Integrand size = 24, antiderivative size = 73 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5}{160 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^7}{112 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3} \] Output:
1/160*(-4*a*c+b^2)^2*d^4*(2*c*x+b)^5/c^3-1/112*(-4*a*c+b^2)*d^4*(2*c*x+b)^ 7/c^3+1/288*d^4*(2*c*x+b)^9/c^3
Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(73)=146\).
Time = 0.02 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.45 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=d^4 \left (a^2 b^4 x+a b^3 \left (b^2+4 a c\right ) x^2+\frac {1}{3} b^2 \left (b^4+18 a b^2 c+24 a^2 c^2\right ) x^3+\frac {1}{2} b c \left (5 b^4+32 a b^2 c+16 a^2 c^2\right ) x^4+\frac {1}{5} c^2 \left (41 b^4+112 a b^2 c+16 a^2 c^2\right ) x^5+\frac {4}{3} b c^3 \left (11 b^2+12 a c\right ) x^6+\frac {8}{7} c^4 \left (13 b^2+4 a c\right ) x^7+8 b c^5 x^8+\frac {16 c^6 x^9}{9}\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]
Output:
d^4*(a^2*b^4*x + a*b^3*(b^2 + 4*a*c)*x^2 + (b^2*(b^4 + 18*a*b^2*c + 24*a^2 *c^2)*x^3)/3 + (b*c*(5*b^4 + 32*a*b^2*c + 16*a^2*c^2)*x^4)/2 + (c^2*(41*b^ 4 + 112*a*b^2*c + 16*a^2*c^2)*x^5)/5 + (4*b*c^3*(11*b^2 + 12*a*c)*x^6)/3 + (8*c^4*(13*b^2 + 4*a*c)*x^7)/7 + 8*b*c^5*x^8 + (16*c^6*x^9)/9)
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 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.
\(\displaystyle \int \left (a+b x+c x^2\right )^2 (b d+2 c d x)^4 \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right ) (b d+2 c d x)^6}{8 c^2 d^2}+\frac {\left (4 a c-b^2\right )^2 (b d+2 c d x)^4}{16 c^2}+\frac {(b d+2 c d x)^8}{16 c^2 d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d^4 \left (b^2-4 a c\right ) (b+2 c x)^7}{112 c^3}+\frac {d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^5}{160 c^3}+\frac {d^4 (b+2 c x)^9}{288 c^3}\) |
Input:
Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x]
Output:
((b^2 - 4*a*c)^2*d^4*(b + 2*c*x)^5)/(160*c^3) - ((b^2 - 4*a*c)*d^4*(b + 2* c*x)^7)/(112*c^3) + (d^4*(b + 2*c*x)^9)/(288*c^3)
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] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(189\) vs. \(2(67)=134\).
Time = 0.77 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.60
method | result | size |
gosper | \(\frac {x \left (1120 c^{6} x^{8}+5040 b \,c^{5} x^{7}+2880 x^{6} a \,c^{5}+9360 x^{6} b^{2} c^{4}+10080 x^{5} a b \,c^{4}+9240 b^{3} c^{3} x^{5}+2016 x^{4} a^{2} c^{4}+14112 a \,b^{2} c^{3} x^{4}+5166 b^{4} c^{2} x^{4}+5040 x^{3} a^{2} b \,c^{3}+10080 a \,b^{3} c^{2} x^{3}+1575 b^{5} c \,x^{3}+5040 a^{2} b^{2} c^{2} x^{2}+3780 a \,b^{4} c \,x^{2}+210 x^{2} b^{6}+2520 a^{2} b^{3} c x +630 a \,b^{5} x +630 a^{2} b^{4}\right ) d^{4}}{630}\) | \(190\) |
orering | \(\frac {x \left (1120 c^{6} x^{8}+5040 b \,c^{5} x^{7}+2880 x^{6} a \,c^{5}+9360 x^{6} b^{2} c^{4}+10080 x^{5} a b \,c^{4}+9240 b^{3} c^{3} x^{5}+2016 x^{4} a^{2} c^{4}+14112 a \,b^{2} c^{3} x^{4}+5166 b^{4} c^{2} x^{4}+5040 x^{3} a^{2} b \,c^{3}+10080 a \,b^{3} c^{2} x^{3}+1575 b^{5} c \,x^{3}+5040 a^{2} b^{2} c^{2} x^{2}+3780 a \,b^{4} c \,x^{2}+210 x^{2} b^{6}+2520 a^{2} b^{3} c x +630 a \,b^{5} x +630 a^{2} b^{4}\right ) \left (2 c d x +b d \right )^{4}}{630 \left (2 c x +b \right )^{4}}\) | \(206\) |
norman | \(\left (\frac {32}{7} a \,c^{5} d^{4}+\frac {104}{7} b^{2} c^{4} d^{4}\right ) x^{7}+\left (16 a b \,c^{4} d^{4}+\frac {44}{3} b^{3} c^{3} d^{4}\right ) x^{6}+\left (\frac {16}{5} a^{2} c^{4} d^{4}+\frac {112}{5} b^{2} c^{3} d^{4} a +\frac {41}{5} b^{4} d^{4} c^{2}\right ) x^{5}+\left (8 a^{2} b^{2} c^{2} d^{4}+6 b^{4} c \,d^{4} a +\frac {1}{3} b^{6} d^{4}\right ) x^{3}+\left (8 c^{3} b \,a^{2} d^{4}+16 a \,b^{3} c^{2} d^{4}+\frac {5}{2} b^{5} d^{4} c \right ) x^{4}+\left (4 a^{2} b^{3} c \,d^{4}+b^{5} d^{4} a \right ) x^{2}+a^{2} b^{4} d^{4} x +\frac {16 d^{4} c^{6} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}\) | \(226\) |
risch | \(\frac {16}{9} d^{4} c^{6} x^{9}+8 b \,c^{5} d^{4} x^{8}+\frac {32}{7} d^{4} x^{7} a \,c^{5}+\frac {104}{7} d^{4} x^{7} b^{2} c^{4}+16 d^{4} x^{6} a b \,c^{4}+\frac {44}{3} d^{4} b^{3} c^{3} x^{6}+\frac {16}{5} d^{4} x^{5} a^{2} c^{4}+\frac {112}{5} d^{4} x^{5} a \,b^{2} c^{3}+\frac {41}{5} d^{4} b^{4} c^{2} x^{5}+8 d^{4} x^{4} a^{2} b \,c^{3}+16 d^{4} a \,b^{3} c^{2} x^{4}+\frac {5}{2} b^{5} c \,d^{4} x^{4}+8 d^{4} a^{2} b^{2} c^{2} x^{3}+6 a \,b^{4} c \,d^{4} x^{3}+\frac {1}{3} d^{4} x^{3} b^{6}+4 a^{2} b^{3} c \,d^{4} x^{2}+d^{4} a \,b^{5} x^{2}+a^{2} b^{4} d^{4} x\) | \(241\) |
parallelrisch | \(\frac {16}{9} d^{4} c^{6} x^{9}+8 b \,c^{5} d^{4} x^{8}+\frac {32}{7} d^{4} x^{7} a \,c^{5}+\frac {104}{7} d^{4} x^{7} b^{2} c^{4}+16 d^{4} x^{6} a b \,c^{4}+\frac {44}{3} d^{4} b^{3} c^{3} x^{6}+\frac {16}{5} d^{4} x^{5} a^{2} c^{4}+\frac {112}{5} d^{4} x^{5} a \,b^{2} c^{3}+\frac {41}{5} d^{4} b^{4} c^{2} x^{5}+8 d^{4} x^{4} a^{2} b \,c^{3}+16 d^{4} a \,b^{3} c^{2} x^{4}+\frac {5}{2} b^{5} c \,d^{4} x^{4}+8 d^{4} a^{2} b^{2} c^{2} x^{3}+6 a \,b^{4} c \,d^{4} x^{3}+\frac {1}{3} d^{4} x^{3} b^{6}+4 a^{2} b^{3} c \,d^{4} x^{2}+d^{4} a \,b^{5} x^{2}+a^{2} b^{4} d^{4} x\) | \(241\) |
default | \(\frac {16 d^{4} c^{6} x^{9}}{9}+8 b \,c^{5} d^{4} x^{8}+\frac {\left (88 b^{2} c^{4} d^{4}+16 d^{4} c^{4} \left (2 a c +b^{2}\right )\right ) x^{7}}{7}+\frac {\left (56 b^{3} c^{3} d^{4}+32 b \,c^{3} d^{4} \left (2 a c +b^{2}\right )+32 a b \,c^{4} d^{4}\right ) x^{6}}{6}+\frac {\left (17 b^{4} d^{4} c^{2}+24 b^{2} d^{4} c^{2} \left (2 a c +b^{2}\right )+64 b^{2} c^{3} d^{4} a +16 a^{2} c^{4} d^{4}\right ) x^{5}}{5}+\frac {\left (2 b^{5} d^{4} c +8 b^{3} c \,d^{4} \left (2 a c +b^{2}\right )+48 a \,b^{3} c^{2} d^{4}+32 c^{3} b \,a^{2} d^{4}\right ) x^{4}}{4}+\frac {\left (b^{4} d^{4} \left (2 a c +b^{2}\right )+16 b^{4} c \,d^{4} a +24 a^{2} b^{2} c^{2} d^{4}\right ) x^{3}}{3}+\frac {\left (8 a^{2} b^{3} c \,d^{4}+2 b^{5} d^{4} a \right ) x^{2}}{2}+a^{2} b^{4} d^{4} x\) | \(300\) |
Input:
int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/630*x*(1120*c^6*x^8+5040*b*c^5*x^7+2880*a*c^5*x^6+9360*b^2*c^4*x^6+10080 *a*b*c^4*x^5+9240*b^3*c^3*x^5+2016*a^2*c^4*x^4+14112*a*b^2*c^3*x^4+5166*b^ 4*c^2*x^4+5040*a^2*b*c^3*x^3+10080*a*b^3*c^2*x^3+1575*b^5*c*x^3+5040*a^2*b ^2*c^2*x^2+3780*a*b^4*c*x^2+210*b^6*x^2+2520*a^2*b^3*c*x+630*a*b^5*x+630*a ^2*b^4)*d^4
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (67) = 134\).
Time = 0.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.75 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 8/7*(13*b^2*c^4 + 4*a*c^5)*d^4*x^7 + a^2*b^4*d^4*x + 4/3*(11*b^3*c^3 + 12*a*b*c^4)*d^4*x^6 + 1/5*(41*b^4*c^2 + 112*a*b^2*c^3 + 16*a^2*c^4)*d^4*x^5 + 1/2*(5*b^5*c + 32*a*b^3*c^2 + 16*a^2 *b*c^3)*d^4*x^4 + 1/3*(b^6 + 18*a*b^4*c + 24*a^2*b^2*c^2)*d^4*x^3 + (a*b^5 + 4*a^2*b^3*c)*d^4*x^2
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (68) = 136\).
Time = 0.04 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.40 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^{2} b^{4} d^{4} x + 8 b c^{5} d^{4} x^{8} + \frac {16 c^{6} d^{4} x^{9}}{9} + x^{7} \cdot \left (\frac {32 a c^{5} d^{4}}{7} + \frac {104 b^{2} c^{4} d^{4}}{7}\right ) + x^{6} \cdot \left (16 a b c^{4} d^{4} + \frac {44 b^{3} c^{3} d^{4}}{3}\right ) + x^{5} \cdot \left (\frac {16 a^{2} c^{4} d^{4}}{5} + \frac {112 a b^{2} c^{3} d^{4}}{5} + \frac {41 b^{4} c^{2} d^{4}}{5}\right ) + x^{4} \cdot \left (8 a^{2} b c^{3} d^{4} + 16 a b^{3} c^{2} d^{4} + \frac {5 b^{5} c d^{4}}{2}\right ) + x^{3} \cdot \left (8 a^{2} b^{2} c^{2} d^{4} + 6 a b^{4} c d^{4} + \frac {b^{6} d^{4}}{3}\right ) + x^{2} \cdot \left (4 a^{2} b^{3} c d^{4} + a b^{5} d^{4}\right ) \] Input:
integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**2,x)
Output:
a**2*b**4*d**4*x + 8*b*c**5*d**4*x**8 + 16*c**6*d**4*x**9/9 + x**7*(32*a*c **5*d**4/7 + 104*b**2*c**4*d**4/7) + x**6*(16*a*b*c**4*d**4 + 44*b**3*c**3 *d**4/3) + x**5*(16*a**2*c**4*d**4/5 + 112*a*b**2*c**3*d**4/5 + 41*b**4*c* *2*d**4/5) + x**4*(8*a**2*b*c**3*d**4 + 16*a*b**3*c**2*d**4 + 5*b**5*c*d** 4/2) + x**3*(8*a**2*b**2*c**2*d**4 + 6*a*b**4*c*d**4 + b**6*d**4/3) + x**2 *(4*a**2*b**3*c*d**4 + a*b**5*d**4)
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (67) = 134\).
Time = 0.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.75 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {8}{7} \, {\left (13 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{4} x^{7} + a^{2} b^{4} d^{4} x + \frac {4}{3} \, {\left (11 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} d^{4} x^{6} + \frac {1}{5} \, {\left (41 \, b^{4} c^{2} + 112 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{4} x^{5} + \frac {1}{2} \, {\left (5 \, b^{5} c + 32 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{4} x^{4} + \frac {1}{3} \, {\left (b^{6} + 18 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2}\right )} d^{4} x^{3} + {\left (a b^{5} + 4 \, a^{2} b^{3} c\right )} d^{4} x^{2} \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 8/7*(13*b^2*c^4 + 4*a*c^5)*d^4*x^7 + a^2*b^4*d^4*x + 4/3*(11*b^3*c^3 + 12*a*b*c^4)*d^4*x^6 + 1/5*(41*b^4*c^2 + 112*a*b^2*c^3 + 16*a^2*c^4)*d^4*x^5 + 1/2*(5*b^5*c + 32*a*b^3*c^2 + 16*a^2 *b*c^3)*d^4*x^4 + 1/3*(b^6 + 18*a*b^4*c + 24*a^2*b^2*c^2)*d^4*x^3 + (a*b^5 + 4*a^2*b^3*c)*d^4*x^2
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (67) = 134\).
Time = 0.14 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.29 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16}{9} \, c^{6} d^{4} x^{9} + 8 \, b c^{5} d^{4} x^{8} + \frac {104}{7} \, b^{2} c^{4} d^{4} x^{7} + \frac {32}{7} \, a c^{5} d^{4} x^{7} + \frac {44}{3} \, b^{3} c^{3} d^{4} x^{6} + 16 \, a b c^{4} d^{4} x^{6} + \frac {41}{5} \, b^{4} c^{2} d^{4} x^{5} + \frac {112}{5} \, a b^{2} c^{3} d^{4} x^{5} + \frac {16}{5} \, a^{2} c^{4} d^{4} x^{5} + \frac {5}{2} \, b^{5} c d^{4} x^{4} + 16 \, a b^{3} c^{2} d^{4} x^{4} + 8 \, a^{2} b c^{3} d^{4} x^{4} + \frac {1}{3} \, b^{6} d^{4} x^{3} + 6 \, a b^{4} c d^{4} x^{3} + 8 \, a^{2} b^{2} c^{2} d^{4} x^{3} + a b^{5} d^{4} x^{2} + 4 \, a^{2} b^{3} c d^{4} x^{2} + a^{2} b^{4} d^{4} x \] Input:
integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
16/9*c^6*d^4*x^9 + 8*b*c^5*d^4*x^8 + 104/7*b^2*c^4*d^4*x^7 + 32/7*a*c^5*d^ 4*x^7 + 44/3*b^3*c^3*d^4*x^6 + 16*a*b*c^4*d^4*x^6 + 41/5*b^4*c^2*d^4*x^5 + 112/5*a*b^2*c^3*d^4*x^5 + 16/5*a^2*c^4*d^4*x^5 + 5/2*b^5*c*d^4*x^4 + 16*a *b^3*c^2*d^4*x^4 + 8*a^2*b*c^3*d^4*x^4 + 1/3*b^6*d^4*x^3 + 6*a*b^4*c*d^4*x ^3 + 8*a^2*b^2*c^2*d^4*x^3 + a*b^5*d^4*x^2 + 4*a^2*b^3*c*d^4*x^2 + a^2*b^4 *d^4*x
Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.60 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {16\,c^6\,d^4\,x^9}{9}+\frac {c^2\,d^4\,x^5\,\left (16\,a^2\,c^2+112\,a\,b^2\,c+41\,b^4\right )}{5}+a^2\,b^4\,d^4\,x+8\,b\,c^5\,d^4\,x^8+\frac {8\,c^4\,d^4\,x^7\,\left (13\,b^2+4\,a\,c\right )}{7}+\frac {b^2\,d^4\,x^3\,\left (24\,a^2\,c^2+18\,a\,b^2\,c+b^4\right )}{3}+\frac {b\,c\,d^4\,x^4\,\left (16\,a^2\,c^2+32\,a\,b^2\,c+5\,b^4\right )}{2}+a\,b^3\,d^4\,x^2\,\left (b^2+4\,a\,c\right )+\frac {4\,b\,c^3\,d^4\,x^6\,\left (11\,b^2+12\,a\,c\right )}{3} \] Input:
int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^2,x)
Output:
(16*c^6*d^4*x^9)/9 + (c^2*d^4*x^5*(41*b^4 + 16*a^2*c^2 + 112*a*b^2*c))/5 + a^2*b^4*d^4*x + 8*b*c^5*d^4*x^8 + (8*c^4*d^4*x^7*(4*a*c + 13*b^2))/7 + (b ^2*d^4*x^3*(b^4 + 24*a^2*c^2 + 18*a*b^2*c))/3 + (b*c*d^4*x^4*(5*b^4 + 16*a ^2*c^2 + 32*a*b^2*c))/2 + a*b^3*d^4*x^2*(4*a*c + b^2) + (4*b*c^3*d^4*x^6*( 12*a*c + 11*b^2))/3
Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.59 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {d^{4} x \left (1120 c^{6} x^{8}+5040 b \,c^{5} x^{7}+2880 a \,c^{5} x^{6}+9360 b^{2} c^{4} x^{6}+10080 a b \,c^{4} x^{5}+9240 b^{3} c^{3} x^{5}+2016 a^{2} c^{4} x^{4}+14112 a \,b^{2} c^{3} x^{4}+5166 b^{4} c^{2} x^{4}+5040 a^{2} b \,c^{3} x^{3}+10080 a \,b^{3} c^{2} x^{3}+1575 b^{5} c \,x^{3}+5040 a^{2} b^{2} c^{2} x^{2}+3780 a \,b^{4} c \,x^{2}+210 b^{6} x^{2}+2520 a^{2} b^{3} c x +630 a \,b^{5} x +630 a^{2} b^{4}\right )}{630} \] Input:
int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^2,x)
Output:
(d**4*x*(630*a**2*b**4 + 2520*a**2*b**3*c*x + 5040*a**2*b**2*c**2*x**2 + 5 040*a**2*b*c**3*x**3 + 2016*a**2*c**4*x**4 + 630*a*b**5*x + 3780*a*b**4*c* x**2 + 10080*a*b**3*c**2*x**3 + 14112*a*b**2*c**3*x**4 + 10080*a*b*c**4*x* *5 + 2880*a*c**5*x**6 + 210*b**6*x**2 + 1575*b**5*c*x**3 + 5166*b**4*c**2* x**4 + 9240*b**3*c**3*x**5 + 9360*b**2*c**4*x**6 + 5040*b*c**5*x**7 + 1120 *c**6*x**8))/630