3.1481 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=158 \[ -\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7} \]

[Out]

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

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Rubi [A]  time = 0.417392, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{2 b^5 (d+e x)^3 (b d-a e)}{e^7}+\frac{15 b^4 (d+e x)^2 (b d-a e)^2}{2 e^7}-\frac{20 b^3 x (b d-a e)^3}{e^6}+\frac{15 b^2 (b d-a e)^4 \log (d+e x)}{e^7}+\frac{6 b (b d-a e)^5}{e^7 (d+e x)}-\frac{(b d-a e)^6}{2 e^7 (d+e x)^2}+\frac{b^6 (d+e x)^4}{4 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 89.5038, size = 144, normalized size = 0.91 \[ \frac{b^{6} \left (d + e x\right )^{4}}{4 e^{7}} + \frac{2 b^{5} \left (d + e x\right )^{3} \left (a e - b d\right )}{e^{7}} + \frac{15 b^{4} \left (d + e x\right )^{2} \left (a e - b d\right )^{2}}{2 e^{7}} + \frac{20 b^{3} x \left (a e - b d\right )^{3}}{e^{6}} + \frac{15 b^{2} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{7}} - \frac{6 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )} - \frac{\left (a e - b d\right )^{6}}{2 e^{7} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**3,x)

[Out]

b**6*(d + e*x)**4/(4*e**7) + 2*b**5*(d + e*x)**3*(a*e - b*d)/e**7 + 15*b**4*(d +
e*x)**2*(a*e - b*d)**2/(2*e**7) + 20*b**3*x*(a*e - b*d)**3/e**6 + 15*b**2*(a*e
- b*d)**4*log(d + e*x)/e**7 - 6*b*(a*e - b*d)**5/(e**7*(d + e*x)) - (a*e - b*d)*
*6/(2*e**7*(d + e*x)**2)

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Mathematica [A]  time = 0.2073, size = 303, normalized size = 1.92 \[ \frac{-2 a^6 e^6-12 a^5 b e^5 (d+2 e x)+30 a^4 b^2 d e^4 (3 d+4 e x)+40 a^3 b^3 e^3 \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+30 a^2 b^4 e^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 a b^5 e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+60 b^2 (d+e x)^2 (b d-a e)^4 \log (d+e x)+b^6 \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )}{4 e^7 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^3,x]

[Out]

(-2*a^6*e^6 - 12*a^5*b*e^5*(d + 2*e*x) + 30*a^4*b^2*d*e^4*(3*d + 4*e*x) + 40*a^3
*b^3*e^3*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + 30*a^2*b^4*e^2*(7*d^4
+ 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + 4*a*b^5*e*(-27*d^5 + 6*d
^4*e*x + 63*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + b^6*(22*d^
6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 2*d*e^5*x^5 +
e^6*x^6) + 60*b^2*(b*d - a*e)^4*(d + e*x)^2*Log[d + e*x])/(4*e^7*(d + e*x)^2)

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Maple [B]  time = 0.016, size = 464, normalized size = 2.9 \[{\frac{{b}^{6}{x}^{4}}{4\,{e}^{3}}}-{\frac{{a}^{6}}{2\,e \left ( ex+d \right ) ^{2}}}-60\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+60\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-9\,{\frac{{b}^{5}{x}^{2}ad}{{e}^{4}}}-45\,{\frac{d{a}^{2}{b}^{4}x}{{e}^{4}}}+36\,{\frac{{d}^{2}a{b}^{5}x}{{e}^{5}}}-60\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}d}{{e}^{4}}}+90\,{\frac{{b}^{4}\ln \left ( ex+d \right ){d}^{2}{a}^{2}}{{e}^{5}}}-60\,{\frac{{b}^{5}\ln \left ( ex+d \right ){d}^{3}a}{{e}^{6}}}+3\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{2}{b}^{2}{a}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{d}^{3}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{4}{a}^{2}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-30\,{\frac{{d}^{4}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+30\,{\frac{d{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{15\,{b}^{4}{x}^{2}{a}^{2}}{2\,{e}^{3}}}+3\,{\frac{{b}^{6}{x}^{2}{d}^{2}}{{e}^{5}}}+20\,{\frac{{a}^{3}{b}^{3}x}{{e}^{3}}}-10\,{\frac{{d}^{3}{b}^{6}x}{{e}^{6}}}+15\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{4}}{{e}^{3}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{4}}{{e}^{7}}}-{\frac{{d}^{6}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-6\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) }}+6\,{\frac{{d}^{5}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{{d}^{5}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{b}^{5}{x}^{3}a}{{e}^{3}}}-{\frac{{b}^{6}{x}^{3}d}{{e}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^3,x)

[Out]

1/4*b^6/e^3*x^4-1/2/e/(e*x+d)^2*a^6-60*b^3/e^4/(e*x+d)*a^3*d^2+60*b^4/e^5/(e*x+d
)*a^2*d^3-9*b^5/e^4*x^2*a*d-45*b^4/e^4*a^2*d*x+36*b^5/e^5*a*d^2*x-60*b^3/e^4*ln(
e*x+d)*a^3*d+90*b^4/e^5*ln(e*x+d)*d^2*a^2-60*b^5/e^6*ln(e*x+d)*d^3*a+3/e^2/(e*x+
d)^2*d*a^5*b-15/2/e^3/(e*x+d)^2*d^2*b^2*a^4+10/e^4/(e*x+d)^2*d^3*a^3*b^3-15/2/e^
5/(e*x+d)^2*d^4*a^2*b^4-30*b^5/e^6/(e*x+d)*a*d^4+30*b^2/e^3/(e*x+d)*a^4*d+15/2*b
^4/e^3*x^2*a^2+3*b^6/e^5*x^2*d^2+20*b^3/e^3*a^3*x-10*b^6/e^6*d^3*x+15*b^2/e^3*ln
(e*x+d)*a^4+15*b^6/e^7*ln(e*x+d)*d^4-1/2/e^7/(e*x+d)^2*d^6*b^6-6*b/e^2/(e*x+d)*a
^5+6*b^6/e^7/(e*x+d)*d^5+3/e^6/(e*x+d)^2*d^5*a*b^5+2*b^5/e^3*x^3*a-b^6/e^4*x^3*d

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Maxima [A]  time = 0.692839, size = 491, normalized size = 3.11 \[ \frac{11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x}{2 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} + \frac{b^{6} e^{3} x^{4} - 4 \,{\left (b^{6} d e^{2} - 2 \, a b^{5} e^{3}\right )} x^{3} + 6 \,{\left (2 \, b^{6} d^{2} e - 6 \, a b^{5} d e^{2} + 5 \, a^{2} b^{4} e^{3}\right )} x^{2} - 4 \,{\left (10 \, b^{6} d^{3} - 36 \, a b^{5} d^{2} e + 45 \, a^{2} b^{4} d e^{2} - 20 \, a^{3} b^{3} e^{3}\right )} x}{4 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^3,x, algorithm="maxima")

[Out]

1/2*(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 4
5*a^4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a^6*e^6 + 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 +
10*a^2*b^4*d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*x)/(e^9*x
^2 + 2*d*e^8*x + d^2*e^7) + 1/4*(b^6*e^3*x^4 - 4*(b^6*d*e^2 - 2*a*b^5*e^3)*x^3 +
6*(2*b^6*d^2*e - 6*a*b^5*d*e^2 + 5*a^2*b^4*e^3)*x^2 - 4*(10*b^6*d^3 - 36*a*b^5*
d^2*e + 45*a^2*b^4*d*e^2 - 20*a^3*b^3*e^3)*x)/e^6 + 15*(b^6*d^4 - 4*a*b^5*d^3*e
+ 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*log(e*x + d)/e^7

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Fricas [A]  time = 0.202022, size = 740, normalized size = 4.68 \[ \frac{b^{6} e^{6} x^{6} + 22 \, b^{6} d^{6} - 108 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} d^{4} e^{2} - 200 \, a^{3} b^{3} d^{3} e^{3} + 90 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 2 \,{\left (b^{6} d e^{5} - 4 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 4 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (b^{6} d^{3} e^{3} - 4 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} - 4 \, a^{3} b^{3} e^{6}\right )} x^{3} - 2 \,{\left (34 \, b^{6} d^{4} e^{2} - 126 \, a b^{5} d^{3} e^{3} + 165 \, a^{2} b^{4} d^{2} e^{4} - 80 \, a^{3} b^{3} d e^{5}\right )} x^{2} - 4 \,{\left (4 \, b^{6} d^{5} e - 6 \, a b^{5} d^{4} e^{2} - 15 \, a^{2} b^{4} d^{3} e^{3} + 40 \, a^{3} b^{3} d^{2} e^{4} - 30 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 4 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} - 4 \, a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} +{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 2 \,{\left (b^{6} d^{5} e - 4 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} - 4 \, a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/4*(b^6*e^6*x^6 + 22*b^6*d^6 - 108*a*b^5*d^5*e + 210*a^2*b^4*d^4*e^2 - 200*a^3*
b^3*d^3*e^3 + 90*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 2*a^6*e^6 - 2*(b^6*d*e^5 - 4
*a*b^5*e^6)*x^5 + 5*(b^6*d^2*e^4 - 4*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 - 20*(b^6*
d^3*e^3 - 4*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 - 4*a^3*b^3*e^6)*x^3 - 2*(34*b^6*d^4
*e^2 - 126*a*b^5*d^3*e^3 + 165*a^2*b^4*d^2*e^4 - 80*a^3*b^3*d*e^5)*x^2 - 4*(4*b^
6*d^5*e - 6*a*b^5*d^4*e^2 - 15*a^2*b^4*d^3*e^3 + 40*a^3*b^3*d^2*e^4 - 30*a^4*b^2
*d*e^5 + 6*a^5*b*e^6)*x + 60*(b^6*d^6 - 4*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 - 4*a^
3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2
*e^4 - 4*a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 2*(b^6*d^5*e - 4*a*b^5*d^4*e^2 + 6*a
^2*b^4*d^3*e^3 - 4*a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5)*x)*log(e*x + d))/(e^9*x^2 +
2*d*e^8*x + d^2*e^7)

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Sympy [A]  time = 10.233, size = 335, normalized size = 2.12 \[ \frac{b^{6} x^{4}}{4 e^{3}} + \frac{15 b^{2} \left (a e - b d\right )^{4} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 6 a^{5} b d e^{5} - 45 a^{4} b^{2} d^{2} e^{4} + 100 a^{3} b^{3} d^{3} e^{3} - 105 a^{2} b^{4} d^{4} e^{2} + 54 a b^{5} d^{5} e - 11 b^{6} d^{6} + x \left (12 a^{5} b e^{6} - 60 a^{4} b^{2} d e^{5} + 120 a^{3} b^{3} d^{2} e^{4} - 120 a^{2} b^{4} d^{3} e^{3} + 60 a b^{5} d^{4} e^{2} - 12 b^{6} d^{5} e\right )}{2 d^{2} e^{7} + 4 d e^{8} x + 2 e^{9} x^{2}} + \frac{x^{3} \left (2 a b^{5} e - b^{6} d\right )}{e^{4}} + \frac{x^{2} \left (15 a^{2} b^{4} e^{2} - 18 a b^{5} d e + 6 b^{6} d^{2}\right )}{2 e^{5}} + \frac{x \left (20 a^{3} b^{3} e^{3} - 45 a^{2} b^{4} d e^{2} + 36 a b^{5} d^{2} e - 10 b^{6} d^{3}\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**3,x)

[Out]

b**6*x**4/(4*e**3) + 15*b**2*(a*e - b*d)**4*log(d + e*x)/e**7 - (a**6*e**6 + 6*a
**5*b*d*e**5 - 45*a**4*b**2*d**2*e**4 + 100*a**3*b**3*d**3*e**3 - 105*a**2*b**4*
d**4*e**2 + 54*a*b**5*d**5*e - 11*b**6*d**6 + x*(12*a**5*b*e**6 - 60*a**4*b**2*d
*e**5 + 120*a**3*b**3*d**2*e**4 - 120*a**2*b**4*d**3*e**3 + 60*a*b**5*d**4*e**2
- 12*b**6*d**5*e))/(2*d**2*e**7 + 4*d*e**8*x + 2*e**9*x**2) + x**3*(2*a*b**5*e -
b**6*d)/e**4 + x**2*(15*a**2*b**4*e**2 - 18*a*b**5*d*e + 6*b**6*d**2)/(2*e**5)
+ x*(20*a**3*b**3*e**3 - 45*a**2*b**4*d*e**2 + 36*a*b**5*d**2*e - 10*b**6*d**3)/
e**6

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GIAC/XCAS [A]  time = 0.211781, size = 460, normalized size = 2.91 \[ 15 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{4} \,{\left (b^{6} x^{4} e^{9} - 4 \, b^{6} d x^{3} e^{8} + 12 \, b^{6} d^{2} x^{2} e^{7} - 40 \, b^{6} d^{3} x e^{6} + 8 \, a b^{5} x^{3} e^{9} - 36 \, a b^{5} d x^{2} e^{8} + 144 \, a b^{5} d^{2} x e^{7} + 30 \, a^{2} b^{4} x^{2} e^{9} - 180 \, a^{2} b^{4} d x e^{8} + 80 \, a^{3} b^{3} x e^{9}\right )} e^{\left (-12\right )} + \frac{{\left (11 \, b^{6} d^{6} - 54 \, a b^{5} d^{5} e + 105 \, a^{2} b^{4} d^{4} e^{2} - 100 \, a^{3} b^{3} d^{3} e^{3} + 45 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} - a^{6} e^{6} + 12 \,{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^3,x, algorithm="giac")

[Out]

15*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)
*e^(-7)*ln(abs(x*e + d)) + 1/4*(b^6*x^4*e^9 - 4*b^6*d*x^3*e^8 + 12*b^6*d^2*x^2*e
^7 - 40*b^6*d^3*x*e^6 + 8*a*b^5*x^3*e^9 - 36*a*b^5*d*x^2*e^8 + 144*a*b^5*d^2*x*e
^7 + 30*a^2*b^4*x^2*e^9 - 180*a^2*b^4*d*x*e^8 + 80*a^3*b^3*x*e^9)*e^(-12) + 1/2*
(11*b^6*d^6 - 54*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 - 100*a^3*b^3*d^3*e^3 + 45*a^
4*b^2*d^2*e^4 - 6*a^5*b*d*e^5 - a^6*e^6 + 12*(b^6*d^5*e - 5*a*b^5*d^4*e^2 + 10*a
^2*b^4*d^3*e^3 - 10*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 - a^5*b*e^6)*x)*e^(-7)/(x*
e + d)^2