\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^7} \, dx\) [427]

Optimal result

Integrand size = 20, antiderivative size = 156 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \] Output:

-1/6*(a*e^2-b*d*e+c*d^2)^2/e^5/(e*x+d)^6+2/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d 
^2)/e^5/(e*x+d)^5-1/4*(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))/e^5/(e*x+d)^4 
+2/3*c*(-b*e+2*c*d)/e^5/(e*x+d)^3-1/2*c^2/e^5/(e*x+d)^2
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^5 (d+e x)^6} \] Input:

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

Output:

-1/60*(2*c^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 
) + e^2*(10*a^2*e^2 + 4*a*b*e*(d + 6*e*x) + b^2*(d^2 + 6*d*e*x + 15*e^2*x^ 
2)) + 2*c*e*(a*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b*(d^3 + 6*d^2*e*x + 15*d* 
e^2*x^2 + 20*e^3*x^3)))/(e^5*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1140

\(\displaystyle \int \left (\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^4 (d+e x)^5}+\frac {2 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)^6}+\frac {\left (a e^2-b d e+c d^2\right )^2}{e^4 (d+e x)^7}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^4}+\frac {c^2}{e^4 (d+e x)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{4 e^5 (d+e x)^4}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2}\)

Input:

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

Output:

-1/6*(c*d^2 - b*d*e + a*e^2)^2/(e^5*(d + e*x)^6) + (2*(2*c*d - b*e)*(c*d^2 
 - b*d*e + a*e^2))/(5*e^5*(d + e*x)^5) - (6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b 
*d - a*e))/(4*e^5*(d + e*x)^4) + (2*c*(2*c*d - b*e))/(3*e^5*(d + e*x)^3) - 
 c^2/(2*e^5*(d + e*x)^2)
 

Defintions of rubi rules used

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c \left (b e +c d \right ) x^{3}}{3 e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (4 a \,e^{3} b +2 a d \,e^{2} c +d \,e^{2} b^{2}+2 d^{2} e b c +2 c^{2} d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+2 b c \,d^{3} e +2 c^{2} d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\)	\(178\)
norman	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 \left (b c \,e^{2}+c^{2} d e \right ) x^{3}}{3 e^{3}}-\frac {\left (2 a c \,e^{3}+b^{2} e^{3}+2 d \,e^{2} b c +2 d^{2} e \,c^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 e^{4} a b +2 d \,e^{3} a c +b^{2} d \,e^{3}+2 d^{2} e^{2} b c +2 d^{3} e \,c^{2}\right ) x}{10 e^{5}}-\frac {10 a^{2} e^{5}+4 a b d \,e^{4}+2 a c \,d^{2} e^{3}+b^{2} d^{2} e^{3}+2 b c \,d^{3} e^{2}+2 c^{2} d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(192\)
gosper	\(-\frac {30 c^{2} x^{4} e^{4}+40 x^{3} b c \,e^{4}+40 d \,c^{2} x^{3} e^{3}+30 x^{2} a c \,e^{4}+15 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+30 x^{2} c^{2} d^{2} e^{2}+24 x a b \,e^{4}+12 x a c d \,e^{3}+6 x \,b^{2} d \,e^{3}+12 x b c \,d^{2} e^{2}+12 x \,c^{2} d^{3} e +10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+2 b c \,d^{3} e +2 c^{2} d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(193\)
orering	\(-\frac {30 c^{2} x^{4} e^{4}+40 x^{3} b c \,e^{4}+40 d \,c^{2} x^{3} e^{3}+30 x^{2} a c \,e^{4}+15 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+30 x^{2} c^{2} d^{2} e^{2}+24 x a b \,e^{4}+12 x a c d \,e^{3}+6 x \,b^{2} d \,e^{3}+12 x b c \,d^{2} e^{2}+12 x \,c^{2} d^{3} e +10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}+2 b c \,d^{3} e +2 c^{2} d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(193\)
default	\(-\frac {2 c \left (b e -2 c d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {2 a \,e^{3} b -4 a d \,e^{2} c -2 d \,e^{2} b^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}\)	\(195\)
parallelrisch	\(\frac {-30 c^{2} x^{4} e^{5}-40 b c \,e^{5} x^{3}-40 c^{2} d \,e^{4} x^{3}-30 a c \,e^{5} x^{2}-15 b^{2} e^{5} x^{2}-30 b c d \,e^{4} x^{2}-30 c^{2} d^{2} e^{3} x^{2}-24 a b \,e^{5} x -12 a c d \,e^{4} x -6 b^{2} d \,e^{4} x -12 b c \,d^{2} e^{3} x -12 c^{2} d^{3} e^{2} x -10 a^{2} e^{5}-4 a b d \,e^{4}-2 a c \,d^{2} e^{3}-b^{2} d^{2} e^{3}-2 b c \,d^{3} e^{2}-2 c^{2} d^{4} e}{60 e^{6} \left (e x +d \right )^{6}}\)	\(199\)

Input:

int((c*x^2+b*x+a)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*c^2*x^4/e-2/3*c/e^2*(b*e+c*d)*x^3-1/4/e^3*(2*a*c*e^2+b^2*e^2+2*b*c*d 
*e+2*c^2*d^2)*x^2-1/10/e^4*(4*a*b*e^3+2*a*c*d*e^2+b^2*d*e^2+2*b*c*d^2*e+2* 
c^2*d^3)*x-1/60/e^5*(10*a^2*e^4+4*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2+2*b* 
c*d^3*e+2*c^2*d^4))/(e*x+d)^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \] Input:

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

Output:

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + 4*a*b*d*e^3 + 10*a^2*e^4 
 + (b^2 + 2*a*c)*d^2*e^2 + 40*(c^2*d*e^3 + b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^ 
2 + 2*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 6*(2*c^2*d^3*e + 2*b*c*d^2*e^2 
+ 4*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^ 
9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
 

Sympy [A] (verification not implemented)

Time = 40.17 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {- 10 a^{2} e^{4} - 4 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 2 b c d^{3} e - 2 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 40 b c e^{4} - 40 c^{2} d e^{3}\right ) + x^{2} \left (- 30 a c e^{4} - 15 b^{2} e^{4} - 30 b c d e^{3} - 30 c^{2} d^{2} e^{2}\right ) + x \left (- 24 a b e^{4} - 12 a c d e^{3} - 6 b^{2} d e^{3} - 12 b c d^{2} e^{2} - 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \] Input:

integrate((c*x**2+b*x+a)**2/(e*x+d)**7,x)
 

Output:

(-10*a**2*e**4 - 4*a*b*d*e**3 - 2*a*c*d**2*e**2 - b**2*d**2*e**2 - 2*b*c*d 
**3*e - 2*c**2*d**4 - 30*c**2*e**4*x**4 + x**3*(-40*b*c*e**4 - 40*c**2*d*e 
**3) + x**2*(-30*a*c*e**4 - 15*b**2*e**4 - 30*b*c*d*e**3 - 30*c**2*d**2*e* 
*2) + x*(-24*a*b*e**4 - 12*a*c*d*e**3 - 6*b**2*d*e**3 - 12*b*c*d**2*e**2 - 
 12*c**2*d**3*e))/(60*d**6*e**5 + 360*d**5*e**6*x + 900*d**4*e**7*x**2 + 1 
200*d**3*e**8*x**3 + 900*d**2*e**9*x**4 + 360*d*e**10*x**5 + 60*e**11*x**6 
)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \] Input:

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

Output:

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + 4*a*b*d*e^3 + 10*a^2*e^4 
 + (b^2 + 2*a*c)*d^2*e^2 + 40*(c^2*d*e^3 + b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^ 
2 + 2*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 6*(2*c^2*d^3*e + 2*b*c*d^2*e^2 
+ 4*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^ 
9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 40 \, c^{2} d e^{3} x^{3} + 40 \, b c e^{4} x^{3} + 30 \, c^{2} d^{2} e^{2} x^{2} + 30 \, b c d e^{3} x^{2} + 15 \, b^{2} e^{4} x^{2} + 30 \, a c e^{4} x^{2} + 12 \, c^{2} d^{3} e x + 12 \, b c d^{2} e^{2} x + 6 \, b^{2} d e^{3} x + 12 \, a c d e^{3} x + 24 \, a b e^{4} x + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 4 \, a b d e^{3} + 10 \, a^{2} e^{4}}{60 \, {\left (e x + d\right )}^{6} e^{5}} \] Input:

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

Output:

-1/60*(30*c^2*e^4*x^4 + 40*c^2*d*e^3*x^3 + 40*b*c*e^4*x^3 + 30*c^2*d^2*e^2 
*x^2 + 30*b*c*d*e^3*x^2 + 15*b^2*e^4*x^2 + 30*a*c*e^4*x^2 + 12*c^2*d^3*e*x 
 + 12*b*c*d^2*e^2*x + 6*b^2*d*e^3*x + 12*a*c*d*e^3*x + 24*a*b*e^4*x + 2*c^ 
2*d^4 + 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 4*a*b*d*e^3 + 10*a^2*e 
^4)/((e*x + d)^6*e^5)
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.49 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {10\,a^2\,e^4+4\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+2\,b\,c\,d^3\,e+2\,c^2\,d^4}{60\,e^5}+\frac {x\,\left (b^2\,d\,e^2+2\,b\,c\,d^2\,e+4\,a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{10\,e^4}+\frac {c^2\,x^4}{2\,e}+\frac {x^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2+2\,a\,c\,e^2\right )}{4\,e^3}+\frac {2\,c\,x^3\,\left (b\,e+c\,d\right )}{3\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {-30 c^{2} e^{4} x^{4}-40 b c \,e^{4} x^{3}-40 c^{2} d \,e^{3} x^{3}-30 a c \,e^{4} x^{2}-15 b^{2} e^{4} x^{2}-30 b c d \,e^{3} x^{2}-30 c^{2} d^{2} e^{2} x^{2}-24 a b \,e^{4} x -12 a c d \,e^{3} x -6 b^{2} d \,e^{3} x -12 b c \,d^{2} e^{2} x -12 c^{2} d^{3} e x -10 a^{2} e^{4}-4 a b d \,e^{3}-2 a c \,d^{2} e^{2}-b^{2} d^{2} e^{2}-2 b c \,d^{3} e -2 c^{2} d^{4}}{60 e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

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

Output:

( - 10*a**2*e**4 - 4*a*b*d*e**3 - 24*a*b*e**4*x - 2*a*c*d**2*e**2 - 12*a*c 
*d*e**3*x - 30*a*c*e**4*x**2 - b**2*d**2*e**2 - 6*b**2*d*e**3*x - 15*b**2* 
e**4*x**2 - 2*b*c*d**3*e - 12*b*c*d**2*e**2*x - 30*b*c*d*e**3*x**2 - 40*b* 
c*e**4*x**3 - 2*c**2*d**4 - 12*c**2*d**3*e*x - 30*c**2*d**2*e**2*x**2 - 40 
*c**2*d*e**3*x**3 - 30*c**2*e**4*x**4)/(60*e**5*(d**6 + 6*d**5*e*x + 15*d* 
*4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d**2*e**4*x**4 + 6*d*e**5*x**5 + e** 
6*x**6))