3.2130 \(\int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=256 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]

[Out]

(c^3*x)/e^6 - (c*d^2 - b*d*e + a*e^2)^3/(5*e^7*(d + e*x)^5) + (3*(2*c*d - b*e)*(
c*d^2 - b*d*e + a*e^2)^2)/(4*e^7*(d + e*x)^4) - ((c*d^2 - b*d*e + a*e^2)*(5*c^2*
d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)^3) + ((2*c*d - b*e)*(10*c^2*d
^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 + b
^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - (3*c^2*(2*c*d - b*e)*Log[d + e*x]
)/e^7

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Rubi [A]  time = 0.759283, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^7 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac{3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac{c^3 x}{e^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

(c^3*x)/e^6 - (c*d^2 - b*d*e + a*e^2)^3/(5*e^7*(d + e*x)^5) + (3*(2*c*d - b*e)*(
c*d^2 - b*d*e + a*e^2)^2)/(4*e^7*(d + e*x)^4) - ((c*d^2 - b*d*e + a*e^2)*(5*c^2*
d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)^3) + ((2*c*d - b*e)*(10*c^2*d
^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(2*e^7*(d + e*x)^2) - (3*c*(5*c^2*d^2 + b
^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - (3*c^2*(2*c*d - b*e)*Log[d + e*x]
)/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 c^{2} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \left (d + e x\right )} + \frac{\int c^{3}\, dx}{e^{6}} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \left (d + e x\right )^{3}} - \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{4 e^{7} \left (d + e x\right )^{4}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**6,x)

[Out]

3*c**2*(b*e - 2*c*d)*log(d + e*x)/e**7 - 3*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e +
5*c**2*d**2)/(e**7*(d + e*x)) + Integral(c**3, x)/e**6 - (b*e - 2*c*d)*(6*a*c*e
**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/(2*e**7*(d + e*x)**2) - (a*e**2 - b
*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**7*(d + e*x)*
*3) - 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(4*e**7*(d + e*x)**4) - (a*e*
*2 - b*d*e + c*d**2)**3/(5*e**7*(d + e*x)**5)

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Mathematica [A]  time = 1.67705, size = 396, normalized size = 1.55 \[ -\frac{2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+c^2 e \left (12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (d+e x)^5 (2 c d-b e) \log (d+e x)+2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )}{20 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-(2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x
^4 - 50*d*e^5*x^5 - 10*e^6*x^6) + e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a
*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e
^3*x^3)) + 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*
e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*
d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(12*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d
*e^3*x^3 + 5*e^4*x^4) - b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^
3*x^3 + 300*e^4*x^4)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(20*e^7*(
d + e*x)^5)

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Maple [B]  time = 0.017, size = 688, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(e*x+d)^6,x)

[Out]

-15/4/e^6/(e*x+d)^4*b*c^2*d^4+3/5/e^2/(e*x+d)^5*d*a^2*b-3/5/e^3/(e*x+d)^5*a^2*c*
d^2-3/5/e^3/(e*x+d)^5*d^2*a*b^2-3/5/e^5/(e*x+d)^5*c^2*d^4*a-3/5/e^5/(e*x+d)^5*d^
4*b^2*c+3/5/e^6/(e*x+d)^5*d^5*b*c^2+15*c^2/e^6/(e*x+d)*b*d+3/2/e^3/(e*x+d)^4*a^2
*c*d+6/5/e^4/(e*x+d)^5*d^3*a*b*c-1/2/e^4/(e*x+d)^2*b^3-1/5/e/(e*x+d)^5*a^3-6*c^3
*d*ln(e*x+d)/e^7+c^3*x/e^6+6/e^4/(e*x+d)^3*a*b*c*d-9/2/e^4/(e*x+d)^4*a*b*c*d^2+1
0/e^7/(e*x+d)^2*c^3*d^3-1/e^3/(e*x+d)^3*a^2*c-1/e^3/(e*x+d)^3*a*b^2+1/e^4/(e*x+d
)^3*b^3*d-5/e^7/(e*x+d)^3*c^3*d^4-3/4/e^4/(e*x+d)^4*b^3*d^2+3/2/e^7/(e*x+d)^4*c^
3*d^5+1/5/e^4/(e*x+d)^5*d^3*b^3-1/5/e^7/(e*x+d)^5*c^3*d^6+3*c^2/e^6*ln(e*x+d)*b-
3*c^2/e^5/(e*x+d)*a-3*c/e^5/(e*x+d)*b^2-15*c^3/e^7/(e*x+d)*d^2+3/e^5/(e*x+d)^4*a
*c^2*d^3+3/e^5/(e*x+d)^4*b^2*c*d^3+3/2/e^3/(e*x+d)^4*a*b^2*d-6/e^5/(e*x+d)^3*a*c
^2*d^2-6/e^5/(e*x+d)^3*b^2*c*d^2+10/e^6/(e*x+d)^3*d^3*b*c^2-3/e^4/(e*x+d)^2*a*b*
c+6/e^5/(e*x+d)^2*a*c^2*d+6/e^5/(e*x+d)^2*b^2*c*d-15/e^6/(e*x+d)^2*b*c^2*d^2-3/4
/e^2/(e*x+d)^4*a^2*b

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Maxima [A]  time = 0.835939, size = 606, normalized size = 2.37 \[ -\frac{174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 4 \, a^{3} e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} +{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} + \frac{c^{3} x}{e^{6}} - \frac{3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^6,x, algorithm="maxima")

[Out]

-1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 3*a^2*b*d*e^5 + 4*a^3*e^6 + 12*(b^2*c + a
*c^2)*d^4*e^2 + (b^3 + 6*a*b*c)*d^3*e^3 + 2*(a*b^2 + a^2*c)*d^2*e^4 + 60*(5*c^3*
d^2*e^4 - 5*b*c^2*d*e^5 + (b^2*c + a*c^2)*e^6)*x^4 + 10*(100*c^3*d^3*e^3 - 90*b*
c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 10*(130*c^3*
d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5
+ 2*(a*b^2 + a^2*c)*e^6)*x^2 + 5*(154*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e
^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*
e^5)*x)/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3 + 10*d^3*e^9*x^2 + 5*d^4*e^8*
x + d^5*e^7) + c^3*x/e^6 - 3*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^7

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Fricas [A]  time = 0.207503, size = 810, normalized size = 3.16 \[ \frac{20 \, c^{3} e^{6} x^{6} + 100 \, c^{3} d e^{5} x^{5} - 174 \, c^{3} d^{6} + 137 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 4 \, a^{3} e^{6} - 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 2 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 20 \,{\left (5 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \,{\left (80 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 10 \,{\left (120 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \,{\left (150 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{6} - b c^{2} d^{5} e +{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (2 \, c^{3} d^{2} e^{4} - b c^{2} d e^{5}\right )} x^{4} + 10 \,{\left (2 \, c^{3} d^{3} e^{3} - b c^{2} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (2 \, c^{3} d^{4} e^{2} - b c^{2} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (2 \, c^{3} d^{5} e - b c^{2} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^6,x, algorithm="fricas")

[Out]

1/20*(20*c^3*e^6*x^6 + 100*c^3*d*e^5*x^5 - 174*c^3*d^6 + 137*b*c^2*d^5*e - 3*a^2
*b*d*e^5 - 4*a^3*e^6 - 12*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 2*
(a*b^2 + a^2*c)*d^2*e^4 - 20*(5*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 3*(b^2*c + a*c^2)
*e^6)*x^4 - 10*(80*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 + (
b^3 + 6*a*b*c)*e^6)*x^3 - 10*(120*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*(b^2*c +
a*c^2)*d^2*e^4 + (b^3 + 6*a*b*c)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*x^2 - 5*(150*c^3
*d^5*e - 125*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 12*(b^2*c + a*c^2)*d^3*e^3 + (b^3 + 6
*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - b*c^2*d^5*e + (2*
c^3*d*e^5 - b*c^2*e^6)*x^5 + 5*(2*c^3*d^2*e^4 - b*c^2*d*e^5)*x^4 + 10*(2*c^3*d^3
*e^3 - b*c^2*d^2*e^4)*x^3 + 10*(2*c^3*d^4*e^2 - b*c^2*d^3*e^3)*x^2 + 5*(2*c^3*d^
5*e - b*c^2*d^4*e^2)*x)*log(e*x + d))/(e^12*x^5 + 5*d*e^11*x^4 + 10*d^2*e^10*x^3
+ 10*d^3*e^9*x^2 + 5*d^4*e^8*x + d^5*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(e*x+d)**6,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.205789, size = 560, normalized size = 2.19 \[ c^{3} x e^{\left (-6\right )} - 3 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 60 \,{\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 3 \, a^{2} b d e^{5} + 10 \,{\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} + b^{3} e^{6} + 6 \, a b c e^{6}\right )} x^{3} + 4 \, a^{3} e^{6} + 10 \,{\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \,{\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{20 \,{\left (x e + d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^6,x, algorithm="giac")

[Out]

c^3*x*e^(-6) - 3*(2*c^3*d - b*c^2*e)*e^(-7)*ln(abs(x*e + d)) - 1/20*(174*c^3*d^6
- 137*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 12*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c
*d^3*e^3 + 2*a*b^2*d^2*e^4 + 2*a^2*c*d^2*e^4 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5
+ b^2*c*e^6 + a*c^2*e^6)*x^4 + 3*a^2*b*d*e^5 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d
^2*e^4 + 12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 + b^3*e^6 + 6*a*b*c*e^6)*x^3 + 4*a^3*e^
6 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 + 12*a*c^2*d^2*e^
4 + b^3*d*e^5 + 6*a*b*c*d*e^5 + 2*a*b^2*e^6 + 2*a^2*c*e^6)*x^2 + 5*(154*c^3*d^5*
e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 12*a*c^2*d^3*e^3 + b^3*d^2*e^4 + 6*a*
b*c*d^2*e^4 + 2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 + 3*a^2*b*e^6)*x)*e^(-7)/(x*e + d)^5