Integrand size = 26, antiderivative size = 98 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {16}{315} \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{5/2}+\frac {8}{63} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2} \] Output:
16/315*(-4*a*c+b^2)^2*d^5*(c*x^2+b*x+a)^(5/2)+8/63*(-4*a*c+b^2)*d^5*(2*c*x +b)^2*(c*x^2+b*x+a)^(5/2)+2/9*d^5*(2*c*x+b)^4*(c*x^2+b*x+a)^(5/2)
Time = 2.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2}{315} d^5 (a+x (b+c x))^{5/2} \left (63 b^4+360 b^3 c x+160 b c^2 x \left (-2 a+7 c x^2\right )+8 b^2 c \left (-18 a+115 c x^2\right )+16 c^2 \left (8 a^2-20 a c x^2+35 c^2 x^4\right )\right ) \] Input:
Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^(3/2),x]
Output:
(2*d^5*(a + x*(b + c*x))^(5/2)*(63*b^4 + 360*b^3*c*x + 160*b*c^2*x*(-2*a + 7*c*x^2) + 8*b^2*c*(-18*a + 115*c*x^2) + 16*c^2*(8*a^2 - 20*a*c*x^2 + 35* c^2*x^4)))/315
Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1116, 27, 1116, 1104}
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.
\(\displaystyle \int \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^5 \, dx\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle \frac {4}{9} d^2 \left (b^2-4 a c\right ) \int d^3 (b+2 c x)^3 \left (c x^2+b x+a\right )^{3/2}dx+\frac {2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4}{9} d^5 \left (b^2-4 a c\right ) \int (b+2 c x)^3 \left (c x^2+b x+a\right )^{3/2}dx+\frac {2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle \frac {4}{9} d^5 \left (b^2-4 a c\right ) \left (\frac {2}{7} \left (b^2-4 a c\right ) \int (b+2 c x) \left (c x^2+b x+a\right )^{3/2}dx+\frac {2}{7} (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}\right )+\frac {2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {4}{9} d^5 \left (b^2-4 a c\right ) \left (\frac {4}{35} \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac {2}{7} (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}\right )+\frac {2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}\) |
Input:
Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^(3/2),x]
Output:
(2*d^5*(b + 2*c*x)^4*(a + b*x + c*x^2)^(5/2))/9 + (4*(b^2 - 4*a*c)*d^5*((4 *(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2))/35 + (2*(b + 2*c*x)^2*(a + b*x + c *x^2)^(5/2))/7))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Time = 0.88 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {256 d^{5} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (\frac {35 c^{4} x^{4}}{8}-\frac {5 x^{2} \left (-\frac {7 b x}{2}+a \right ) c^{3}}{2}+\left (\frac {115}{16} b^{2} x^{2}-\frac {5}{2} a b x +a^{2}\right ) c^{2}-\frac {9 \left (-\frac {5 b x}{2}+a \right ) b^{2} c}{8}+\frac {63 b^{4}}{128}\right )}{315}\) | \(79\) |
gosper | \(\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (560 c^{4} x^{4}+1120 b \,c^{3} x^{3}-320 a \,c^{3} x^{2}+920 b^{2} c^{2} x^{2}-320 a b \,c^{2} x +360 b^{3} c x +128 a^{2} c^{2}-144 c a \,b^{2}+63 b^{4}\right ) d^{5}}{315}\) | \(91\) |
orering | \(\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (560 c^{4} x^{4}+1120 b \,c^{3} x^{3}-320 a \,c^{3} x^{2}+920 b^{2} c^{2} x^{2}-320 a b \,c^{2} x +360 b^{3} c x +128 a^{2} c^{2}-144 c a \,b^{2}+63 b^{4}\right ) \left (2 c d x +b d \right )^{5}}{315 \left (2 c x +b \right )^{5}}\) | \(107\) |
trager | \(d^{5} \left (\frac {32}{9} c^{6} x^{8}+\frac {128}{9} b \,c^{5} x^{7}+\frac {320}{63} a \,c^{5} x^{6}+\frac {496}{21} b^{2} c^{4} x^{6}+\frac {320}{21} x^{5} a b \,c^{4}+\frac {1328}{63} b^{3} c^{3} x^{5}+\frac {32}{105} a^{2} c^{4} x^{4}+\frac {1984}{105} a \,b^{2} c^{3} x^{4}+\frac {3406}{315} b^{4} c^{2} x^{4}+\frac {64}{105} a^{2} b \,c^{3} x^{3}+\frac {3904}{315} a \,b^{3} c^{2} x^{3}+\frac {108}{35} b^{5} c \,x^{3}-\frac {128}{315} a^{3} c^{3} x^{2}+\frac {16}{21} a^{2} b^{2} c^{2} x^{2}+\frac {156}{35} a \,b^{4} c \,x^{2}+\frac {2}{5} b^{6} x^{2}-\frac {128}{315} a^{3} b \,c^{2} x +\frac {16}{35} a^{2} b^{3} c x +\frac {4}{5} a \,b^{5} x +\frac {256}{315} a^{4} c^{2}-\frac {32}{35} a^{3} b^{2} c +\frac {2}{5} a^{2} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}\) | \(238\) |
risch | \(\frac {2 d^{5} \left (560 c^{6} x^{8}+2240 b \,c^{5} x^{7}+800 a \,c^{5} x^{6}+3720 b^{2} c^{4} x^{6}+2400 x^{5} a b \,c^{4}+3320 b^{3} c^{3} x^{5}+48 a^{2} c^{4} x^{4}+2976 a \,b^{2} c^{3} x^{4}+1703 b^{4} c^{2} x^{4}+96 a^{2} b \,c^{3} x^{3}+1952 a \,b^{3} c^{2} x^{3}+486 b^{5} c \,x^{3}-64 a^{3} c^{3} x^{2}+120 a^{2} b^{2} c^{2} x^{2}+702 a \,b^{4} c \,x^{2}+63 b^{6} x^{2}-64 a^{3} b \,c^{2} x +72 a^{2} b^{3} c x +126 a \,b^{5} x +128 a^{4} c^{2}-144 a^{3} b^{2} c +63 a^{2} b^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{315}\) | \(239\) |
default | \(\text {Expression too large to display}\) | \(2839\) |
Input:
int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
256/315*d^5*(c*x^2+b*x+a)^(5/2)*(35/8*c^4*x^4-5/2*x^2*(-7/2*b*x+a)*c^3+(11 5/16*b^2*x^2-5/2*a*b*x+a^2)*c^2-9/8*(-5/2*b*x+a)*b^2*c+63/128*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (86) = 172\).
Time = 0.11 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.56 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2}{315} \, {\left (560 \, c^{6} d^{5} x^{8} + 2240 \, b c^{5} d^{5} x^{7} + 40 \, {\left (93 \, b^{2} c^{4} + 20 \, a c^{5}\right )} d^{5} x^{6} + 40 \, {\left (83 \, b^{3} c^{3} + 60 \, a b c^{4}\right )} d^{5} x^{5} + {\left (1703 \, b^{4} c^{2} + 2976 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{5} x^{4} + 2 \, {\left (243 \, b^{5} c + 976 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} d^{5} x^{3} + {\left (63 \, b^{6} + 702 \, a b^{4} c + 120 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{5} x^{2} + 2 \, {\left (63 \, a b^{5} + 36 \, a^{2} b^{3} c - 32 \, a^{3} b c^{2}\right )} d^{5} x + {\left (63 \, a^{2} b^{4} - 144 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} d^{5}\right )} \sqrt {c x^{2} + b x + a} \] Input:
integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
2/315*(560*c^6*d^5*x^8 + 2240*b*c^5*d^5*x^7 + 40*(93*b^2*c^4 + 20*a*c^5)*d ^5*x^6 + 40*(83*b^3*c^3 + 60*a*b*c^4)*d^5*x^5 + (1703*b^4*c^2 + 2976*a*b^2 *c^3 + 48*a^2*c^4)*d^5*x^4 + 2*(243*b^5*c + 976*a*b^3*c^2 + 48*a^2*b*c^3)* d^5*x^3 + (63*b^6 + 702*a*b^4*c + 120*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*x^2 + 2*(63*a*b^5 + 36*a^2*b^3*c - 32*a^3*b*c^2)*d^5*x + (63*a^2*b^4 - 144*a^3*b ^2*c + 128*a^4*c^2)*d^5)*sqrt(c*x^2 + b*x + a)
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (97) = 194\).
Time = 0.24 (sec) , antiderivative size = 656, normalized size of antiderivative = 6.69 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {256 a^{4} c^{2} d^{5} \sqrt {a + b x + c x^{2}}}{315} - \frac {32 a^{3} b^{2} c d^{5} \sqrt {a + b x + c x^{2}}}{35} - \frac {128 a^{3} b c^{2} d^{5} x \sqrt {a + b x + c x^{2}}}{315} - \frac {128 a^{3} c^{3} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{315} + \frac {2 a^{2} b^{4} d^{5} \sqrt {a + b x + c x^{2}}}{5} + \frac {16 a^{2} b^{3} c d^{5} x \sqrt {a + b x + c x^{2}}}{35} + \frac {16 a^{2} b^{2} c^{2} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{21} + \frac {64 a^{2} b c^{3} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{105} + \frac {32 a^{2} c^{4} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{105} + \frac {4 a b^{5} d^{5} x \sqrt {a + b x + c x^{2}}}{5} + \frac {156 a b^{4} c d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{35} + \frac {3904 a b^{3} c^{2} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{315} + \frac {1984 a b^{2} c^{3} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{105} + \frac {320 a b c^{4} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{21} + \frac {320 a c^{5} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{63} + \frac {2 b^{6} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{5} + \frac {108 b^{5} c d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{35} + \frac {3406 b^{4} c^{2} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{315} + \frac {1328 b^{3} c^{3} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{63} + \frac {496 b^{2} c^{4} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{21} + \frac {128 b c^{5} d^{5} x^{7} \sqrt {a + b x + c x^{2}}}{9} + \frac {32 c^{6} d^{5} x^{8} \sqrt {a + b x + c x^{2}}}{9} \] Input:
integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**(3/2),x)
Output:
256*a**4*c**2*d**5*sqrt(a + b*x + c*x**2)/315 - 32*a**3*b**2*c*d**5*sqrt(a + b*x + c*x**2)/35 - 128*a**3*b*c**2*d**5*x*sqrt(a + b*x + c*x**2)/315 - 128*a**3*c**3*d**5*x**2*sqrt(a + b*x + c*x**2)/315 + 2*a**2*b**4*d**5*sqrt (a + b*x + c*x**2)/5 + 16*a**2*b**3*c*d**5*x*sqrt(a + b*x + c*x**2)/35 + 1 6*a**2*b**2*c**2*d**5*x**2*sqrt(a + b*x + c*x**2)/21 + 64*a**2*b*c**3*d**5 *x**3*sqrt(a + b*x + c*x**2)/105 + 32*a**2*c**4*d**5*x**4*sqrt(a + b*x + c *x**2)/105 + 4*a*b**5*d**5*x*sqrt(a + b*x + c*x**2)/5 + 156*a*b**4*c*d**5* x**2*sqrt(a + b*x + c*x**2)/35 + 3904*a*b**3*c**2*d**5*x**3*sqrt(a + b*x + c*x**2)/315 + 1984*a*b**2*c**3*d**5*x**4*sqrt(a + b*x + c*x**2)/105 + 320 *a*b*c**4*d**5*x**5*sqrt(a + b*x + c*x**2)/21 + 320*a*c**5*d**5*x**6*sqrt( a + b*x + c*x**2)/63 + 2*b**6*d**5*x**2*sqrt(a + b*x + c*x**2)/5 + 108*b** 5*c*d**5*x**3*sqrt(a + b*x + c*x**2)/35 + 3406*b**4*c**2*d**5*x**4*sqrt(a + b*x + c*x**2)/315 + 1328*b**3*c**3*d**5*x**5*sqrt(a + b*x + c*x**2)/63 + 496*b**2*c**4*d**5*x**6*sqrt(a + b*x + c*x**2)/21 + 128*b*c**5*d**5*x**7* sqrt(a + b*x + c*x**2)/9 + 32*c**6*d**5*x**8*sqrt(a + b*x + c*x**2)/9
Exception generated. \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),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(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} b^{4} d^{5} + \frac {16}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} b^{2} c d^{5} - \frac {16}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} a b^{2} c d^{5} + \frac {32}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} c^{2} d^{5} - \frac {64}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a c^{2} d^{5} + \frac {32}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} a^{2} c^{2} d^{5} \] Input:
integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
2/5*(c*x^2 + b*x + a)^(5/2)*b^4*d^5 + 16/7*(c*x^2 + b*x + a)^(7/2)*b^2*c*d ^5 - 16/5*(c*x^2 + b*x + a)^(5/2)*a*b^2*c*d^5 + 32/9*(c*x^2 + b*x + a)^(9/ 2)*c^2*d^5 - 64/7*(c*x^2 + b*x + a)^(7/2)*a*c^2*d^5 + 32/5*(c*x^2 + b*x + a)^(5/2)*a^2*c^2*d^5
Time = 5.73 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.46 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\sqrt {c\,x^2+b\,x+a}\,\left (\frac {2\,a^2\,d^5\,\left (128\,a^2\,c^2-144\,a\,b^2\,c+63\,b^4\right )}{315}+\frac {32\,c^6\,d^5\,x^8}{9}+\frac {2\,d^5\,x^2\,\left (-64\,a^3\,c^3+120\,a^2\,b^2\,c^2+702\,a\,b^4\,c+63\,b^6\right )}{315}+\frac {2\,c^2\,d^5\,x^4\,\left (48\,a^2\,c^2+2976\,a\,b^2\,c+1703\,b^4\right )}{315}+\frac {128\,b\,c^5\,d^5\,x^7}{9}+\frac {16\,c^4\,d^5\,x^6\,\left (93\,b^2+20\,a\,c\right )}{63}+\frac {4\,b\,c\,d^5\,x^3\,\left (48\,a^2\,c^2+976\,a\,b^2\,c+243\,b^4\right )}{315}+\frac {4\,a\,b\,d^5\,x\,\left (-32\,a^2\,c^2+36\,a\,b^2\,c+63\,b^4\right )}{315}+\frac {16\,b\,c^3\,d^5\,x^5\,\left (83\,b^2+60\,a\,c\right )}{63}\right ) \] Input:
int((b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^(3/2),x)
Output:
(a + b*x + c*x^2)^(1/2)*((2*a^2*d^5*(63*b^4 + 128*a^2*c^2 - 144*a*b^2*c))/ 315 + (32*c^6*d^5*x^8)/9 + (2*d^5*x^2*(63*b^6 - 64*a^3*c^3 + 120*a^2*b^2*c ^2 + 702*a*b^4*c))/315 + (2*c^2*d^5*x^4*(1703*b^4 + 48*a^2*c^2 + 2976*a*b^ 2*c))/315 + (128*b*c^5*d^5*x^7)/9 + (16*c^4*d^5*x^6*(20*a*c + 93*b^2))/63 + (4*b*c*d^5*x^3*(243*b^4 + 48*a^2*c^2 + 976*a*b^2*c))/315 + (4*a*b*d^5*x* (63*b^4 - 32*a^2*c^2 + 36*a*b^2*c))/315 + (16*b*c^3*d^5*x^5*(60*a*c + 83*b ^2))/63)
Time = 0.21 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.42 \[ \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c \,x^{2}+b x +a}\, d^{5} \left (560 c^{6} x^{8}+2240 b \,c^{5} x^{7}+800 a \,c^{5} x^{6}+3720 b^{2} c^{4} x^{6}+2400 a b \,c^{4} x^{5}+3320 b^{3} c^{3} x^{5}+48 a^{2} c^{4} x^{4}+2976 a \,b^{2} c^{3} x^{4}+1703 b^{4} c^{2} x^{4}+96 a^{2} b \,c^{3} x^{3}+1952 a \,b^{3} c^{2} x^{3}+486 b^{5} c \,x^{3}-64 a^{3} c^{3} x^{2}+120 a^{2} b^{2} c^{2} x^{2}+702 a \,b^{4} c \,x^{2}+63 b^{6} x^{2}-64 a^{3} b \,c^{2} x +72 a^{2} b^{3} c x +126 a \,b^{5} x +128 a^{4} c^{2}-144 a^{3} b^{2} c +63 a^{2} b^{4}\right )}{315} \] Input:
int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x)
Output:
(2*sqrt(a + b*x + c*x**2)*d**5*(128*a**4*c**2 - 144*a**3*b**2*c - 64*a**3* b*c**2*x - 64*a**3*c**3*x**2 + 63*a**2*b**4 + 72*a**2*b**3*c*x + 120*a**2* b**2*c**2*x**2 + 96*a**2*b*c**3*x**3 + 48*a**2*c**4*x**4 + 126*a*b**5*x + 702*a*b**4*c*x**2 + 1952*a*b**3*c**2*x**3 + 2976*a*b**2*c**3*x**4 + 2400*a *b*c**4*x**5 + 800*a*c**5*x**6 + 63*b**6*x**2 + 486*b**5*c*x**3 + 1703*b** 4*c**2*x**4 + 3320*b**3*c**3*x**5 + 3720*b**2*c**4*x**6 + 2240*b*c**5*x**7 + 560*c**6*x**8))/315