3.2131 \(\int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx\)

Optimal. Leaf size=266 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]  time = 0.758804, antiderivative size = 266, 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 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac{3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac{3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac{c^3 \log (d+e x)}{e^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 90.9621, size = 260, normalized size = 0.98 \[ \frac{c^{3} \log{\left (d + e x \right )}}{e^{7}} - \frac{3 c^{2} \left (b e - 2 c d\right )}{e^{7} \left (d + e x\right )} - \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{2}} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{3 e^{7} \left (d + e x\right )^{3}} - \frac{3 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} - \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{5 e^{7} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{6 e^{7} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**7,x)

[Out]

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

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Mathematica [A]  time = 0.588208, size = 385, normalized size = 1.45 \[ \frac{-3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-e^3 \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )-6 c^2 e \left (a e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(d + e*x)^7,x]

[Out]

(c^3*d*(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) - e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2
+ 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) -
3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e*(d^3 + 6*d^2*e*x + 15*d*
e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 +
15*e^4*x^4)) - 6*c^2*e*(a*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 +
15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*
x^4 + 6*e^5*x^5)) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)

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Maple [B]  time = 0.013, size = 695, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^3/(e*x+d)^7,x)

[Out]

-1/2/e^3/(e*x+d)^6*a^2*c*d^2-1/2/e^3/(e*x+d)^6*a*b^2*d^2-1/2/e^5/(e*x+d)^6*c^2*d
^4*a-1/2/e^5/(e*x+d)^6*d^4*b^2*c+1/2/e^6/(e*x+d)^6*b*c^2*d^5+6*c^3*d/e^7/(e*x+d)
-18/5/e^4/(e*x+d)^5*a*b*c*d^2+1/e^4/(e*x+d)^6*d^3*a*c*b+9/2/e^4/(e*x+d)^4*c*a*b*
d+15/2*c^2/e^6/(e*x+d)^2*b*d-2/e^4/(e*x+d)^3*a*b*c+12/5/e^5/(e*x+d)^5*a*c^2*d^3-
15/2*c^3/e^7/(e*x+d)^2*d^2+12/5/e^5/(e*x+d)^5*d^3*b^2*c-3/e^6/(e*x+d)^5*d^4*b*c^
2+1/2/e^2/(e*x+d)^6*a^2*b*d-9/2/e^5/(e*x+d)^4*a*c^2*d^2-9/2/e^5/(e*x+d)^4*c*b^2*
d^2+15/2/e^6/(e*x+d)^4*d^3*b*c^2+4/e^5/(e*x+d)^3*a*c^2*d+4/e^5/(e*x+d)^3*b^2*c*d
-10/e^6/(e*x+d)^3*b*c^2*d^2+6/5/e^3/(e*x+d)^5*a^2*c*d+6/5/e^3/(e*x+d)^5*a*b^2*d-
1/6/e/(e*x+d)^6*a^3-1/3/e^4/(e*x+d)^3*b^3-3/5/e^2/(e*x+d)^5*a^2*b-3/5/e^4/(e*x+d
)^5*b^3*d^2+6/5/e^7/(e*x+d)^5*c^3*d^5+1/6/e^4/(e*x+d)^6*b^3*d^3+c^3*ln(e*x+d)/e^
7-3/4/e^3/(e*x+d)^4*a^2*c-3*c^2/e^6/(e*x+d)*b-3/4/e^3/(e*x+d)^4*a*b^2+3/4/e^4/(e
*x+d)^4*d*b^3-15/4/e^7/(e*x+d)^4*c^3*d^4-1/6/e^7/(e*x+d)^6*c^3*d^6+20/3/e^7/(e*x
+d)^3*c^3*d^3-3/2*c^2/e^5/(e*x+d)^2*a-3/2*c/e^5/(e*x+d)^2*b^2

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Maxima [A]  time = 0.831928, size = 633, normalized size = 2.38 \[ \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} -{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac{c^{3} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^7,x, algorithm="maxima")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c
^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d
*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - (b^2*c + a*c^2)*e^6
)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2*c + a*c^2)*d*e^5 - (b^3
+ 6*a*b*c)*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*(b^2*c + a*c^2)
*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - 3*(a*b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e
- 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*
d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^5)*x)/(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) + c^3*log(e*x + d)/
e^7

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Fricas [A]  time = 0.210189, size = 736, normalized size = 2.77 \[ \frac{147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \,{\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} -{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} -{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \,{\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^7,x, algorithm="fricas")

[Out]

1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c
^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d
*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - (b^2*c + a*c^2)*e^6
)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2*c + a*c^2)*d*e^5 - (b^3
+ 6*a*b*c)*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*(b^2*c + a*c^2)
*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - 3*(a*b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e
- 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*
d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^5)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 + 15*c^
3*d^2*e^4*x^4 + 20*c^3*d^3*e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**3/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.206418, size = 574, normalized size = 2.16 \[ c^{3} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (180 \,{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \,{\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5} - a c^{2} e^{5}\right )} x^{4} + 20 \,{\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - 6 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{3} + 15 \,{\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - 3 \, a b^{2} e^{5} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \,{\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - 6 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - 3 \, a b^{2} d e^{4} - 3 \, a^{2} c d e^{4} - 6 \, a^{2} b e^{5}\right )} x +{\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - 6 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3/(e*x + d)^7,x, algorithm="giac")

[Out]

c^3*e^(-7)*ln(abs(x*e + d)) + 1/60*(180*(2*c^3*d*e^4 - b*c^2*e^5)*x^5 + 90*(15*c
^3*d^2*e^3 - 5*b*c^2*d*e^4 - b^2*c*e^5 - a*c^2*e^5)*x^4 + 20*(110*c^3*d^3*e^2 -
30*b*c^2*d^2*e^3 - 6*b^2*c*d*e^4 - 6*a*c^2*d*e^4 - b^3*e^5 - 6*a*b*c*e^5)*x^3 +
15*(125*c^3*d^4*e - 30*b*c^2*d^3*e^2 - 6*b^2*c*d^2*e^3 - 6*a*c^2*d^2*e^3 - b^3*d
*e^4 - 6*a*b*c*d*e^4 - 3*a*b^2*e^5 - 3*a^2*c*e^5)*x^2 + 6*(137*c^3*d^5 - 30*b*c^
2*d^4*e - 6*b^2*c*d^3*e^2 - 6*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 - 3*
a*b^2*d*e^4 - 3*a^2*c*d*e^4 - 6*a^2*b*e^5)*x + (147*c^3*d^6 - 30*b*c^2*d^5*e - 6
*b^2*c*d^4*e^2 - 6*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 - 3*a*b^2*d^2*e
^4 - 3*a^2*c*d^2*e^4 - 6*a^2*b*d*e^5 - 10*a^3*e^6)*e^(-1))*e^(-6)/(x*e + d)^6