3.1482 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=156 \[ -\frac{3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac{15 b^4 x (b d-a e)^2}{e^6}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{b^6 (d+e x)^3}{3 e^7} \]

[Out]

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

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Rubi [A]  time = 0.406509, antiderivative size = 156, 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 \[ -\frac{3 b^5 (d+e x)^2 (b d-a e)}{e^7}+\frac{15 b^4 x (b d-a e)^2}{e^6}-\frac{20 b^3 (b d-a e)^3 \log (d+e x)}{e^7}-\frac{15 b^2 (b d-a e)^4}{e^7 (d+e x)}+\frac{3 b (b d-a e)^5}{e^7 (d+e x)^2}-\frac{(b d-a e)^6}{3 e^7 (d+e x)^3}+\frac{b^6 (d+e x)^3}{3 e^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 84.9865, size = 143, normalized size = 0.92 \[ \frac{b^{6} \left (d + e x\right )^{3}}{3 e^{7}} + \frac{3 b^{5} \left (d + e x\right )^{2} \left (a e - b d\right )}{e^{7}} + \frac{15 b^{4} x \left (a e - b d\right )^{2}}{e^{6}} + \frac{20 b^{3} \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{15 b^{2} \left (a e - b d\right )^{4}}{e^{7} \left (d + e x\right )} - \frac{3 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{2}} - \frac{\left (a e - b d\right )^{6}}{3 e^{7} \left (d + e x\right )^{3}} \]

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)**4,x)

[Out]

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

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Mathematica [A]  time = 0.214116, size = 302, normalized size = 1.94 \[ \frac{-a^6 e^6-3 a^5 b e^5 (d+3 e x)-15 a^4 b^2 e^4 \left (d^2+3 d e x+3 e^2 x^2\right )+10 a^3 b^3 d e^3 \left (11 d^2+27 d e x+18 e^2 x^2\right )+15 a^2 b^4 e^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )+3 a b^5 e \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )-60 b^3 (d+e x)^3 (b d-a e)^3 \log (d+e x)+b^6 \left (-37 d^6-51 d^5 e x+39 d^4 e^2 x^2+73 d^3 e^3 x^3+15 d^2 e^4 x^4-3 d e^5 x^5+e^6 x^6\right )}{3 e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^4,x]

[Out]

(-(a^6*e^6) - 3*a^5*b*e^5*(d + 3*e*x) - 15*a^4*b^2*e^4*(d^2 + 3*d*e*x + 3*e^2*x^
2) + 10*a^3*b^3*d*e^3*(11*d^2 + 27*d*e*x + 18*e^2*x^2) + 15*a^2*b^4*e^2*(-13*d^4
- 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) + 3*a*b^5*e*(47*d^5 + 8
1*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) + b^6*(-3
7*d^6 - 51*d^5*e*x + 39*d^4*e^2*x^2 + 73*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 3*d*e^5*
x^5 + e^6*x^6) - 60*b^3*(b*d - a*e)^3*(d + e*x)^3*Log[d + e*x])/(3*e^7*(d + e*x)
^3)

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Maple [B]  time = 0.016, size = 483, normalized size = 3.1 \[ -5\,{\frac{{d}^{4}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{3}}}-15\,{\frac{{d}^{4}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-30\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+30\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-60\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}d}{{e}^{5}}}+60\,{\frac{d{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-90\,{\frac{{d}^{2}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+60\,{\frac{{d}^{3}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+{\frac{20\,{d}^{3}{a}^{3}{b}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{d}^{5}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}-3\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{d}^{5}{b}^{6}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-15\,{\frac{{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) }}-15\,{\frac{{d}^{4}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}+3\,{\frac{{b}^{5}{x}^{2}a}{{e}^{4}}}-2\,{\frac{{b}^{6}{x}^{2}d}{{e}^{5}}}+15\,{\frac{{a}^{2}{b}^{4}x}{{e}^{4}}}+10\,{\frac{{b}^{6}{d}^{2}x}{{e}^{6}}}-{\frac{{d}^{6}{b}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}+20\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}}{{e}^{4}}}+2\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{d}^{2}{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-20\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{3}}{{e}^{7}}}+60\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{2}}{{e}^{6}}}-24\,{\frac{da{b}^{5}x}{{e}^{5}}}+{\frac{{b}^{6}{x}^{3}}{3\,{e}^{4}}}-{\frac{{a}^{6}}{3\,e \left ( ex+d \right ) ^{3}}}+15\,{\frac{d{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{2}}} \]

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)^4,x)

[Out]

-5/e^5/(e*x+d)^3*d^4*a^2*b^4-15*b^5/e^6/(e*x+d)^2*a*d^4-30*b^3/e^4/(e*x+d)^2*a^3
*d^2+30*b^4/e^5/(e*x+d)^2*a^2*d^3-60*b^4/e^5*ln(e*x+d)*a^2*d+60*b^3/e^4/(e*x+d)*
a^3*d-90*b^4/e^5/(e*x+d)*d^2*a^2+60*b^5/e^6/(e*x+d)*d^3*a+20/3/e^4/(e*x+d)^3*d^3
*a^3*b^3+2/e^6/(e*x+d)^3*d^5*a*b^5-3*b/e^2/(e*x+d)^2*a^5+3*b^6/e^7/(e*x+d)^2*d^5
-15*b^2/e^3/(e*x+d)*a^4-15*b^6/e^7/(e*x+d)*d^4+3*b^5/e^4*x^2*a-2*b^6/e^5*x^2*d+1
5*b^4/e^4*a^2*x+10*b^6/e^6*d^2*x-1/3/e^7/(e*x+d)^3*d^6*b^6+20*b^3/e^4*ln(e*x+d)*
a^3+2/e^2/(e*x+d)^3*d*a^5*b-5/e^3/(e*x+d)^3*d^2*b^2*a^4-20*b^6/e^7*ln(e*x+d)*d^3
+60*b^5/e^6*ln(e*x+d)*a*d^2-24*b^5/e^5*d*a*x+1/3*b^6/e^4*x^3-1/3/e/(e*x+d)^3*a^6
+15*b^2/e^3/(e*x+d)^2*a^4*d

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Maxima [A]  time = 0.696425, size = 505, normalized size = 3.24 \[ -\frac{37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac{b^{6} e^{2} x^{3} - 3 \,{\left (2 \, b^{6} d e - 3 \, a b^{5} e^{2}\right )} x^{2} + 3 \,{\left (10 \, b^{6} d^{2} - 24 \, a b^{5} d e + 15 \, a^{2} b^{4} e^{2}\right )} x}{3 \, e^{6}} - \frac{20 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \]

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)^4,x, algorithm="maxima")

[Out]

-1/3*(37*b^6*d^6 - 141*a*b^5*d^5*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 +
15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*e^5 + a^6*e^6 + 45*(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 + 9*(9*b^6*d^5*e - 35
*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*e^3 - 30*a^3*b^3*d^2*e^4 + 5*a^4*b^2*d*e^5 + a^5
*b*e^6)*x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7) + 1/3*(b^6*e^2*x^3 -
3*(2*b^6*d*e - 3*a*b^5*e^2)*x^2 + 3*(10*b^6*d^2 - 24*a*b^5*d*e + 15*a^2*b^4*e^2
)*x)/e^6 - 20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*log(e*x
+ d)/e^7

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Fricas [A]  time = 0.204531, size = 778, normalized size = 4.99 \[ \frac{b^{6} e^{6} x^{6} - 37 \, b^{6} d^{6} + 141 \, a b^{5} d^{5} e - 195 \, a^{2} b^{4} d^{4} e^{2} + 110 \, a^{3} b^{3} d^{3} e^{3} - 15 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} + 15 \,{\left (b^{6} d^{2} e^{4} - 3 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} +{\left (73 \, b^{6} d^{3} e^{3} - 189 \, a b^{5} d^{2} e^{4} + 135 \, a^{2} b^{4} d e^{5}\right )} x^{3} + 3 \,{\left (13 \, b^{6} d^{4} e^{2} - 9 \, a b^{5} d^{3} e^{3} - 45 \, a^{2} b^{4} d^{2} e^{4} + 60 \, a^{3} b^{3} d e^{5} - 15 \, a^{4} b^{2} e^{6}\right )} x^{2} - 3 \,{\left (17 \, b^{6} d^{5} e - 81 \, a b^{5} d^{4} e^{2} + 135 \, a^{2} b^{4} d^{3} e^{3} - 90 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - 3 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} - a^{3} b^{3} d^{3} e^{3} +{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (b^{6} d^{4} e^{2} - 3 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} - a^{3} b^{3} d e^{5}\right )} x^{2} + 3 \,{\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} - a^{3} b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

1/3*(b^6*e^6*x^6 - 37*b^6*d^6 + 141*a*b^5*d^5*e - 195*a^2*b^4*d^4*e^2 + 110*a^3*
b^3*d^3*e^3 - 15*a^4*b^2*d^2*e^4 - 3*a^5*b*d*e^5 - a^6*e^6 - 3*(b^6*d*e^5 - 3*a*
b^5*e^6)*x^5 + 15*(b^6*d^2*e^4 - 3*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 + (73*b^6*d^
3*e^3 - 189*a*b^5*d^2*e^4 + 135*a^2*b^4*d*e^5)*x^3 + 3*(13*b^6*d^4*e^2 - 9*a*b^5
*d^3*e^3 - 45*a^2*b^4*d^2*e^4 + 60*a^3*b^3*d*e^5 - 15*a^4*b^2*e^6)*x^2 - 3*(17*b
^6*d^5*e - 81*a*b^5*d^4*e^2 + 135*a^2*b^4*d^3*e^3 - 90*a^3*b^3*d^2*e^4 + 15*a^4*
b^2*d*e^5 + 3*a^5*b*e^6)*x - 60*(b^6*d^6 - 3*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 - a
^3*b^3*d^3*e^3 + (b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 - a^3*b^3*e^6)
*x^3 + 3*(b^6*d^4*e^2 - 3*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 - a^3*b^3*d*e^5)*x^2
+ 3*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*e^3 - a^3*b^3*d^2*e^4)*x)*log(
e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)

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Sympy [A]  time = 18.7287, size = 364, normalized size = 2.33 \[ \frac{b^{6} x^{3}}{3 e^{4}} + \frac{20 b^{3} \left (a e - b d\right )^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 3 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 110 a^{3} b^{3} d^{3} e^{3} + 195 a^{2} b^{4} d^{4} e^{2} - 141 a b^{5} d^{5} e + 37 b^{6} d^{6} + x^{2} \left (45 a^{4} b^{2} e^{6} - 180 a^{3} b^{3} d e^{5} + 270 a^{2} b^{4} d^{2} e^{4} - 180 a b^{5} d^{3} e^{3} + 45 b^{6} d^{4} e^{2}\right ) + x \left (9 a^{5} b e^{6} + 45 a^{4} b^{2} d e^{5} - 270 a^{3} b^{3} d^{2} e^{4} + 450 a^{2} b^{4} d^{3} e^{3} - 315 a b^{5} d^{4} e^{2} + 81 b^{6} d^{5} e\right )}{3 d^{3} e^{7} + 9 d^{2} e^{8} x + 9 d e^{9} x^{2} + 3 e^{10} x^{3}} + \frac{x^{2} \left (3 a b^{5} e - 2 b^{6} d\right )}{e^{5}} + \frac{x \left (15 a^{2} b^{4} e^{2} - 24 a b^{5} d e + 10 b^{6} d^{2}\right )}{e^{6}} \]

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)**4,x)

[Out]

b**6*x**3/(3*e**4) + 20*b**3*(a*e - b*d)**3*log(d + e*x)/e**7 - (a**6*e**6 + 3*a
**5*b*d*e**5 + 15*a**4*b**2*d**2*e**4 - 110*a**3*b**3*d**3*e**3 + 195*a**2*b**4*
d**4*e**2 - 141*a*b**5*d**5*e + 37*b**6*d**6 + x**2*(45*a**4*b**2*e**6 - 180*a**
3*b**3*d*e**5 + 270*a**2*b**4*d**2*e**4 - 180*a*b**5*d**3*e**3 + 45*b**6*d**4*e*
*2) + x*(9*a**5*b*e**6 + 45*a**4*b**2*d*e**5 - 270*a**3*b**3*d**2*e**4 + 450*a**
2*b**4*d**3*e**3 - 315*a*b**5*d**4*e**2 + 81*b**6*d**5*e))/(3*d**3*e**7 + 9*d**2
*e**8*x + 9*d*e**9*x**2 + 3*e**10*x**3) + x**2*(3*a*b**5*e - 2*b**6*d)/e**5 + x*
(15*a**2*b**4*e**2 - 24*a*b**5*d*e + 10*b**6*d**2)/e**6

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GIAC/XCAS [A]  time = 0.21075, size = 452, normalized size = 2.9 \[ -20 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{3} \,{\left (b^{6} x^{3} e^{8} - 6 \, b^{6} d x^{2} e^{7} + 30 \, b^{6} d^{2} x e^{6} + 9 \, a b^{5} x^{2} e^{8} - 72 \, a b^{5} d x e^{7} + 45 \, a^{2} b^{4} x e^{8}\right )} e^{\left (-12\right )} - \frac{{\left (37 \, b^{6} d^{6} - 141 \, a b^{5} d^{5} e + 195 \, a^{2} b^{4} d^{4} e^{2} - 110 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + a^{6} e^{6} + 45 \,{\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (9 \, b^{6} d^{5} e - 35 \, a b^{5} d^{4} e^{2} + 50 \, a^{2} b^{4} d^{3} e^{3} - 30 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

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)^4,x, algorithm="giac")

[Out]

-20*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*e^(-7)*ln(abs(x*e
+ d)) + 1/3*(b^6*x^3*e^8 - 6*b^6*d*x^2*e^7 + 30*b^6*d^2*x*e^6 + 9*a*b^5*x^2*e^8
- 72*a*b^5*d*x*e^7 + 45*a^2*b^4*x*e^8)*e^(-12) - 1/3*(37*b^6*d^6 - 141*a*b^5*d^5
*e + 195*a^2*b^4*d^4*e^2 - 110*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 3*a^5*b*d*
e^5 + a^6*e^6 + 45*(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 + 9*(9*b^6*d^5*e - 35*a*b^5*d^4*e^2 + 50*a^2*b^4*d^3*
e^3 - 30*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)^3