Integrand size = 27, antiderivative size = 111 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=-\frac {\left (c d^2-a e^2\right )^3 (d+e x)^4}{4 e^4}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^5}{5 e^4}-\frac {c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^6}{2 e^4}+\frac {c^3 d^3 (d+e x)^7}{7 e^4} \] Output:
-1/4*(-a*e^2+c*d^2)^3*(e*x+d)^4/e^4+3/5*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^5/e^4 -1/2*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^6/e^4+1/7*c^3*d^3*(e*x+d)^7/e^4
Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.50 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{140} x \left (35 a^3 e^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+21 a^2 c d e^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )+7 a c^2 d^2 e x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+c^3 d^3 x^3 \left (35 d^3+84 d^2 e x+70 d e^2 x^2+20 e^3 x^3\right )\right ) \] Input:
Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(x*(35*a^3*e^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 21*a^2*c*d*e^ 2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3) + 7*a*c^2*d^2*e*x^2*( 20*d^3 + 45*d^2*e*x + 36*d*e^2*x^2 + 10*e^3*x^3) + c^3*d^3*x^3*(35*d^3 + 8 4*d^2*e*x + 70*d*e^2*x^2 + 20*e^3*x^3)))/140
Time = 0.59 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1084, 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.
| \(\displaystyle \int \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 1084 |
| \(\displaystyle \frac {\int \left (\left (c d^2+c e x d\right )^6-3 c^5 d^5 \left (c d^2-a e^2\right ) (d+e x)^5+3 c^4 d^4 \left (c d^2-a e^2\right )^2 (d+e x)^4-c^3 d^3 \left (c d^2-a e^2\right )^3 (d+e x)^3\right )dx}{c^3 d^3 e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {c^5 d^5 (d+e x)^6 \left (c d^2-a e^2\right )}{2 e}+\frac {3 c^4 d^4 (d+e x)^5 \left (c d^2-a e^2\right )^2}{5 e}-\frac {c^3 d^3 (d+e x)^4 \left (c d^2-a e^2\right )^3}{4 e}+\frac {c^6 d^6 (d+e x)^7}{7 e}}{c^3 d^3 e^3}\) |
Input:
Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
Output:
(-1/4*(c^3*d^3*(c*d^2 - a*e^2)^3*(d + e*x)^4)/e + (3*c^4*d^4*(c*d^2 - a*e^ 2)^2*(d + e*x)^5)/(5*e) - (c^5*d^5*(c*d^2 - a*e^2)*(d + e*x)^6)/(2*e) + (c ^6*d^6*(d + e*x)^7)/(7*e))/(c^3*d^3*e^3)
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[(b/2 - q/2 + c*x)^p*(b/2 + q /2 + c*x)^p, x], x], x] /; !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c}, x] && IntegerQ[p] && NiceSqrtQ[b^2 - 4*a*c]
Time = 1.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.78
| method | result | size |
| norman | \(\frac {e^{3} d^{3} c^{3} x^{7}}{7}+\left (\frac {1}{2} a \,c^{2} d^{2} e^{4}+\frac {1}{2} c^{3} d^{4} e^{2}\right ) x^{6}+\left (\frac {3}{5} d \,e^{5} a^{2} c +\frac {9}{5} a \,c^{2} d^{3} e^{3}+\frac {3}{5} c^{3} d^{5} e \right ) x^{5}+\left (\frac {1}{4} e^{6} a^{3}+\frac {9}{4} d^{2} e^{4} a^{2} c +\frac {9}{4} d^{4} e^{2} a \,c^{2}+\frac {1}{4} d^{6} c^{3}\right ) x^{4}+\left (a^{3} d \,e^{5}+3 d^{3} e^{3} c \,a^{2}+d^{5} a e \,c^{2}\right ) x^{3}+\left (\frac {3}{2} d^{2} e^{4} a^{3}+\frac {3}{2} d^{4} a^{2} e^{2} c \right ) x^{2}+d^{3} e^{3} a^{3} x\) | \(198\) |
| risch | \(\frac {1}{7} e^{3} d^{3} c^{3} x^{7}+\frac {1}{2} a \,c^{2} d^{2} e^{4} x^{6}+\frac {1}{2} c^{3} d^{4} e^{2} x^{6}+\frac {3}{5} x^{5} d \,e^{5} a^{2} c +\frac {9}{5} x^{5} a \,c^{2} d^{3} e^{3}+\frac {3}{5} x^{5} c^{3} d^{5} e +\frac {1}{4} x^{4} e^{6} a^{3}+\frac {9}{4} x^{4} d^{2} e^{4} a^{2} c +\frac {9}{4} x^{4} d^{4} e^{2} a \,c^{2}+\frac {1}{4} x^{4} d^{6} c^{3}+a^{3} d \,e^{5} x^{3}+3 a^{2} c \,d^{3} e^{3} x^{3}+a \,c^{2} d^{5} e \,x^{3}+\frac {3}{2} a^{3} d^{2} e^{4} x^{2}+\frac {3}{2} a^{2} c \,d^{4} e^{2} x^{2}+d^{3} e^{3} a^{3} x\) | \(215\) |
| parallelrisch | \(\frac {1}{7} e^{3} d^{3} c^{3} x^{7}+\frac {1}{2} a \,c^{2} d^{2} e^{4} x^{6}+\frac {1}{2} c^{3} d^{4} e^{2} x^{6}+\frac {3}{5} x^{5} d \,e^{5} a^{2} c +\frac {9}{5} x^{5} a \,c^{2} d^{3} e^{3}+\frac {3}{5} x^{5} c^{3} d^{5} e +\frac {1}{4} x^{4} e^{6} a^{3}+\frac {9}{4} x^{4} d^{2} e^{4} a^{2} c +\frac {9}{4} x^{4} d^{4} e^{2} a \,c^{2}+\frac {1}{4} x^{4} d^{6} c^{3}+a^{3} d \,e^{5} x^{3}+3 a^{2} c \,d^{3} e^{3} x^{3}+a \,c^{2} d^{5} e \,x^{3}+\frac {3}{2} a^{3} d^{2} e^{4} x^{2}+\frac {3}{2} a^{2} c \,d^{4} e^{2} x^{2}+d^{3} e^{3} a^{3} x\) | \(215\) |
| gosper | \(\frac {x \left (20 e^{3} d^{3} c^{3} x^{6}+70 x^{5} a \,c^{2} d^{2} e^{4}+70 x^{5} c^{3} d^{4} e^{2}+84 x^{4} d \,e^{5} a^{2} c +252 x^{4} a \,c^{2} d^{3} e^{3}+84 x^{4} c^{3} d^{5} e +35 x^{3} e^{6} a^{3}+315 x^{3} d^{2} e^{4} a^{2} c +315 x^{3} d^{4} e^{2} a \,c^{2}+35 x^{3} d^{6} c^{3}+140 a^{3} d \,e^{5} x^{2}+420 a^{2} c \,d^{3} e^{3} x^{2}+140 a \,c^{2} d^{5} e \,x^{2}+210 x \,d^{2} e^{4} a^{3}+210 x \,d^{4} a^{2} e^{2} c +140 d^{3} e^{3} a^{3}\right )}{140}\) | \(216\) |
| orering | \(\frac {x \left (20 e^{3} d^{3} c^{3} x^{6}+70 x^{5} a \,c^{2} d^{2} e^{4}+70 x^{5} c^{3} d^{4} e^{2}+84 x^{4} d \,e^{5} a^{2} c +252 x^{4} a \,c^{2} d^{3} e^{3}+84 x^{4} c^{3} d^{5} e +35 x^{3} e^{6} a^{3}+315 x^{3} d^{2} e^{4} a^{2} c +315 x^{3} d^{4} e^{2} a \,c^{2}+35 x^{3} d^{6} c^{3}+140 a^{3} d \,e^{5} x^{2}+420 a^{2} c \,d^{3} e^{3} x^{2}+140 a \,c^{2} d^{5} e \,x^{2}+210 x \,d^{2} e^{4} a^{3}+210 x \,d^{4} a^{2} e^{2} c +140 d^{3} e^{3} a^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{3}}{140 \left (e x +d \right )^{3} \left (c d x +a e \right )^{3}}\) | \(260\) |
| default | \(\frac {e^{3} d^{3} c^{3} x^{7}}{7}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) e^{2} c^{2} d^{2} x^{6}}{2}+\frac {\left (a \,c^{2} d^{3} e^{3}+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} d e c +d e c \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) x^{5}}{5}+\frac {\left (4 a \,d^{2} e^{2} c \left (a \,e^{2}+c \,d^{2}\right )+\left (a \,e^{2}+c \,d^{2}\right ) \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )\right ) x^{4}}{4}+\frac {\left (a d e \left (2 a c \,d^{2} e^{2}+\left (a \,e^{2}+c \,d^{2}\right )^{2}\right )+2 \left (a \,e^{2}+c \,d^{2}\right )^{2} a d e +d^{3} e^{3} c \,a^{2}\right ) x^{3}}{3}+\frac {3 a^{2} d^{2} e^{2} \left (a \,e^{2}+c \,d^{2}\right ) x^{2}}{2}+d^{3} e^{3} a^{3} x\) | \(266\) |
Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERBOSE)
Output:
1/7*e^3*d^3*c^3*x^7+(1/2*a*c^2*d^2*e^4+1/2*c^3*d^4*e^2)*x^6+(3/5*d*e^5*a^2 *c+9/5*a*c^2*d^3*e^3+3/5*c^3*d^5*e)*x^5+(1/4*e^6*a^3+9/4*d^2*e^4*a^2*c+9/4 *d^4*e^2*a*c^2+1/4*d^6*c^3)*x^4+(a^3*d*e^5+3*a^2*c*d^3*e^3+a*c^2*d^5*e)*x^ 3+(3/2*d^2*e^4*a^3+3/2*d^4*a^2*e^2*c)*x^2+d^3*e^3*a^3*x
Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.74 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} d^{3} e^{3} x^{7} + a^{3} d^{3} e^{3} x + \frac {1}{2} \, {\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{6} + \frac {3}{5} \, {\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{5} + \frac {1}{4} \, {\left (c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{4} + {\left (a c^{2} d^{5} e + 3 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{3} + \frac {3}{2} \, {\left (a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\right )} x^{2} \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")
Output:
1/7*c^3*d^3*e^3*x^7 + a^3*d^3*e^3*x + 1/2*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^ 6 + 3/5*(c^3*d^5*e + 3*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x^5 + 1/4*(c^3*d^6 + 9 *a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + a^3*e^6)*x^4 + (a*c^2*d^5*e + 3*a^2*c*d ^3*e^3 + a^3*d*e^5)*x^3 + 3/2*(a^2*c*d^4*e^2 + a^3*d^2*e^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (99) = 198\).
Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.96 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=a^{3} d^{3} e^{3} x + \frac {c^{3} d^{3} e^{3} x^{7}}{7} + x^{6} \left (\frac {a c^{2} d^{2} e^{4}}{2} + \frac {c^{3} d^{4} e^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c d e^{5}}{5} + \frac {9 a c^{2} d^{3} e^{3}}{5} + \frac {3 c^{3} d^{5} e}{5}\right ) + x^{4} \left (\frac {a^{3} e^{6}}{4} + \frac {9 a^{2} c d^{2} e^{4}}{4} + \frac {9 a c^{2} d^{4} e^{2}}{4} + \frac {c^{3} d^{6}}{4}\right ) + x^{3} \left (a^{3} d e^{5} + 3 a^{2} c d^{3} e^{3} + a c^{2} d^{5} e\right ) + x^{2} \cdot \left (\frac {3 a^{3} d^{2} e^{4}}{2} + \frac {3 a^{2} c d^{4} e^{2}}{2}\right ) \] Input:
integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
Output:
a**3*d**3*e**3*x + c**3*d**3*e**3*x**7/7 + x**6*(a*c**2*d**2*e**4/2 + c**3 *d**4*e**2/2) + x**5*(3*a**2*c*d*e**5/5 + 9*a*c**2*d**3*e**3/5 + 3*c**3*d* *5*e/5) + x**4*(a**3*e**6/4 + 9*a**2*c*d**2*e**4/4 + 9*a*c**2*d**4*e**2/4 + c**3*d**6/4) + x**3*(a**3*d*e**5 + 3*a**2*c*d**3*e**3 + a*c**2*d**5*e) + x**2*(3*a**3*d**2*e**4/2 + 3*a**2*c*d**4*e**2/2)
Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.65 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} d^{3} e^{3} x^{7} + \frac {1}{2} \, {\left (c d^{2} + a e^{2}\right )} c^{2} d^{2} e^{2} x^{6} + a^{3} d^{3} e^{3} x + \frac {3}{5} \, {\left (c d^{2} + a e^{2}\right )}^{2} c d e x^{5} + \frac {1}{2} \, {\left (2 \, c d e x^{3} + 3 \, {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} a^{2} d^{2} e^{2} + \frac {1}{4} \, {\left (c d^{2} + a e^{2}\right )}^{3} x^{4} + \frac {1}{10} \, {\left (6 \, c^{2} d^{2} e^{2} x^{5} + 15 \, {\left (c d^{2} + a e^{2}\right )} c d e x^{4} + 10 \, {\left (c d^{2} + a e^{2}\right )}^{2} x^{3}\right )} a d e \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")
Output:
1/7*c^3*d^3*e^3*x^7 + 1/2*(c*d^2 + a*e^2)*c^2*d^2*e^2*x^6 + a^3*d^3*e^3*x + 3/5*(c*d^2 + a*e^2)^2*c*d*e*x^5 + 1/2*(2*c*d*e*x^3 + 3*(c*d^2 + a*e^2)*x ^2)*a^2*d^2*e^2 + 1/4*(c*d^2 + a*e^2)^3*x^4 + 1/10*(6*c^2*d^2*e^2*x^5 + 15 *(c*d^2 + a*e^2)*c*d*e*x^4 + 10*(c*d^2 + a*e^2)^2*x^3)*a*d*e
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (103) = 206\).
Time = 0.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.93 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {1}{7} \, c^{3} d^{3} e^{3} x^{7} + \frac {1}{2} \, c^{3} d^{4} e^{2} x^{6} + \frac {1}{2} \, a c^{2} d^{2} e^{4} x^{6} + \frac {3}{5} \, c^{3} d^{5} e x^{5} + \frac {9}{5} \, a c^{2} d^{3} e^{3} x^{5} + \frac {3}{5} \, a^{2} c d e^{5} x^{5} + \frac {1}{4} \, c^{3} d^{6} x^{4} + \frac {9}{4} \, a c^{2} d^{4} e^{2} x^{4} + \frac {9}{4} \, a^{2} c d^{2} e^{4} x^{4} + \frac {1}{4} \, a^{3} e^{6} x^{4} + a c^{2} d^{5} e x^{3} + 3 \, a^{2} c d^{3} e^{3} x^{3} + a^{3} d e^{5} x^{3} + \frac {3}{2} \, a^{2} c d^{4} e^{2} x^{2} + \frac {3}{2} \, a^{3} d^{2} e^{4} x^{2} + a^{3} d^{3} e^{3} x \] Input:
integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")
Output:
1/7*c^3*d^3*e^3*x^7 + 1/2*c^3*d^4*e^2*x^6 + 1/2*a*c^2*d^2*e^4*x^6 + 3/5*c^ 3*d^5*e*x^5 + 9/5*a*c^2*d^3*e^3*x^5 + 3/5*a^2*c*d*e^5*x^5 + 1/4*c^3*d^6*x^ 4 + 9/4*a*c^2*d^4*e^2*x^4 + 9/4*a^2*c*d^2*e^4*x^4 + 1/4*a^3*e^6*x^4 + a*c^ 2*d^5*e*x^3 + 3*a^2*c*d^3*e^3*x^3 + a^3*d*e^5*x^3 + 3/2*a^2*c*d^4*e^2*x^2 + 3/2*a^3*d^2*e^4*x^2 + a^3*d^3*e^3*x
Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.68 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=x^4\,\left (\frac {a^3\,e^6}{4}+\frac {9\,a^2\,c\,d^2\,e^4}{4}+\frac {9\,a\,c^2\,d^4\,e^2}{4}+\frac {c^3\,d^6}{4}\right )+a^3\,d^3\,e^3\,x+\frac {c^3\,d^3\,e^3\,x^7}{7}+a\,d\,e\,x^3\,\left (a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4\right )+\frac {3\,c\,d\,e\,x^5\,\left (a^2\,e^4+3\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{5}+\frac {3\,a^2\,d^2\,e^2\,x^2\,\left (c\,d^2+a\,e^2\right )}{2}+\frac {c^2\,d^2\,e^2\,x^6\,\left (c\,d^2+a\,e^2\right )}{2} \] Input:
int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)
Output:
x^4*((a^3*e^6)/4 + (c^3*d^6)/4 + (9*a*c^2*d^4*e^2)/4 + (9*a^2*c*d^2*e^4)/4 ) + a^3*d^3*e^3*x + (c^3*d^3*e^3*x^7)/7 + a*d*e*x^3*(a^2*e^4 + c^2*d^4 + 3 *a*c*d^2*e^2) + (3*c*d*e*x^5*(a^2*e^4 + c^2*d^4 + 3*a*c*d^2*e^2))/5 + (3*a ^2*d^2*e^2*x^2*(a*e^2 + c*d^2))/2 + (c^2*d^2*e^2*x^6*(a*e^2 + c*d^2))/2
Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.94 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3 \, dx=\frac {x \left (20 c^{3} d^{3} e^{3} x^{6}+70 a \,c^{2} d^{2} e^{4} x^{5}+70 c^{3} d^{4} e^{2} x^{5}+84 a^{2} c d \,e^{5} x^{4}+252 a \,c^{2} d^{3} e^{3} x^{4}+84 c^{3} d^{5} e \,x^{4}+35 a^{3} e^{6} x^{3}+315 a^{2} c \,d^{2} e^{4} x^{3}+315 a \,c^{2} d^{4} e^{2} x^{3}+35 c^{3} d^{6} x^{3}+140 a^{3} d \,e^{5} x^{2}+420 a^{2} c \,d^{3} e^{3} x^{2}+140 a \,c^{2} d^{5} e \,x^{2}+210 a^{3} d^{2} e^{4} x +210 a^{2} c \,d^{4} e^{2} x +140 a^{3} d^{3} e^{3}\right )}{140} \] Input:
int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
Output:
(x*(140*a**3*d**3*e**3 + 210*a**3*d**2*e**4*x + 140*a**3*d*e**5*x**2 + 35* a**3*e**6*x**3 + 210*a**2*c*d**4*e**2*x + 420*a**2*c*d**3*e**3*x**2 + 315* a**2*c*d**2*e**4*x**3 + 84*a**2*c*d*e**5*x**4 + 140*a*c**2*d**5*e*x**2 + 3 15*a*c**2*d**4*e**2*x**3 + 252*a*c**2*d**3*e**3*x**4 + 70*a*c**2*d**2*e**4 *x**5 + 35*c**3*d**6*x**3 + 84*c**3*d**5*e*x**4 + 70*c**3*d**4*e**2*x**5 + 20*c**3*d**3*e**3*x**6))/140