Integrand size = 26, antiderivative size = 171 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=8 A x+2 (9 A+2 B) x^2+\frac {2}{3} (51 A+18 B+4 C) x^3+\frac {3}{4} (61 A+34 B+12 C) x^4+\frac {3}{5} (80 A+61 B+34 C) x^5+\frac {1}{2} (73 A+80 B+61 C) x^6+\frac {1}{7} (142 A+219 B+240 C) x^7+\frac {1}{8} (57 A+142 B+219 C) x^8+\frac {1}{9} (12 A+57 B+142 C) x^9+\frac {1}{10} (A+12 B+57 C) x^{10}+\frac {1}{11} (B+12 C) x^{11}+\frac {C x^{12}}{12} \] Output:
8*A*x+2*(9*A+2*B)*x^2+2/3*(51*A+18*B+4*C)*x^3+3/4*(61*A+34*B+12*C)*x^4+3/5 *(80*A+61*B+34*C)*x^5+1/2*(73*A+80*B+61*C)*x^6+1/7*(142*A+219*B+240*C)*x^7 +1/8*(57*A+142*B+219*C)*x^8+1/9*(12*A+57*B+142*C)*x^9+1/10*(A+12*B+57*C)*x ^10+1/11*(B+12*C)*x^11+1/12*C*x^12
Time = 0.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=8 A x+2 (9 A+2 B) x^2+\frac {2}{3} (51 A+18 B+4 C) x^3+\frac {3}{4} (61 A+34 B+12 C) x^4+\frac {3}{5} (80 A+61 B+34 C) x^5+\frac {1}{2} (73 A+80 B+61 C) x^6+\frac {1}{7} (142 A+219 B+240 C) x^7+\frac {1}{8} (57 A+142 B+219 C) x^8+\frac {1}{9} (12 A+57 B+142 C) x^9+\frac {1}{10} (A+12 B+57 C) x^{10}+\frac {1}{11} (B+12 C) x^{11}+\frac {C x^{12}}{12} \] Input:
Integrate[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^3,x]
Output:
8*A*x + 2*(9*A + 2*B)*x^2 + (2*(51*A + 18*B + 4*C)*x^3)/3 + (3*(61*A + 34* B + 12*C)*x^4)/4 + (3*(80*A + 61*B + 34*C)*x^5)/5 + ((73*A + 80*B + 61*C)* x^6)/2 + ((142*A + 219*B + 240*C)*x^7)/7 + ((57*A + 142*B + 219*C)*x^8)/8 + ((12*A + 57*B + 142*C)*x^9)/9 + ((A + 12*B + 57*C)*x^10)/10 + ((B + 12*C )*x^11)/11 + (C*x^12)/12
Time = 0.40 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2188, 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.
| \(\displaystyle \int \left (x^3+4 x^2+3 x+2\right )^3 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \int \left (x^9 (A+12 B+57 C)+x^8 (12 A+57 B+142 C)+x^7 (57 A+142 B+219 C)+x^6 (142 A+219 B+240 C)+3 x^5 (73 A+80 B+61 C)+3 x^4 (80 A+61 B+34 C)+3 x^3 (61 A+34 B+12 C)+2 x^2 (51 A+18 B+4 C)+4 x (9 A+2 B)+8 A+x^{10} (B+12 C)+C x^{11}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{10} x^{10} (A+12 B+57 C)+\frac {1}{9} x^9 (12 A+57 B+142 C)+\frac {1}{8} x^8 (57 A+142 B+219 C)+\frac {1}{7} x^7 (142 A+219 B+240 C)+\frac {1}{2} x^6 (73 A+80 B+61 C)+\frac {3}{5} x^5 (80 A+61 B+34 C)+\frac {3}{4} x^4 (61 A+34 B+12 C)+\frac {2}{3} x^3 (51 A+18 B+4 C)+2 x^2 (9 A+2 B)+8 A x+\frac {1}{11} x^{11} (B+12 C)+\frac {C x^{12}}{12}\) |
Input:
Int[(A + B*x + C*x^2)*(2 + 3*x + 4*x^2 + x^3)^3,x]
Output:
8*A*x + 2*(9*A + 2*B)*x^2 + (2*(51*A + 18*B + 4*C)*x^3)/3 + (3*(61*A + 34* B + 12*C)*x^4)/4 + (3*(80*A + 61*B + 34*C)*x^5)/5 + ((73*A + 80*B + 61*C)* x^6)/2 + ((142*A + 219*B + 240*C)*x^7)/7 + ((57*A + 142*B + 219*C)*x^8)/8 + ((12*A + 57*B + 142*C)*x^9)/9 + ((A + 12*B + 57*C)*x^10)/10 + ((B + 12*C )*x^11)/11 + (C*x^12)/12
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85
| method | result | size |
| norman | \(\frac {C \,x^{12}}{12}+\left (\frac {B}{11}+\frac {12 C}{11}\right ) x^{11}+\left (\frac {A}{10}+\frac {6 B}{5}+\frac {57 C}{10}\right ) x^{10}+\left (\frac {4 A}{3}+\frac {19 B}{3}+\frac {142 C}{9}\right ) x^{9}+\left (\frac {57 A}{8}+\frac {71 B}{4}+\frac {219 C}{8}\right ) x^{8}+\left (\frac {142 A}{7}+\frac {219 B}{7}+\frac {240 C}{7}\right ) x^{7}+\left (\frac {73 A}{2}+40 B +\frac {61 C}{2}\right ) x^{6}+\left (48 A +\frac {183 B}{5}+\frac {102 C}{5}\right ) x^{5}+\left (\frac {183 A}{4}+\frac {51 B}{2}+9 C \right ) x^{4}+\left (34 A +12 B +\frac {8 C}{3}\right ) x^{3}+\left (18 A +4 B \right ) x^{2}+8 A x\) | \(146\) |
| default | \(\frac {C \,x^{12}}{12}+\frac {\left (B +12 C \right ) x^{11}}{11}+\frac {\left (A +12 B +57 C \right ) x^{10}}{10}+\frac {\left (12 A +57 B +142 C \right ) x^{9}}{9}+\frac {\left (57 A +142 B +219 C \right ) x^{8}}{8}+\frac {\left (142 A +219 B +240 C \right ) x^{7}}{7}+\frac {\left (219 A +240 B +183 C \right ) x^{6}}{6}+\frac {\left (240 A +183 B +102 C \right ) x^{5}}{5}+\frac {\left (183 A +102 B +36 C \right ) x^{4}}{4}+\frac {\left (102 A +36 B +8 C \right ) x^{3}}{3}+\frac {\left (36 A +8 B \right ) x^{2}}{2}+8 A x\) | \(152\) |
| orering | \(\frac {x \left (2310 C \,x^{11}+2520 B \,x^{10}+30240 x^{10} C +2772 A \,x^{9}+33264 x^{9} B +158004 C \,x^{9}+36960 x^{8} A +175560 B \,x^{8}+437360 x^{8} C +197505 x^{7} A +492030 x^{7} B +758835 x^{7} C +562320 x^{6} A +867240 x^{6} B +950400 C \,x^{6}+1011780 x^{5} A +1108800 B \,x^{5}+845460 x^{5} C +1330560 x^{4} A +1014552 x^{4} B +565488 C \,x^{4}+1268190 x^{3} A +706860 B \,x^{3}+249480 C \,x^{3}+942480 A \,x^{2}+332640 B \,x^{2}+73920 C \,x^{2}+498960 A x +110880 B x +221760 A \right )}{27720}\) | \(178\) |
| gosper | \(9 C \,x^{4}+\frac {8}{3} C \,x^{3}+4 B \,x^{2}+18 A \,x^{2}+\frac {183}{4} x^{4} A +48 x^{5} A +\frac {4}{3} A \,x^{9}+\frac {6}{5} B \,x^{10}+\frac {219}{8} x^{8} C +\frac {142}{7} x^{7} A +\frac {19}{3} x^{9} B +\frac {57}{8} x^{8} A +\frac {219}{7} x^{7} B +\frac {240}{7} x^{7} C +\frac {73}{2} x^{6} A +40 x^{6} B +\frac {1}{10} x^{10} A +\frac {57}{10} x^{10} C +8 A x +\frac {1}{11} B \,x^{11}+\frac {1}{12} C \,x^{12}+\frac {71}{4} B \,x^{8}+\frac {183}{5} B \,x^{5}+\frac {61}{2} C \,x^{6}+12 B \,x^{3}+\frac {12}{11} C \,x^{11}+\frac {142}{9} C \,x^{9}+34 x^{3} A +\frac {51}{2} x^{4} B +\frac {102}{5} x^{5} C\) | \(180\) |
| risch | \(9 C \,x^{4}+\frac {8}{3} C \,x^{3}+4 B \,x^{2}+18 A \,x^{2}+\frac {183}{4} x^{4} A +48 x^{5} A +\frac {4}{3} A \,x^{9}+\frac {6}{5} B \,x^{10}+\frac {219}{8} x^{8} C +\frac {142}{7} x^{7} A +\frac {19}{3} x^{9} B +\frac {57}{8} x^{8} A +\frac {219}{7} x^{7} B +\frac {240}{7} x^{7} C +\frac {73}{2} x^{6} A +40 x^{6} B +\frac {1}{10} x^{10} A +\frac {57}{10} x^{10} C +8 A x +\frac {1}{11} B \,x^{11}+\frac {1}{12} C \,x^{12}+\frac {71}{4} B \,x^{8}+\frac {183}{5} B \,x^{5}+\frac {61}{2} C \,x^{6}+12 B \,x^{3}+\frac {12}{11} C \,x^{11}+\frac {142}{9} C \,x^{9}+34 x^{3} A +\frac {51}{2} x^{4} B +\frac {102}{5} x^{5} C\) | \(180\) |
| parallelrisch | \(9 C \,x^{4}+\frac {8}{3} C \,x^{3}+4 B \,x^{2}+18 A \,x^{2}+\frac {183}{4} x^{4} A +48 x^{5} A +\frac {4}{3} A \,x^{9}+\frac {6}{5} B \,x^{10}+\frac {219}{8} x^{8} C +\frac {142}{7} x^{7} A +\frac {19}{3} x^{9} B +\frac {57}{8} x^{8} A +\frac {219}{7} x^{7} B +\frac {240}{7} x^{7} C +\frac {73}{2} x^{6} A +40 x^{6} B +\frac {1}{10} x^{10} A +\frac {57}{10} x^{10} C +8 A x +\frac {1}{11} B \,x^{11}+\frac {1}{12} C \,x^{12}+\frac {71}{4} B \,x^{8}+\frac {183}{5} B \,x^{5}+\frac {61}{2} C \,x^{6}+12 B \,x^{3}+\frac {12}{11} C \,x^{11}+\frac {142}{9} C \,x^{9}+34 x^{3} A +\frac {51}{2} x^{4} B +\frac {102}{5} x^{5} C\) | \(180\) |
Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^3,x,method=_RETURNVERBOSE)
Output:
1/12*C*x^12+(1/11*B+12/11*C)*x^11+(1/10*A+6/5*B+57/10*C)*x^10+(4/3*A+19/3* B+142/9*C)*x^9+(57/8*A+71/4*B+219/8*C)*x^8+(142/7*A+219/7*B+240/7*C)*x^7+( 73/2*A+40*B+61/2*C)*x^6+(48*A+183/5*B+102/5*C)*x^5+(183/4*A+51/2*B+9*C)*x^ 4+(34*A+12*B+8/3*C)*x^3+(18*A+4*B)*x^2+8*A*x
Time = 0.06 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=\frac {1}{12} \, C x^{12} + \frac {1}{11} \, {\left (B + 12 \, C\right )} x^{11} + \frac {1}{10} \, {\left (A + 12 \, B + 57 \, C\right )} x^{10} + \frac {1}{9} \, {\left (12 \, A + 57 \, B + 142 \, C\right )} x^{9} + \frac {1}{8} \, {\left (57 \, A + 142 \, B + 219 \, C\right )} x^{8} + \frac {1}{7} \, {\left (142 \, A + 219 \, B + 240 \, C\right )} x^{7} + \frac {1}{2} \, {\left (73 \, A + 80 \, B + 61 \, C\right )} x^{6} + \frac {3}{5} \, {\left (80 \, A + 61 \, B + 34 \, C\right )} x^{5} + \frac {3}{4} \, {\left (61 \, A + 34 \, B + 12 \, C\right )} x^{4} + \frac {2}{3} \, {\left (51 \, A + 18 \, B + 4 \, C\right )} x^{3} + 2 \, {\left (9 \, A + 2 \, B\right )} x^{2} + 8 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^3,x, algorithm="fricas")
Output:
1/12*C*x^12 + 1/11*(B + 12*C)*x^11 + 1/10*(A + 12*B + 57*C)*x^10 + 1/9*(12 *A + 57*B + 142*C)*x^9 + 1/8*(57*A + 142*B + 219*C)*x^8 + 1/7*(142*A + 219 *B + 240*C)*x^7 + 1/2*(73*A + 80*B + 61*C)*x^6 + 3/5*(80*A + 61*B + 34*C)* x^5 + 3/4*(61*A + 34*B + 12*C)*x^4 + 2/3*(51*A + 18*B + 4*C)*x^3 + 2*(9*A + 2*B)*x^2 + 8*A*x
Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=8 A x + \frac {C x^{12}}{12} + x^{11} \left (\frac {B}{11} + \frac {12 C}{11}\right ) + x^{10} \left (\frac {A}{10} + \frac {6 B}{5} + \frac {57 C}{10}\right ) + x^{9} \cdot \left (\frac {4 A}{3} + \frac {19 B}{3} + \frac {142 C}{9}\right ) + x^{8} \cdot \left (\frac {57 A}{8} + \frac {71 B}{4} + \frac {219 C}{8}\right ) + x^{7} \cdot \left (\frac {142 A}{7} + \frac {219 B}{7} + \frac {240 C}{7}\right ) + x^{6} \cdot \left (\frac {73 A}{2} + 40 B + \frac {61 C}{2}\right ) + x^{5} \cdot \left (48 A + \frac {183 B}{5} + \frac {102 C}{5}\right ) + x^{4} \cdot \left (\frac {183 A}{4} + \frac {51 B}{2} + 9 C\right ) + x^{3} \cdot \left (34 A + 12 B + \frac {8 C}{3}\right ) + x^{2} \cdot \left (18 A + 4 B\right ) \] Input:
integrate((C*x**2+B*x+A)*(x**3+4*x**2+3*x+2)**3,x)
Output:
8*A*x + C*x**12/12 + x**11*(B/11 + 12*C/11) + x**10*(A/10 + 6*B/5 + 57*C/1 0) + x**9*(4*A/3 + 19*B/3 + 142*C/9) + x**8*(57*A/8 + 71*B/4 + 219*C/8) + x**7*(142*A/7 + 219*B/7 + 240*C/7) + x**6*(73*A/2 + 40*B + 61*C/2) + x**5* (48*A + 183*B/5 + 102*C/5) + x**4*(183*A/4 + 51*B/2 + 9*C) + x**3*(34*A + 12*B + 8*C/3) + x**2*(18*A + 4*B)
Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=\frac {1}{12} \, C x^{12} + \frac {1}{11} \, {\left (B + 12 \, C\right )} x^{11} + \frac {1}{10} \, {\left (A + 12 \, B + 57 \, C\right )} x^{10} + \frac {1}{9} \, {\left (12 \, A + 57 \, B + 142 \, C\right )} x^{9} + \frac {1}{8} \, {\left (57 \, A + 142 \, B + 219 \, C\right )} x^{8} + \frac {1}{7} \, {\left (142 \, A + 219 \, B + 240 \, C\right )} x^{7} + \frac {1}{2} \, {\left (73 \, A + 80 \, B + 61 \, C\right )} x^{6} + \frac {3}{5} \, {\left (80 \, A + 61 \, B + 34 \, C\right )} x^{5} + \frac {3}{4} \, {\left (61 \, A + 34 \, B + 12 \, C\right )} x^{4} + \frac {2}{3} \, {\left (51 \, A + 18 \, B + 4 \, C\right )} x^{3} + 2 \, {\left (9 \, A + 2 \, B\right )} x^{2} + 8 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^3,x, algorithm="maxima")
Output:
1/12*C*x^12 + 1/11*(B + 12*C)*x^11 + 1/10*(A + 12*B + 57*C)*x^10 + 1/9*(12 *A + 57*B + 142*C)*x^9 + 1/8*(57*A + 142*B + 219*C)*x^8 + 1/7*(142*A + 219 *B + 240*C)*x^7 + 1/2*(73*A + 80*B + 61*C)*x^6 + 3/5*(80*A + 61*B + 34*C)* x^5 + 3/4*(61*A + 34*B + 12*C)*x^4 + 2/3*(51*A + 18*B + 4*C)*x^3 + 2*(9*A + 2*B)*x^2 + 8*A*x
Time = 0.13 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.05 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=\frac {1}{12} \, C x^{12} + \frac {1}{11} \, B x^{11} + \frac {12}{11} \, C x^{11} + \frac {1}{10} \, A x^{10} + \frac {6}{5} \, B x^{10} + \frac {57}{10} \, C x^{10} + \frac {4}{3} \, A x^{9} + \frac {19}{3} \, B x^{9} + \frac {142}{9} \, C x^{9} + \frac {57}{8} \, A x^{8} + \frac {71}{4} \, B x^{8} + \frac {219}{8} \, C x^{8} + \frac {142}{7} \, A x^{7} + \frac {219}{7} \, B x^{7} + \frac {240}{7} \, C x^{7} + \frac {73}{2} \, A x^{6} + 40 \, B x^{6} + \frac {61}{2} \, C x^{6} + 48 \, A x^{5} + \frac {183}{5} \, B x^{5} + \frac {102}{5} \, C x^{5} + \frac {183}{4} \, A x^{4} + \frac {51}{2} \, B x^{4} + 9 \, C x^{4} + 34 \, A x^{3} + 12 \, B x^{3} + \frac {8}{3} \, C x^{3} + 18 \, A x^{2} + 4 \, B x^{2} + 8 \, A x \] Input:
integrate((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^3,x, algorithm="giac")
Output:
1/12*C*x^12 + 1/11*B*x^11 + 12/11*C*x^11 + 1/10*A*x^10 + 6/5*B*x^10 + 57/1 0*C*x^10 + 4/3*A*x^9 + 19/3*B*x^9 + 142/9*C*x^9 + 57/8*A*x^8 + 71/4*B*x^8 + 219/8*C*x^8 + 142/7*A*x^7 + 219/7*B*x^7 + 240/7*C*x^7 + 73/2*A*x^6 + 40* B*x^6 + 61/2*C*x^6 + 48*A*x^5 + 183/5*B*x^5 + 102/5*C*x^5 + 183/4*A*x^4 + 51/2*B*x^4 + 9*C*x^4 + 34*A*x^3 + 12*B*x^3 + 8/3*C*x^3 + 18*A*x^2 + 4*B*x^ 2 + 8*A*x
Time = 0.14 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=\frac {C\,x^{12}}{12}+\left (\frac {B}{11}+\frac {12\,C}{11}\right )\,x^{11}+\left (\frac {A}{10}+\frac {6\,B}{5}+\frac {57\,C}{10}\right )\,x^{10}+\left (\frac {4\,A}{3}+\frac {19\,B}{3}+\frac {142\,C}{9}\right )\,x^9+\left (\frac {57\,A}{8}+\frac {71\,B}{4}+\frac {219\,C}{8}\right )\,x^8+\left (\frac {142\,A}{7}+\frac {219\,B}{7}+\frac {240\,C}{7}\right )\,x^7+\left (\frac {73\,A}{2}+40\,B+\frac {61\,C}{2}\right )\,x^6+\left (48\,A+\frac {183\,B}{5}+\frac {102\,C}{5}\right )\,x^5+\left (\frac {183\,A}{4}+\frac {51\,B}{2}+9\,C\right )\,x^4+\left (34\,A+12\,B+\frac {8\,C}{3}\right )\,x^3+\left (18\,A+4\,B\right )\,x^2+8\,A\,x \] Input:
int((A + B*x + C*x^2)*(3*x + 4*x^2 + x^3 + 2)^3,x)
Output:
8*A*x + (C*x^12)/12 + x^3*(34*A + 12*B + (8*C)/3) + x^10*(A/10 + (6*B)/5 + (57*C)/10) + x^6*((73*A)/2 + 40*B + (61*C)/2) + x^9*((4*A)/3 + (19*B)/3 + (142*C)/9) + x^4*((183*A)/4 + (51*B)/2 + 9*C) + x^5*(48*A + (183*B)/5 + ( 102*C)/5) + x^8*((57*A)/8 + (71*B)/4 + (219*C)/8) + x^7*((142*A)/7 + (219* B)/7 + (240*C)/7) + x^2*(18*A + 4*B) + x^11*(B/11 + (12*C)/11)
Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.04 \[ \int \left (A+B x+C x^2\right ) \left (2+3 x+4 x^2+x^3\right )^3 \, dx=\frac {x \left (2310 c \,x^{11}+2520 b \,x^{10}+30240 c \,x^{10}+2772 a \,x^{9}+33264 b \,x^{9}+158004 c \,x^{9}+36960 a \,x^{8}+175560 b \,x^{8}+437360 c \,x^{8}+197505 a \,x^{7}+492030 b \,x^{7}+758835 c \,x^{7}+562320 a \,x^{6}+867240 b \,x^{6}+950400 c \,x^{6}+1011780 a \,x^{5}+1108800 b \,x^{5}+845460 c \,x^{5}+1330560 a \,x^{4}+1014552 b \,x^{4}+565488 c \,x^{4}+1268190 a \,x^{3}+706860 b \,x^{3}+249480 c \,x^{3}+942480 a \,x^{2}+332640 b \,x^{2}+73920 c \,x^{2}+498960 a x +110880 b x +221760 a \right )}{27720} \] Input:
int((C*x^2+B*x+A)*(x^3+4*x^2+3*x+2)^3,x)
Output:
(x*(2772*a*x**9 + 36960*a*x**8 + 197505*a*x**7 + 562320*a*x**6 + 1011780*a *x**5 + 1330560*a*x**4 + 1268190*a*x**3 + 942480*a*x**2 + 498960*a*x + 221 760*a + 2520*b*x**10 + 33264*b*x**9 + 175560*b*x**8 + 492030*b*x**7 + 8672 40*b*x**6 + 1108800*b*x**5 + 1014552*b*x**4 + 706860*b*x**3 + 332640*b*x** 2 + 110880*b*x + 2310*c*x**11 + 30240*c*x**10 + 158004*c*x**9 + 437360*c*x **8 + 758835*c*x**7 + 950400*c*x**6 + 845460*c*x**5 + 565488*c*x**4 + 2494 80*c*x**3 + 73920*c*x**2))/27720