\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^{10}} \, dx\) [245]

Optimal result

Integrand size = 37, antiderivative size = 111 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{11 \left (c d^2-a e^2\right ) (d+e x)^{10}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{99 \left (c d^2-a e^2\right )^2 (d+e x)^9} \] Output:

2/11*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)/(e*x+d)^10+4/9 
9*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)^2/(e*x+d)^9
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {2 (a e+c d x)^4 \sqrt {(a e+c d x) (d+e x)} \left (11 c d^2-9 a e^2+2 c d e x\right )}{99 \left (c d^2-a e^2\right )^2 (d+e x)^6} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^10,x]
 

Output:

(2*(a*e + c*d*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*(11*c*d^2 - 9*a*e^2 + 2*c 
*d*e*x))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {1129, 1123}

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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^9}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{99 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^10,x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(11*(c*d^2 - a*e^2)*(d + 
 e*x)^10) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(99*(c*d 
^2 - a*e^2)^2*(d + e*x)^9)
 

Defintions of rubi rules used

rule 1123

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

Maple [A] (verified)

Time = 10.61 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d x e +9 a \,e^{2}-11 c \,d^{2}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {7}{2}}}{99 \left (e x +d \right )^{9} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\)	\(90\)
orering	\(-\frac {2 \left (-2 c d x e +9 a \,e^{2}-11 c \,d^{2}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {7}{2}}}{99 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{9}}\)	\(91\)
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{9}}}{e^{10}}\)	\(131\)
trager	\(-\frac {2 \left (-2 c^{5} d^{5} e \,x^{5}+a \,c^{4} d^{4} e^{2} x^{4}-11 c^{5} d^{6} x^{4}+24 a^{2} c^{3} d^{3} e^{3} x^{3}-44 a \,c^{4} d^{5} e \,x^{3}+46 a^{3} c^{2} d^{2} e^{4} x^{2}-66 a^{2} c^{3} d^{4} e^{2} x^{2}+34 a^{4} c d \,e^{5} x -44 a^{3} c^{2} d^{3} e^{3} x +9 a^{5} e^{6}-11 c \,d^{2} e^{4} a^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{99 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{6}}\)	\(211\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^10,x,method=_RETURNVER 
BOSE)
 

Output:

-2/99*(c*d*x+a*e)*(-2*c*d*e*x+9*a*e^2-11*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x 
+a*d*e)^(7/2)/(e*x+d)^9/(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (103) = 206\).

Time = 23.34 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.74 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {2 \, {\left (2 \, c^{5} d^{5} e x^{5} + 11 \, a^{4} c d^{2} e^{4} - 9 \, a^{5} e^{6} + {\left (11 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} x^{4} + 4 \, {\left (11 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} x^{3} + 2 \, {\left (33 \, a^{2} c^{3} d^{4} e^{2} - 23 \, a^{3} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (22 \, a^{3} c^{2} d^{3} e^{3} - 17 \, a^{4} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{99 \, {\left (c^{2} d^{10} - 2 \, a c d^{8} e^{2} + a^{2} d^{6} e^{4} + {\left (c^{2} d^{4} e^{6} - 2 \, a c d^{2} e^{8} + a^{2} e^{10}\right )} x^{6} + 6 \, {\left (c^{2} d^{5} e^{5} - 2 \, a c d^{3} e^{7} + a^{2} d e^{9}\right )} x^{5} + 15 \, {\left (c^{2} d^{6} e^{4} - 2 \, a c d^{4} e^{6} + a^{2} d^{2} e^{8}\right )} x^{4} + 20 \, {\left (c^{2} d^{7} e^{3} - 2 \, a c d^{5} e^{5} + a^{2} d^{3} e^{7}\right )} x^{3} + 15 \, {\left (c^{2} d^{8} e^{2} - 2 \, a c d^{6} e^{4} + a^{2} d^{4} e^{6}\right )} x^{2} + 6 \, {\left (c^{2} d^{9} e - 2 \, a c d^{7} e^{3} + a^{2} d^{5} e^{5}\right )} x\right )}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^10,x, algorithm= 
"fricas")
 

Output:

2/99*(2*c^5*d^5*e*x^5 + 11*a^4*c*d^2*e^4 - 9*a^5*e^6 + (11*c^5*d^6 - a*c^4 
*d^4*e^2)*x^4 + 4*(11*a*c^4*d^5*e - 6*a^2*c^3*d^3*e^3)*x^3 + 2*(33*a^2*c^3 
*d^4*e^2 - 23*a^3*c^2*d^2*e^4)*x^2 + 2*(22*a^3*c^2*d^3*e^3 - 17*a^4*c*d*e^ 
5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^2*d^10 - 2*a*c*d^8*e^ 
2 + a^2*d^6*e^4 + (c^2*d^4*e^6 - 2*a*c*d^2*e^8 + a^2*e^10)*x^6 + 6*(c^2*d^ 
5*e^5 - 2*a*c*d^3*e^7 + a^2*d*e^9)*x^5 + 15*(c^2*d^6*e^4 - 2*a*c*d^4*e^6 + 
 a^2*d^2*e^8)*x^4 + 20*(c^2*d^7*e^3 - 2*a*c*d^5*e^5 + a^2*d^3*e^7)*x^3 + 1 
5*(c^2*d^8*e^2 - 2*a*c*d^6*e^4 + a^2*d^4*e^6)*x^2 + 6*(c^2*d^9*e - 2*a*c*d 
^7*e^3 + a^2*d^5*e^5)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**10,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^10,x, algorithm= 
"maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^10,x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[0,0,6]%%%},[12]%%%}+%%%{%%{[%%%{-12,[0,1,5]%%%},0]: 
[1,0,%%%{
 

Mupad [B] (verification not implemented)

Time = 11.11 (sec) , antiderivative size = 5860, normalized size of antiderivative = 52.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^10,x)
 

Output:

(((d*((8*c^6*d^7)/(99*e^2*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4*c^ 
5*d^5*(21*a*e^2 - 17*c*d^2))/(99*e^2*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2* 
e))))/e + (4*c^4*d^4*(66*a^2*e^4 - 67*c^2*d^4 + 15*a*c*d^2*e^2))/(693*e^3* 
(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d 
*e*x^2)^(1/2))/(d + e*x)^2 - (((286*c^5*d^6 - 706*a*c^4*d^4*e^2)/(1155*e^3 
*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) + (4*c^5*d^6)/(11*e^3*(a*e^2 - c*d 
^2)*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)) 
/(d + e*x)^2 - (((d*((d*((d*((64*c^9*d^10)/(10395*e*(a*e^2 - c*d^2)^6) - ( 
256*c^8*d^8*(6*a*e^2 - 5*c*d^2))/(10395*e*(a*e^2 - c*d^2)^6)))/e + (64*c^7 
*d^7*(241*a^2*e^4 + 175*c^2*d^4 - 410*a*c*d^2*e^2))/(10395*e^2*(a*e^2 - c* 
d^2)^6)))/e - (128*c^6*d^6*(662*a^3*e^6 - 455*c^3*d^6 + 1540*a*c^2*d^4*e^2 
 - 1745*a^2*c*d^2*e^4))/(10395*e^3*(a*e^2 - c*d^2)^6)))/e + (64*a*c^5*d^5* 
(1106*a^3*e^6 - 910*c^3*d^6 + 2905*a*c^2*d^4*e^2 - 3100*a^2*c*d^2*e^4))/(1 
0395*e^2*(a*e^2 - c*d^2)^6))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) 
)/(d + e*x) - (((d*((d*((d*((4*c^5*d^6)/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c 
*d^2*e)) - (8*c^4*d^4*(5*a*e^2 - 3*c*d^2))/(11*(a*e^2 - c*d^2)*(9*a*e^3 - 
9*c*d^2*e))))/e + (24*c^3*d^3*(5*a^2*e^4 + c^2*d^4 - 5*a*c*d^2*e^2))/(11*e 
*(a*e^2 - c*d^2)*(9*a*e^3 - 9*c*d^2*e))))/e + (24*c^5*d^8 - 120*a*c^4*d^6* 
e^2 + 240*a^2*c^3*d^4*e^4 - 160*a^3*c^2*d^2*e^6)/(11*e^2*(a*e^2 - c*d^2)*( 
9*a*e^3 - 9*c*d^2*e))))/e + (4*a*c*d*(19*a^3*e^6 - 6*c^3*d^6 + 24*a*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 717, normalized size of antiderivative = 6.46 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{5} e^{11}}{11}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{4} c \,d^{2} e^{9}}{9}-\frac {68 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{4} c d \,e^{10} x}{99}+\frac {8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c^{2} d^{3} e^{8} x}{9}-\frac {92 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{3} c^{2} d^{2} e^{9} x^{2}}{99}+\frac {4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{3} d^{4} e^{7} x^{2}}{3}-\frac {16 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a^{2} c^{3} d^{3} e^{8} x^{3}}{33}+\frac {8 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{4} d^{5} e^{6} x^{3}}{9}-\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a \,c^{4} d^{4} e^{7} x^{4}}{99}+\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{5} d^{6} e^{5} x^{4}}{9}+\frac {4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{5} d^{5} e^{6} x^{5}}{99}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{11}}{99}-\frac {8 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{10} e x}{33}-\frac {20 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{9} e^{2} x^{2}}{33}-\frac {80 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{8} e^{3} x^{3}}{99}-\frac {20 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{7} e^{4} x^{4}}{33}-\frac {8 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{6} e^{5} x^{5}}{33}-\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{5} d^{5} e^{6} x^{6}}{99}}{e^{5} \left (a^{2} e^{10} x^{6}-2 a c \,d^{2} e^{8} x^{6}+c^{2} d^{4} e^{6} x^{6}+6 a^{2} d \,e^{9} x^{5}-12 a c \,d^{3} e^{7} x^{5}+6 c^{2} d^{5} e^{5} x^{5}+15 a^{2} d^{2} e^{8} x^{4}-30 a c \,d^{4} e^{6} x^{4}+15 c^{2} d^{6} e^{4} x^{4}+20 a^{2} d^{3} e^{7} x^{3}-40 a c \,d^{5} e^{5} x^{3}+20 c^{2} d^{7} e^{3} x^{3}+15 a^{2} d^{4} e^{6} x^{2}-30 a c \,d^{6} e^{4} x^{2}+15 c^{2} d^{8} e^{2} x^{2}+6 a^{2} d^{5} e^{5} x -12 a c \,d^{7} e^{3} x +6 c^{2} d^{9} e x +a^{2} d^{6} e^{4}-2 a c \,d^{8} e^{2}+c^{2} d^{10}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^10,x)
 

Output:

(2*( - 9*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*e**11 + 11*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**4*c*d**2*e**9 - 34*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4* 
c*d*e**10*x + 44*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**3*e**8*x - 4 
6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**2*d**2*e**9*x**2 + 66*sqrt(d + e 
*x)*sqrt(a*e + c*d*x)*a**2*c**3*d**4*e**7*x**2 - 24*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a**2*c**3*d**3*e**8*x**3 + 44*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
c**4*d**5*e**6*x**3 - sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e**7*x** 
4 + 11*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**5*d**6*e**5*x**4 + 2*sqrt(d + e* 
x)*sqrt(a*e + c*d*x)*c**5*d**5*e**6*x**5 - 2*sqrt(e)*sqrt(d)*sqrt(c)*c**5* 
d**11 - 12*sqrt(e)*sqrt(d)*sqrt(c)*c**5*d**10*e*x - 30*sqrt(e)*sqrt(d)*sqr 
t(c)*c**5*d**9*e**2*x**2 - 40*sqrt(e)*sqrt(d)*sqrt(c)*c**5*d**8*e**3*x**3 
- 30*sqrt(e)*sqrt(d)*sqrt(c)*c**5*d**7*e**4*x**4 - 12*sqrt(e)*sqrt(d)*sqrt 
(c)*c**5*d**6*e**5*x**5 - 2*sqrt(e)*sqrt(d)*sqrt(c)*c**5*d**5*e**6*x**6))/ 
(99*e**5*(a**2*d**6*e**4 + 6*a**2*d**5*e**5*x + 15*a**2*d**4*e**6*x**2 + 2 
0*a**2*d**3*e**7*x**3 + 15*a**2*d**2*e**8*x**4 + 6*a**2*d*e**9*x**5 + a**2 
*e**10*x**6 - 2*a*c*d**8*e**2 - 12*a*c*d**7*e**3*x - 30*a*c*d**6*e**4*x**2 
 - 40*a*c*d**5*e**5*x**3 - 30*a*c*d**4*e**6*x**4 - 12*a*c*d**3*e**7*x**5 - 
 2*a*c*d**2*e**8*x**6 + c**2*d**10 + 6*c**2*d**9*e*x + 15*c**2*d**8*e**2*x 
**2 + 20*c**2*d**7*e**3*x**3 + 15*c**2*d**6*e**4*x**4 + 6*c**2*d**5*e**5*x 
**5 + c**2*d**4*e**6*x**6))