3*e^3+10*a^2*b^4*d^4*e^2+4*a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2
)^(5/2)/(e*x+d)^10/(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)^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.57303, size = 960, normalized size = 4.8 \begin{align*} -\frac{210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \,{\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \,{\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \,{\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \,{\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} 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)^11,x, algorithm="fricas")

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*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)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.17577, size = 702, normalized size = 3.51 \begin{align*} -\frac{{\left (210 \, b^{6} x^{6} e^{6} \mathrm{sgn}\left (b x + a\right ) + 252 \, b^{6} d x^{5} e^{5} \mathrm{sgn}\left (b x + a\right ) + 210 \, b^{6} d^{2} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 120 \, b^{6} d^{3} x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 45 \, b^{6} d^{4} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 10 \, b^{6} d^{5} x e \mathrm{sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) + 1008 \, a b^{5} x^{5} e^{6} \mathrm{sgn}\left (b x + a\right ) + 840 \, a b^{5} d x^{4} e^{5} \mathrm{sgn}\left (b x + a\right ) + 480 \, a b^{5} d^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 180 \, a b^{5} d^{3} x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 40 \, a b^{5} d^{4} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 4 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) + 2100 \, a^{2} b^{4} x^{4} e^{6} \mathrm{sgn}\left (b x + a\right ) + 1200 \, a^{2} b^{4} d x^{3} e^{5} \mathrm{sgn}\left (b x + a\right ) + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 100 \, a^{2} b^{4} d^{3} x e^{3} \mathrm{sgn}\left (b x + a\right ) + 10 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) + 2400 \, a^{3} b^{3} x^{3} e^{6} \mathrm{sgn}\left (b x + a\right ) + 900 \, a^{3} b^{3} d x^{2} e^{5} \mathrm{sgn}\left (b x + a\right ) + 200 \, a^{3} b^{3} d^{2} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 1575 \, a^{4} b^{2} x^{2} e^{6} \mathrm{sgn}\left (b x + a\right ) + 350 \, a^{4} b^{2} d x e^{5} \mathrm{sgn}\left (b x + a\right ) + 35 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 560 \, a^{5} b x e^{6} \mathrm{sgn}\left (b x + a\right ) + 56 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) + 84 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{840 \,{\left (x e + d\right )}^{10}} \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)^11,x, algorithm="giac")

[Out]

-1/840*(210*b^6*x^6*e^6*sgn(b*x + a) + 252*b^6*d*x^5*e^5*sgn(b*x + a) + 210*b^6*d^2*x^4*e^4*sgn(b*x + a) + 120
*b^6*d^3*x^3*e^3*sgn(b*x + a) + 45*b^6*d^4*x^2*e^2*sgn(b*x + a) + 10*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*
x + a) + 1008*a*b^5*x^5*e^6*sgn(b*x + a) + 840*a*b^5*d*x^4*e^5*sgn(b*x + a) + 480*a*b^5*d^2*x^3*e^4*sgn(b*x +
a) + 180*a*b^5*d^3*x^2*e^3*sgn(b*x + a) + 40*a*b^5*d^4*x*e^2*sgn(b*x + a) + 4*a*b^5*d^5*e*sgn(b*x + a) + 2100*
a^2*b^4*x^4*e^6*sgn(b*x + a) + 1200*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 450*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 10
0*a^2*b^4*d^3*x*e^3*sgn(b*x + a) + 10*a^2*b^4*d^4*e^2*sgn(b*x + a) + 2400*a^3*b^3*x^3*e^6*sgn(b*x + a) + 900*a
^3*b^3*d*x^2*e^5*sgn(b*x + a) + 200*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 1575*a^
4*b^2*x^2*e^6*sgn(b*x + a) + 350*a^4*b^2*d*x*e^5*sgn(b*x + a) + 35*a^4*b^2*d^2*e^4*sgn(b*x + a) + 560*a^5*b*x*
e^6*sgn(b*x + a) + 56*a^5*b*d*e^5*sgn(b*x + a) + 84*a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^10