3.2004 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=356 \[ \frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^7*(a + b*x)*(d + e*x)^6) + (6*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)^5) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7
*(a + b*x)*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^3) -
(15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b^5*(b*d - a*e)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(e^7*(a
+ b*x))

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Rubi [A]  time = 0.210466, antiderivative size = 356, 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} \[ \frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{e^7 (a+b x) (d+e x)}-\frac{15 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{2 e^7 (a+b x) (d+e x)^2}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^3}-\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{4 e^7 (a+b x) (d+e x)^4}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{6 e^7 (a+b x) (d+e x)^6}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^7*(a + b*x)*(d + e*x)^6) + (6*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)^5) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7
*(a + b*x)*(d + e*x)^4) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)*(d + e*x)^3) -
(15*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + (6*b^5*(b*d - a*e)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(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 \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^7} \, 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)^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 \frac{(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}{e^6 (d+e x)^7}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^6}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^5}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^4}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^3}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^2}+\frac{b^6}{e^6 (d+e x)}\right ) \, dx}{a b+b^2 x}\\ &=-\frac{(b d-a e)^6 \sqrt{a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}+\frac{6 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)^5}-\frac{15 b^2 (b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^4}+\frac{20 b^3 (b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^3}-\frac{15 b^4 (b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac{6 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.178555, size = 258, normalized size = 0.72 \[ \frac{\sqrt{(a+b x)^2} \left ((b d-a e) \left (a^2 b^3 e^2 \left (282 d^2 e x+57 d^3+525 d e^2 x^2+400 e^3 x^3\right )+a^3 b^2 e^3 \left (37 d^2+162 d e x+225 e^2 x^2\right )+2 a^4 b e^4 (11 d+36 e x)+10 a^5 e^5+a b^4 e \left (975 d^2 e^2 x^2+462 d^3 e x+87 d^4+1000 d e^3 x^3+450 e^4 x^4\right )+b^5 \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )\right )+60 b^6 (d+e x)^6 \log (d+e x)\right )}{60 e^7 (a+b x) (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[(a + b*x)^2]*((b*d - a*e)*(10*a^5*e^5 + 2*a^4*b*e^4*(11*d + 36*e*x) + a^3*b^2*e^3*(37*d^2 + 162*d*e*x +
225*e^2*x^2) + a^2*b^3*e^2*(57*d^3 + 282*d^2*e*x + 525*d*e^2*x^2 + 400*e^3*x^3) + a*b^4*e*(87*d^4 + 462*d^3*e*
x + 975*d^2*e^2*x^2 + 1000*d*e^3*x^3 + 450*e^4*x^4) + b^5*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2
*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + 60*b^6*(d + e*x)^6*Log[d + e*x]))/(60*e^7*(a + b*x)*(d + e*x)^6)

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Maple [A]  time = 0.013, size = 507, normalized size = 1.4 \begin{align*}{\frac{-10\,{a}^{6}{e}^{6}+147\,{b}^{6}{d}^{6}-450\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-1200\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-300\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-600\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-360\,xa{b}^{5}{d}^{4}{e}^{2}-900\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-900\,{x}^{4}a{b}^{5}d{e}^{5}-120\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-180\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-90\,x{a}^{4}{b}^{2}d{e}^{5}+60\,\ln \left ( ex+d \right ){x}^{6}{b}^{6}{e}^{6}+1875\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-72\,x{a}^{5}b{e}^{6}+822\,x{b}^{6}{d}^{5}e-360\,{x}^{5}a{b}^{5}{e}^{6}+360\,{x}^{5}{b}^{6}d{e}^{5}-450\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+1350\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2200\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-225\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-12\,d{e}^{5}{a}^{5}b-20\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-30\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-60\,a{b}^{5}{d}^{5}e-15\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+60\,\ln \left ( ex+d \right ){b}^{6}{d}^{6}+1200\,\ln \left ( ex+d \right ){x}^{3}{b}^{6}{d}^{3}{e}^{3}+900\,\ln \left ( ex+d \right ){x}^{4}{b}^{6}{d}^{2}{e}^{4}+360\,\ln \left ( ex+d \right ) x{b}^{6}{d}^{5}e+360\,\ln \left ( ex+d \right ){x}^{5}{b}^{6}d{e}^{5}+900\,\ln \left ( ex+d \right ){x}^{2}{b}^{6}{d}^{4}{e}^{2}}{60\, \left ( bx+a \right ) ^{5}{e}^{7} \left ( ex+d \right ) ^{6}} \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)^7,x)

[Out]

1/60*((b*x+a)^2)^(5/2)*(-10*a^6*e^6+147*b^6*d^6-450*x^2*a^2*b^4*d^2*e^4-1200*x^3*a*b^5*d^2*e^4-300*x^2*a^3*b^3
*d*e^5-600*x^3*a^2*b^4*d*e^5-360*x*a*b^5*d^4*e^2-900*x^2*a*b^5*d^3*e^3-900*x^4*a*b^5*d*e^5-120*x*a^3*b^3*d^2*e
^4-180*x*a^2*b^4*d^3*e^3-90*x*a^4*b^2*d*e^5+60*ln(e*x+d)*x^6*b^6*e^6+1875*x^2*b^6*d^4*e^2-72*x*a^5*b*e^6+822*x
*b^6*d^5*e-360*x^5*a*b^5*e^6+360*x^5*b^6*d*e^5-450*x^4*a^2*b^4*e^6+1350*x^4*b^6*d^2*e^4-400*x^3*a^3*b^3*e^6+22
00*x^3*b^6*d^3*e^3-225*x^2*a^4*b^2*e^6-12*d*e^5*a^5*b-20*a^3*b^3*d^3*e^3-30*a^2*b^4*d^4*e^2-60*a*b^5*d^5*e-15*
a^4*b^2*d^2*e^4+60*ln(e*x+d)*b^6*d^6+1200*ln(e*x+d)*x^3*b^6*d^3*e^3+900*ln(e*x+d)*x^4*b^6*d^2*e^4+360*ln(e*x+d
)*x*b^6*d^5*e+360*ln(e*x+d)*x^5*b^6*d*e^5+900*ln(e*x+d)*x^2*b^6*d^4*e^2)/(b*x+a)^5/e^7/(e*x+d)^6

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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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55238, size = 1022, normalized size = 2.87 \begin{align*} \frac{147 \, b^{6} d^{6} - 60 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} + 360 \,{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} - a^{2} b^{4} e^{6}\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{3} - 6 \, a b^{5} d^{2} e^{4} - 3 \, a^{2} b^{4} d e^{5} - 2 \, a^{3} b^{3} e^{6}\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e^{2} - 12 \, a b^{5} d^{3} e^{3} - 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - 3 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} e - 60 \, a b^{5} d^{4} e^{2} - 30 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 15 \, a^{4} b^{2} d e^{5} - 12 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} e^{6} x^{6} + 6 \, b^{6} d e^{5} x^{5} + 15 \, b^{6} d^{2} e^{4} x^{4} + 20 \, b^{6} d^{3} e^{3} x^{3} + 15 \, b^{6} d^{4} e^{2} x^{2} + 6 \, b^{6} d^{5} e x + b^{6} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

1/60*(147*b^6*d^6 - 60*a*b^5*d^5*e - 30*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 12*a^5*b*d
*e^5 - 10*a^6*e^6 + 360*(b^6*d*e^5 - a*b^5*e^6)*x^5 + 450*(3*b^6*d^2*e^4 - 2*a*b^5*d*e^5 - a^2*b^4*e^6)*x^4 +
200*(11*b^6*d^3*e^3 - 6*a*b^5*d^2*e^4 - 3*a^2*b^4*d*e^5 - 2*a^3*b^3*e^6)*x^3 + 75*(25*b^6*d^4*e^2 - 12*a*b^5*d
^3*e^3 - 6*a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - 3*a^4*b^2*e^6)*x^2 + 6*(137*b^6*d^5*e - 60*a*b^5*d^4*e^2 - 30*a
^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 15*a^4*b^2*d*e^5 - 12*a^5*b*e^6)*x + 60*(b^6*e^6*x^6 + 6*b^6*d*e^5*x^5 +
15*b^6*d^2*e^4*x^4 + 20*b^6*d^3*e^3*x^3 + 15*b^6*d^4*e^2*x^2 + 6*b^6*d^5*e*x + b^6*d^6)*log(e*x + d))/(e^13*x
^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*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)**7,x)

[Out]

Timed out

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Giac [A]  time = 1.11605, size = 684, normalized size = 1.92 \begin{align*} b^{6} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (360 \,{\left (b^{6} d e^{4} \mathrm{sgn}\left (b x + a\right ) - a b^{5} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{5} + 450 \,{\left (3 \, b^{6} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 2 \, a b^{5} d e^{4} \mathrm{sgn}\left (b x + a\right ) - a^{2} b^{4} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{4} + 200 \,{\left (11 \, b^{6} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 6 \, a b^{5} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{2} b^{4} d e^{4} \mathrm{sgn}\left (b x + a\right ) - 2 \, a^{3} b^{3} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{3} + 75 \,{\left (25 \, b^{6} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 12 \, a b^{5} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 6 \, a^{2} b^{4} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 4 \, a^{3} b^{3} d e^{4} \mathrm{sgn}\left (b x + a\right ) - 3 \, a^{4} b^{2} e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x^{2} + 6 \,{\left (137 \, b^{6} d^{5} \mathrm{sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{4} e \mathrm{sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{3} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d e^{4} \mathrm{sgn}\left (b x + a\right ) - 12 \, a^{5} b e^{5} \mathrm{sgn}\left (b x + a\right )\right )} x +{\left (147 \, b^{6} d^{6} \mathrm{sgn}\left (b x + a\right ) - 60 \, a b^{5} d^{5} e \mathrm{sgn}\left (b x + a\right ) - 30 \, a^{2} b^{4} d^{4} e^{2} \mathrm{sgn}\left (b x + a\right ) - 20 \, a^{3} b^{3} d^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) - 15 \, a^{4} b^{2} d^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) - 12 \, a^{5} b d e^{5} \mathrm{sgn}\left (b x + a\right ) - 10 \, a^{6} e^{6} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \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)^7,x, algorithm="giac")

[Out]

b^6*e^(-7)*log(abs(x*e + d))*sgn(b*x + a) + 1/60*(360*(b^6*d*e^4*sgn(b*x + a) - a*b^5*e^5*sgn(b*x + a))*x^5 +
450*(3*b^6*d^2*e^3*sgn(b*x + a) - 2*a*b^5*d*e^4*sgn(b*x + a) - a^2*b^4*e^5*sgn(b*x + a))*x^4 + 200*(11*b^6*d^3
*e^2*sgn(b*x + a) - 6*a*b^5*d^2*e^3*sgn(b*x + a) - 3*a^2*b^4*d*e^4*sgn(b*x + a) - 2*a^3*b^3*e^5*sgn(b*x + a))*
x^3 + 75*(25*b^6*d^4*e*sgn(b*x + a) - 12*a*b^5*d^3*e^2*sgn(b*x + a) - 6*a^2*b^4*d^2*e^3*sgn(b*x + a) - 4*a^3*b
^3*d*e^4*sgn(b*x + a) - 3*a^4*b^2*e^5*sgn(b*x + a))*x^2 + 6*(137*b^6*d^5*sgn(b*x + a) - 60*a*b^5*d^4*e*sgn(b*x
+ a) - 30*a^2*b^4*d^3*e^2*sgn(b*x + a) - 20*a^3*b^3*d^2*e^3*sgn(b*x + a) - 15*a^4*b^2*d*e^4*sgn(b*x + a) - 12
*a^5*b*e^5*sgn(b*x + a))*x + (147*b^6*d^6*sgn(b*x + a) - 60*a*b^5*d^5*e*sgn(b*x + a) - 30*a^2*b^4*d^4*e^2*sgn(
b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) - 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 12*a^5*b*d*e^5*sgn(b*x + a) - 1
0*a^6*e^6*sgn(b*x + a))*e^(-1))*e^(-6)/(x*e + d)^6