Integrand size = 36, antiderivative size = 258 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac {(120 d+17 e) (d+e x)^9}{9 e^7}+\frac {2 (d+e x)^{10}}{e^7} \]
1/4*(5*d^2-2*d*e+3*e^2)*(4*d^4+5*d^3*e+3*d^2*e^2-d*e^3+2*e^4)*(e*x+d)^4/e^ 7-1/5*(120*d^5+85*d^4*e+68*d^3*e^2+12*d^2*e^3+42*d*e^4-7*e^5)*(e*x+d)^5/e^ 7+1/6*(300*d^4+170*d^3*e+102*d^2*e^2+12*d*e^3+21*e^4)*(e*x+d)^6/e^7-2/7*(2 00*d^3+85*d^2*e+34*d*e^2+2*e^3)*(e*x+d)^7/e^7+1/8*(300*d^2+85*d*e+17*e^2)* (e*x+d)^8/e^7-1/9*(120*d+17*e)*(e*x+d)^9/e^7+2*(e*x+d)^10/e^7
Time = 0.03 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.82 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=6 d^3 x+\frac {1}{2} d^2 (7 d+18 e) x^2+d \left (7 d^2+7 d e+6 e^2\right ) x^3+\frac {1}{4} \left (-4 d^3+63 d^2 e+21 d e^2+6 e^3\right ) x^4+\frac {1}{5} \left (17 d^3-12 d^2 e+63 d e^2+7 e^3\right ) x^5+\frac {1}{6} \left (-17 d^3+51 d^2 e-12 d e^2+21 e^3\right ) x^6+\frac {1}{7} \left (20 d^3-51 d^2 e+51 d e^2-4 e^3\right ) x^7+\frac {1}{8} e \left (60 d^2-51 d e+17 e^2\right ) x^8+\frac {1}{9} (60 d-17 e) e^2 x^9+2 e^3 x^{10} \]
6*d^3*x + (d^2*(7*d + 18*e)*x^2)/2 + d*(7*d^2 + 7*d*e + 6*e^2)*x^3 + ((-4* d^3 + 63*d^2*e + 21*d*e^2 + 6*e^3)*x^4)/4 + ((17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e^3)*x^5)/5 + ((-17*d^3 + 51*d^2*e - 12*d*e^2 + 21*e^3)*x^6)/6 + ((20 *d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7)/7 + (e*(60*d^2 - 51*d*e + 17*e^2) *x^8)/8 + ((60*d - 17*e)*e^2*x^9)/9 + 2*e^3*x^10
Time = 0.57 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2159, 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.
| \(\displaystyle \int \left (5 x^2+2 x+3\right ) \left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^7}{e^6}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^6}{e^6}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^5}{e^6}+\frac {\left (-120 d^5-85 d^4 e-68 d^3 e^2-12 d^2 e^3-42 d e^4+7 e^5\right ) (d+e x)^4}{e^6}+\frac {\left (20 d^6+17 d^5 e+17 d^4 e^2+4 d^3 e^3+21 d^2 e^4-7 d e^5+6 e^6\right ) (d+e x)^3}{e^6}+\frac {20 (d+e x)^9}{e^6}+\frac {(-120 d-17 e) (d+e x)^8}{e^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (300 d^2+85 d e+17 e^2\right ) (d+e x)^8}{8 e^7}-\frac {2 \left (200 d^3+85 d^2 e+34 d e^2+2 e^3\right ) (d+e x)^7}{7 e^7}+\frac {\left (300 d^4+170 d^3 e+102 d^2 e^2+12 d e^3+21 e^4\right ) (d+e x)^6}{6 e^7}+\frac {\left (5 d^2-2 d e+3 e^2\right ) \left (4 d^4+5 d^3 e+3 d^2 e^2-d e^3+2 e^4\right ) (d+e x)^4}{4 e^7}-\frac {\left (120 d^5+85 d^4 e+68 d^3 e^2+12 d^2 e^3+42 d e^4-7 e^5\right ) (d+e x)^5}{5 e^7}+\frac {2 (d+e x)^{10}}{e^7}-\frac {(120 d+17 e) (d+e x)^9}{9 e^7}\) |
((5*d^2 - 2*d*e + 3*e^2)*(4*d^4 + 5*d^3*e + 3*d^2*e^2 - d*e^3 + 2*e^4)*(d + e*x)^4)/(4*e^7) - ((120*d^5 + 85*d^4*e + 68*d^3*e^2 + 12*d^2*e^3 + 42*d* e^4 - 7*e^5)*(d + e*x)^5)/(5*e^7) + ((300*d^4 + 170*d^3*e + 102*d^2*e^2 + 12*d*e^3 + 21*e^4)*(d + e*x)^6)/(6*e^7) - (2*(200*d^3 + 85*d^2*e + 34*d*e^ 2 + 2*e^3)*(d + e*x)^7)/(7*e^7) + ((300*d^2 + 85*d*e + 17*e^2)*(d + e*x)^8 )/(8*e^7) - ((120*d + 17*e)*(d + e*x)^9)/(9*e^7) + (2*(d + e*x)^10)/e^7
3.3.89.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.48 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.78
| method | result | size |
| norman | \(2 e^{3} x^{10}+\left (\frac {20}{3} d \,e^{2}-\frac {17}{9} e^{3}\right ) x^{9}+\left (\frac {15}{2} d^{2} e -\frac {51}{8} d \,e^{2}+\frac {17}{8} e^{3}\right ) x^{8}+\left (\frac {20}{7} d^{3}-\frac {51}{7} d^{2} e +\frac {51}{7} d \,e^{2}-\frac {4}{7} e^{3}\right ) x^{7}+\left (-\frac {17}{6} d^{3}+\frac {17}{2} d^{2} e -2 d \,e^{2}+\frac {7}{2} e^{3}\right ) x^{6}+\left (\frac {17}{5} d^{3}-\frac {12}{5} d^{2} e +\frac {63}{5} d \,e^{2}+\frac {7}{5} e^{3}\right ) x^{5}+\left (-d^{3}+\frac {63}{4} d^{2} e +\frac {21}{4} d \,e^{2}+\frac {3}{2} e^{3}\right ) x^{4}+\left (7 d^{3}+7 d^{2} e +6 d \,e^{2}\right ) x^{3}+\left (\frac {7}{2} d^{3}+9 d^{2} e \right ) x^{2}+6 x \,d^{3}\) | \(200\) |
| default | \(2 e^{3} x^{10}+\frac {\left (60 d \,e^{2}-17 e^{3}\right ) x^{9}}{9}+\frac {\left (60 d^{2} e -51 d \,e^{2}+17 e^{3}\right ) x^{8}}{8}+\frac {\left (20 d^{3}-51 d^{2} e +51 d \,e^{2}-4 e^{3}\right ) x^{7}}{7}+\frac {\left (-17 d^{3}+51 d^{2} e -12 d \,e^{2}+21 e^{3}\right ) x^{6}}{6}+\frac {\left (17 d^{3}-12 d^{2} e +63 d \,e^{2}+7 e^{3}\right ) x^{5}}{5}+\frac {\left (-4 d^{3}+63 d^{2} e +21 d \,e^{2}+6 e^{3}\right ) x^{4}}{4}+\frac {\left (21 d^{3}+21 d^{2} e +18 d \,e^{2}\right ) x^{3}}{3}+\frac {\left (7 d^{3}+18 d^{2} e \right ) x^{2}}{2}+6 x \,d^{3}\) | \(208\) |
| gosper | \(7 d^{2} e \,x^{3}+\frac {17}{2} x^{6} d^{2} e -2 x^{6} d \,e^{2}-\frac {12}{5} x^{5} d^{2} e +\frac {63}{5} x^{5} d \,e^{2}+\frac {63}{4} x^{4} d^{2} e +\frac {21}{4} x^{4} d \,e^{2}-\frac {51}{7} x^{7} d^{2} e +\frac {51}{7} x^{7} d \,e^{2}+\frac {15}{2} x^{8} d^{2} e -\frac {51}{8} x^{8} d \,e^{2}+\frac {20}{3} x^{9} d \,e^{2}+\frac {3}{2} e^{3} x^{4}+\frac {17}{5} x^{5} d^{3}+\frac {7}{5} x^{5} e^{3}-x^{4} d^{3}+\frac {7}{2} x^{2} d^{3}+7 d^{3} x^{3}-\frac {17}{9} x^{9} e^{3}+\frac {17}{8} x^{8} e^{3}+\frac {20}{7} x^{7} d^{3}-\frac {4}{7} x^{7} e^{3}-\frac {17}{6} x^{6} d^{3}+2 e^{3} x^{10}+9 d^{2} e \,x^{2}+6 x \,d^{3}+6 d \,e^{2} x^{3}+\frac {7}{2} e^{3} x^{6}\) | \(238\) |
| risch | \(7 d^{2} e \,x^{3}+\frac {17}{2} x^{6} d^{2} e -2 x^{6} d \,e^{2}-\frac {12}{5} x^{5} d^{2} e +\frac {63}{5} x^{5} d \,e^{2}+\frac {63}{4} x^{4} d^{2} e +\frac {21}{4} x^{4} d \,e^{2}-\frac {51}{7} x^{7} d^{2} e +\frac {51}{7} x^{7} d \,e^{2}+\frac {15}{2} x^{8} d^{2} e -\frac {51}{8} x^{8} d \,e^{2}+\frac {20}{3} x^{9} d \,e^{2}+\frac {3}{2} e^{3} x^{4}+\frac {17}{5} x^{5} d^{3}+\frac {7}{5} x^{5} e^{3}-x^{4} d^{3}+\frac {7}{2} x^{2} d^{3}+7 d^{3} x^{3}-\frac {17}{9} x^{9} e^{3}+\frac {17}{8} x^{8} e^{3}+\frac {20}{7} x^{7} d^{3}-\frac {4}{7} x^{7} e^{3}-\frac {17}{6} x^{6} d^{3}+2 e^{3} x^{10}+9 d^{2} e \,x^{2}+6 x \,d^{3}+6 d \,e^{2} x^{3}+\frac {7}{2} e^{3} x^{6}\) | \(238\) |
| parallelrisch | \(7 d^{2} e \,x^{3}+\frac {17}{2} x^{6} d^{2} e -2 x^{6} d \,e^{2}-\frac {12}{5} x^{5} d^{2} e +\frac {63}{5} x^{5} d \,e^{2}+\frac {63}{4} x^{4} d^{2} e +\frac {21}{4} x^{4} d \,e^{2}-\frac {51}{7} x^{7} d^{2} e +\frac {51}{7} x^{7} d \,e^{2}+\frac {15}{2} x^{8} d^{2} e -\frac {51}{8} x^{8} d \,e^{2}+\frac {20}{3} x^{9} d \,e^{2}+\frac {3}{2} e^{3} x^{4}+\frac {17}{5} x^{5} d^{3}+\frac {7}{5} x^{5} e^{3}-x^{4} d^{3}+\frac {7}{2} x^{2} d^{3}+7 d^{3} x^{3}-\frac {17}{9} x^{9} e^{3}+\frac {17}{8} x^{8} e^{3}+\frac {20}{7} x^{7} d^{3}-\frac {4}{7} x^{7} e^{3}-\frac {17}{6} x^{6} d^{3}+2 e^{3} x^{10}+9 d^{2} e \,x^{2}+6 x \,d^{3}+6 d \,e^{2} x^{3}+\frac {7}{2} e^{3} x^{6}\) | \(238\) |
2*e^3*x^10+(20/3*d*e^2-17/9*e^3)*x^9+(15/2*d^2*e-51/8*d*e^2+17/8*e^3)*x^8+ (20/7*d^3-51/7*d^2*e+51/7*d*e^2-4/7*e^3)*x^7+(-17/6*d^3+17/2*d^2*e-2*d*e^2 +7/2*e^3)*x^6+(17/5*d^3-12/5*d^2*e+63/5*d*e^2+7/5*e^3)*x^5+(-d^3+63/4*d^2* e+21/4*d*e^2+3/2*e^3)*x^4+(7*d^3+7*d^2*e+6*d*e^2)*x^3+(7/2*d^3+9*d^2*e)*x^ 2+6*x*d^3
Time = 0.24 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.80 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=2 \, e^{3} x^{10} + \frac {1}{9} \, {\left (60 \, d e^{2} - 17 \, e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (60 \, d^{2} e - 51 \, d e^{2} + 17 \, e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{3} - 51 \, d^{2} e + 51 \, d e^{2} - 4 \, e^{3}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{3} - 51 \, d^{2} e + 12 \, d e^{2} - 21 \, e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{3} - 12 \, d^{2} e + 63 \, d e^{2} + 7 \, e^{3}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{3} - 63 \, d^{2} e - 21 \, d e^{2} - 6 \, e^{3}\right )} x^{4} + 6 \, d^{3} x + {\left (7 \, d^{3} + 7 \, d^{2} e + 6 \, d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \]
2*e^3*x^10 + 1/9*(60*d*e^2 - 17*e^3)*x^9 + 1/8*(60*d^2*e - 51*d*e^2 + 17*e ^3)*x^8 + 1/7*(20*d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7 - 1/6*(17*d^3 - 5 1*d^2*e + 12*d*e^2 - 21*e^3)*x^6 + 1/5*(17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e ^3)*x^5 - 1/4*(4*d^3 - 63*d^2*e - 21*d*e^2 - 6*e^3)*x^4 + 6*d^3*x + (7*d^3 + 7*d^2*e + 6*d*e^2)*x^3 + 1/2*(7*d^3 + 18*d^2*e)*x^2
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.89 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=6 d^{3} x + 2 e^{3} x^{10} + x^{9} \cdot \left (\frac {20 d e^{2}}{3} - \frac {17 e^{3}}{9}\right ) + x^{8} \cdot \left (\frac {15 d^{2} e}{2} - \frac {51 d e^{2}}{8} + \frac {17 e^{3}}{8}\right ) + x^{7} \cdot \left (\frac {20 d^{3}}{7} - \frac {51 d^{2} e}{7} + \frac {51 d e^{2}}{7} - \frac {4 e^{3}}{7}\right ) + x^{6} \left (- \frac {17 d^{3}}{6} + \frac {17 d^{2} e}{2} - 2 d e^{2} + \frac {7 e^{3}}{2}\right ) + x^{5} \cdot \left (\frac {17 d^{3}}{5} - \frac {12 d^{2} e}{5} + \frac {63 d e^{2}}{5} + \frac {7 e^{3}}{5}\right ) + x^{4} \left (- d^{3} + \frac {63 d^{2} e}{4} + \frac {21 d e^{2}}{4} + \frac {3 e^{3}}{2}\right ) + x^{3} \cdot \left (7 d^{3} + 7 d^{2} e + 6 d e^{2}\right ) + x^{2} \cdot \left (\frac {7 d^{3}}{2} + 9 d^{2} e\right ) \]
6*d**3*x + 2*e**3*x**10 + x**9*(20*d*e**2/3 - 17*e**3/9) + x**8*(15*d**2*e /2 - 51*d*e**2/8 + 17*e**3/8) + x**7*(20*d**3/7 - 51*d**2*e/7 + 51*d*e**2/ 7 - 4*e**3/7) + x**6*(-17*d**3/6 + 17*d**2*e/2 - 2*d*e**2 + 7*e**3/2) + x* *5*(17*d**3/5 - 12*d**2*e/5 + 63*d*e**2/5 + 7*e**3/5) + x**4*(-d**3 + 63*d **2*e/4 + 21*d*e**2/4 + 3*e**3/2) + x**3*(7*d**3 + 7*d**2*e + 6*d*e**2) + x**2*(7*d**3/2 + 9*d**2*e)
Time = 0.20 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.80 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=2 \, e^{3} x^{10} + \frac {1}{9} \, {\left (60 \, d e^{2} - 17 \, e^{3}\right )} x^{9} + \frac {1}{8} \, {\left (60 \, d^{2} e - 51 \, d e^{2} + 17 \, e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (20 \, d^{3} - 51 \, d^{2} e + 51 \, d e^{2} - 4 \, e^{3}\right )} x^{7} - \frac {1}{6} \, {\left (17 \, d^{3} - 51 \, d^{2} e + 12 \, d e^{2} - 21 \, e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (17 \, d^{3} - 12 \, d^{2} e + 63 \, d e^{2} + 7 \, e^{3}\right )} x^{5} - \frac {1}{4} \, {\left (4 \, d^{3} - 63 \, d^{2} e - 21 \, d e^{2} - 6 \, e^{3}\right )} x^{4} + 6 \, d^{3} x + {\left (7 \, d^{3} + 7 \, d^{2} e + 6 \, d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (7 \, d^{3} + 18 \, d^{2} e\right )} x^{2} \]
2*e^3*x^10 + 1/9*(60*d*e^2 - 17*e^3)*x^9 + 1/8*(60*d^2*e - 51*d*e^2 + 17*e ^3)*x^8 + 1/7*(20*d^3 - 51*d^2*e + 51*d*e^2 - 4*e^3)*x^7 - 1/6*(17*d^3 - 5 1*d^2*e + 12*d*e^2 - 21*e^3)*x^6 + 1/5*(17*d^3 - 12*d^2*e + 63*d*e^2 + 7*e ^3)*x^5 - 1/4*(4*d^3 - 63*d^2*e - 21*d*e^2 - 6*e^3)*x^4 + 6*d^3*x + (7*d^3 + 7*d^2*e + 6*d*e^2)*x^3 + 1/2*(7*d^3 + 18*d^2*e)*x^2
Time = 0.28 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.92 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=2 \, e^{3} x^{10} + \frac {20}{3} \, d e^{2} x^{9} - \frac {17}{9} \, e^{3} x^{9} + \frac {15}{2} \, d^{2} e x^{8} - \frac {51}{8} \, d e^{2} x^{8} + \frac {17}{8} \, e^{3} x^{8} + \frac {20}{7} \, d^{3} x^{7} - \frac {51}{7} \, d^{2} e x^{7} + \frac {51}{7} \, d e^{2} x^{7} - \frac {4}{7} \, e^{3} x^{7} - \frac {17}{6} \, d^{3} x^{6} + \frac {17}{2} \, d^{2} e x^{6} - 2 \, d e^{2} x^{6} + \frac {7}{2} \, e^{3} x^{6} + \frac {17}{5} \, d^{3} x^{5} - \frac {12}{5} \, d^{2} e x^{5} + \frac {63}{5} \, d e^{2} x^{5} + \frac {7}{5} \, e^{3} x^{5} - d^{3} x^{4} + \frac {63}{4} \, d^{2} e x^{4} + \frac {21}{4} \, d e^{2} x^{4} + \frac {3}{2} \, e^{3} x^{4} + 7 \, d^{3} x^{3} + 7 \, d^{2} e x^{3} + 6 \, d e^{2} x^{3} + \frac {7}{2} \, d^{3} x^{2} + 9 \, d^{2} e x^{2} + 6 \, d^{3} x \]
2*e^3*x^10 + 20/3*d*e^2*x^9 - 17/9*e^3*x^9 + 15/2*d^2*e*x^8 - 51/8*d*e^2*x ^8 + 17/8*e^3*x^8 + 20/7*d^3*x^7 - 51/7*d^2*e*x^7 + 51/7*d*e^2*x^7 - 4/7*e ^3*x^7 - 17/6*d^3*x^6 + 17/2*d^2*e*x^6 - 2*d*e^2*x^6 + 7/2*e^3*x^6 + 17/5* d^3*x^5 - 12/5*d^2*e*x^5 + 63/5*d*e^2*x^5 + 7/5*e^3*x^5 - d^3*x^4 + 63/4*d ^2*e*x^4 + 21/4*d*e^2*x^4 + 3/2*e^3*x^4 + 7*d^3*x^3 + 7*d^2*e*x^3 + 6*d*e^ 2*x^3 + 7/2*d^3*x^2 + 9*d^2*e*x^2 + 6*d^3*x
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.76 \[ \int (d+e x)^3 \left (3+2 x+5 x^2\right ) \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=6\,d^3\,x+x^8\,\left (\frac {15\,d^2\,e}{2}-\frac {51\,d\,e^2}{8}+\frac {17\,e^3}{8}\right )-x^6\,\left (\frac {17\,d^3}{6}-\frac {17\,d^2\,e}{2}+2\,d\,e^2-\frac {7\,e^3}{2}\right )+x^4\,\left (-d^3+\frac {63\,d^2\,e}{4}+\frac {21\,d\,e^2}{4}+\frac {3\,e^3}{2}\right )+x^5\,\left (\frac {17\,d^3}{5}-\frac {12\,d^2\,e}{5}+\frac {63\,d\,e^2}{5}+\frac {7\,e^3}{5}\right )+x^7\,\left (\frac {20\,d^3}{7}-\frac {51\,d^2\,e}{7}+\frac {51\,d\,e^2}{7}-\frac {4\,e^3}{7}\right )+2\,e^3\,x^{10}+d\,x^3\,\left (7\,d^2+7\,d\,e+6\,e^2\right )+\frac {d^2\,x^2\,\left (7\,d+18\,e\right )}{2}+\frac {e^2\,x^9\,\left (60\,d-17\,e\right )}{9} \]
6*d^3*x + x^8*((15*d^2*e)/2 - (51*d*e^2)/8 + (17*e^3)/8) - x^6*(2*d*e^2 - (17*d^2*e)/2 + (17*d^3)/6 - (7*e^3)/2) + x^4*((21*d*e^2)/4 + (63*d^2*e)/4 - d^3 + (3*e^3)/2) + x^5*((63*d*e^2)/5 - (12*d^2*e)/5 + (17*d^3)/5 + (7*e^ 3)/5) + x^7*((51*d*e^2)/7 - (51*d^2*e)/7 + (20*d^3)/7 - (4*e^3)/7) + 2*e^3 *x^10 + d*x^3*(7*d*e + 7*d^2 + 6*e^2) + (d^2*x^2*(7*d + 18*e))/2 + (e^2*x^ 9*(60*d - 17*e))/9