Integrand size = 26, antiderivative size = 18 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} \left (a+b x^3+c x^6\right )^{14} \] Output:
1/42*(c*x^6+b*x^3+a)^14
Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(18)=36\).
Time = 0.21 (sec) , antiderivative size = 233, normalized size of antiderivative = 12.94 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} x^3 \left (b+c x^3\right ) \left (14 a^{13}+91 a^{12} x^3 \left (b+c x^3\right )+364 a^{11} x^6 \left (b+c x^3\right )^2+1001 a^{10} x^9 \left (b+c x^3\right )^3+2002 a^9 x^{12} \left (b+c x^3\right )^4+3003 a^8 x^{15} \left (b+c x^3\right )^5+3432 a^7 x^{18} \left (b+c x^3\right )^6+3003 a^6 x^{21} \left (b+c x^3\right )^7+2002 a^5 x^{24} \left (b+c x^3\right )^8+1001 a^4 x^{27} \left (b+c x^3\right )^9+364 a^3 x^{30} \left (b+c x^3\right )^{10}+91 a^2 x^{33} \left (b+c x^3\right )^{11}+14 a x^{36} \left (b+c x^3\right )^{12}+x^{39} \left (b+c x^3\right )^{13}\right ) \] Input:
Integrate[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x]
Output:
(x^3*(b + c*x^3)*(14*a^13 + 91*a^12*x^3*(b + c*x^3) + 364*a^11*x^6*(b + c* x^3)^2 + 1001*a^10*x^9*(b + c*x^3)^3 + 2002*a^9*x^12*(b + c*x^3)^4 + 3003* a^8*x^15*(b + c*x^3)^5 + 3432*a^7*x^18*(b + c*x^3)^6 + 3003*a^6*x^21*(b + c*x^3)^7 + 2002*a^5*x^24*(b + c*x^3)^8 + 1001*a^4*x^27*(b + c*x^3)^9 + 364 *a^3*x^30*(b + c*x^3)^10 + 91*a^2*x^33*(b + c*x^3)^11 + 14*a*x^36*(b + c*x ^3)^12 + x^39*(b + c*x^3)^13))/42
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1798, 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.
\(\displaystyle \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx\) |
\(\Big \downarrow \) 1798 |
\(\displaystyle \frac {1}{3} \int \left (2 c x^3+b\right ) \left (c x^6+b x^3+a\right )^{13}dx^3\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {1}{42} \left (a+b x^3+c x^6\right )^{14}\) |
Input:
Int[x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x]
Output:
(a + b*x^3 + c*x^6)^14/42
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[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[(d + e*x)^q*(a + b *x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{14}}{42}\) | \(17\) |
gosper | \(\text {Expression too large to display}\) | \(1455\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1455\) |
orering | \(\text {Expression too large to display}\) | \(1455\) |
risch | \(\text {Expression too large to display}\) | \(1460\) |
Input:
int(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x,method=_RETURNVERBOSE)
Output:
1/42*(c*x^6+b*x^3+a)^14
Leaf count of result is larger than twice the leaf count of optimal. 1240 vs. \(2 (16) = 32\).
Time = 0.07 (sec) , antiderivative size = 1240, normalized size of antiderivative = 68.89 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x, algorithm="fricas")
Output:
1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 1/6*(13*b^2*c^12 + 2*a*c^13)*x^78 + 13/ 3*(2*b^3*c^11 + a*b*c^12)*x^75 + 13/6*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c ^12)*x^72 + 13/3*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^69 + 13/6*( 33*b^6*c^8 + 110*a*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^66 + 143/21*( 12*b^7*c^7 + 63*a*b^5*c^8 + 70*a^2*b^3*c^9 + 14*a^3*b*c^10)*x^63 + 143/6*( 3*b^8*c^6 + 24*a*b^6*c^7 + 45*a^2*b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)*x^6 0 + 143/3*(b^9*c^5 + 12*a*b^7*c^6 + 36*a^2*b^5*c^7 + 30*a^3*b^3*c^8 + 5*a^ 4*b*c^9)*x^57 + 143/6*(b^10*c^4 + 18*a*b^8*c^5 + 84*a^2*b^6*c^6 + 120*a^3* b^4*c^7 + 45*a^4*b^2*c^8 + 2*a^5*c^9)*x^54 + 13/3*(2*b^11*c^3 + 55*a*b^9*c ^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 + 99*a^5*b*c^8)*x ^51 + 13/6*(b^12*c^2 + 44*a*b^10*c^3 + 495*a^2*b^8*c^4 + 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 + 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^48 + 1/3*(b^13*c + 78 *a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 + 12 012*a^5*b^3*c^6 + 1716*a^6*b*c^7)*x^45 + 1/42*(b^14 + 182*a*b^12*c + 6006* a^2*b^10*c^2 + 60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a^5*b^4*c^5 + 84084*a^6*b^2*c^6 + 3432*a^7*c^7)*x^42 + 1/3*(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2 + 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^ 5 + 1716*a^7*b*c^6)*x^39 + 13/6*(a^2*b^12 + 44*a^3*b^10*c + 495*a^4*b^8*c^ 2 + 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 + 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^ 36 + 13/3*(2*a^3*b^11 + 55*a^4*b^9*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^...
Leaf count of result is larger than twice the leaf count of optimal. 1394 vs. \(2 (14) = 28\).
Time = 0.15 (sec) , antiderivative size = 1394, normalized size of antiderivative = 77.44 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x**2*(2*c*x**3+b)*(c*x**6+b*x**3+a)**13,x)
Output:
a**13*b*x**3/3 + b*c**13*x**81/3 + c**14*x**84/42 + x**78*(a*c**13/3 + 13* b**2*c**12/6) + x**75*(13*a*b*c**12/3 + 26*b**3*c**11/3) + x**72*(13*a**2* c**12/6 + 26*a*b**2*c**11 + 143*b**4*c**10/6) + x**69*(26*a**2*b*c**11 + 2 86*a*b**3*c**10/3 + 143*b**5*c**9/3) + x**66*(26*a**3*c**11/3 + 143*a**2*b **2*c**10 + 715*a*b**4*c**9/3 + 143*b**6*c**8/2) + x**63*(286*a**3*b*c**10 /3 + 1430*a**2*b**3*c**9/3 + 429*a*b**5*c**8 + 572*b**7*c**7/7) + x**60*(1 43*a**4*c**10/6 + 1430*a**3*b**2*c**9/3 + 2145*a**2*b**4*c**8/2 + 572*a*b* *6*c**7 + 143*b**8*c**6/2) + x**57*(715*a**4*b*c**9/3 + 1430*a**3*b**3*c** 8 + 1716*a**2*b**5*c**7 + 572*a*b**7*c**6 + 143*b**9*c**5/3) + x**54*(143* a**5*c**9/3 + 2145*a**4*b**2*c**8/2 + 2860*a**3*b**4*c**7 + 2002*a**2*b**6 *c**6 + 429*a*b**8*c**5 + 143*b**10*c**4/6) + x**51*(429*a**5*b*c**8 + 286 0*a**4*b**3*c**7 + 4004*a**3*b**5*c**6 + 1716*a**2*b**7*c**5 + 715*a*b**9* c**4/3 + 26*b**11*c**3/3) + x**48*(143*a**6*c**8/2 + 1716*a**5*b**2*c**7 + 5005*a**4*b**4*c**6 + 4004*a**3*b**6*c**5 + 2145*a**2*b**8*c**4/2 + 286*a *b**10*c**3/3 + 13*b**12*c**2/6) + x**45*(572*a**6*b*c**7 + 4004*a**5*b**3 *c**6 + 6006*a**4*b**5*c**5 + 2860*a**3*b**7*c**4 + 1430*a**2*b**9*c**3/3 + 26*a*b**11*c**2 + b**13*c/3) + x**42*(572*a**7*c**7/7 + 2002*a**6*b**2*c **6 + 6006*a**5*b**4*c**5 + 5005*a**4*b**6*c**4 + 1430*a**3*b**8*c**3 + 14 3*a**2*b**10*c**2 + 13*a*b**12*c/3 + b**14/42) + x**39*(572*a**7*b*c**6 + 4004*a**6*b**3*c**5 + 6006*a**5*b**5*c**4 + 2860*a**4*b**7*c**3 + 1430*...
Leaf count of result is larger than twice the leaf count of optimal. 1240 vs. \(2 (16) = 32\).
Time = 0.04 (sec) , antiderivative size = 1240, normalized size of antiderivative = 68.89 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x, algorithm="maxima")
Output:
1/42*c^14*x^84 + 1/3*b*c^13*x^81 + 1/6*(13*b^2*c^12 + 2*a*c^13)*x^78 + 13/ 3*(2*b^3*c^11 + a*b*c^12)*x^75 + 13/6*(11*b^4*c^10 + 12*a*b^2*c^11 + a^2*c ^12)*x^72 + 13/3*(11*b^5*c^9 + 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^69 + 13/6*( 33*b^6*c^8 + 110*a*b^4*c^9 + 66*a^2*b^2*c^10 + 4*a^3*c^11)*x^66 + 143/21*( 12*b^7*c^7 + 63*a*b^5*c^8 + 70*a^2*b^3*c^9 + 14*a^3*b*c^10)*x^63 + 143/6*( 3*b^8*c^6 + 24*a*b^6*c^7 + 45*a^2*b^4*c^8 + 20*a^3*b^2*c^9 + a^4*c^10)*x^6 0 + 143/3*(b^9*c^5 + 12*a*b^7*c^6 + 36*a^2*b^5*c^7 + 30*a^3*b^3*c^8 + 5*a^ 4*b*c^9)*x^57 + 143/6*(b^10*c^4 + 18*a*b^8*c^5 + 84*a^2*b^6*c^6 + 120*a^3* b^4*c^7 + 45*a^4*b^2*c^8 + 2*a^5*c^9)*x^54 + 13/3*(2*b^11*c^3 + 55*a*b^9*c ^4 + 396*a^2*b^7*c^5 + 924*a^3*b^5*c^6 + 660*a^4*b^3*c^7 + 99*a^5*b*c^8)*x ^51 + 13/6*(b^12*c^2 + 44*a*b^10*c^3 + 495*a^2*b^8*c^4 + 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 + 792*a^5*b^2*c^7 + 33*a^6*c^8)*x^48 + 1/3*(b^13*c + 78 *a*b^11*c^2 + 1430*a^2*b^9*c^3 + 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 + 12 012*a^5*b^3*c^6 + 1716*a^6*b*c^7)*x^45 + 1/42*(b^14 + 182*a*b^12*c + 6006* a^2*b^10*c^2 + 60060*a^3*b^8*c^3 + 210210*a^4*b^6*c^4 + 252252*a^5*b^4*c^5 + 84084*a^6*b^2*c^6 + 3432*a^7*c^7)*x^42 + 1/3*(a*b^13 + 78*a^2*b^11*c + 1430*a^3*b^9*c^2 + 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 + 12012*a^6*b^3*c^ 5 + 1716*a^7*b*c^6)*x^39 + 13/6*(a^2*b^12 + 44*a^3*b^10*c + 495*a^4*b^8*c^ 2 + 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 + 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^ 36 + 13/3*(2*a^3*b^11 + 55*a^4*b^9*c + 396*a^5*b^7*c^2 + 924*a^6*b^5*c^...
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (16) = 32\).
Time = 0.16 (sec) , antiderivative size = 246, normalized size of antiderivative = 13.67 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\frac {1}{42} \, {\left (c x^{6} + b x^{3}\right )}^{14} + \frac {1}{3} \, {\left (c x^{6} + b x^{3}\right )}^{13} a + \frac {13}{6} \, {\left (c x^{6} + b x^{3}\right )}^{12} a^{2} + \frac {26}{3} \, {\left (c x^{6} + b x^{3}\right )}^{11} a^{3} + \frac {143}{6} \, {\left (c x^{6} + b x^{3}\right )}^{10} a^{4} + \frac {143}{3} \, {\left (c x^{6} + b x^{3}\right )}^{9} a^{5} + \frac {143}{2} \, {\left (c x^{6} + b x^{3}\right )}^{8} a^{6} + \frac {572}{7} \, {\left (c x^{6} + b x^{3}\right )}^{7} a^{7} + \frac {143}{2} \, {\left (c x^{6} + b x^{3}\right )}^{6} a^{8} + \frac {143}{3} \, {\left (c x^{6} + b x^{3}\right )}^{5} a^{9} + \frac {143}{6} \, {\left (c x^{6} + b x^{3}\right )}^{4} a^{10} + \frac {26}{3} \, {\left (c x^{6} + b x^{3}\right )}^{3} a^{11} + \frac {13}{6} \, {\left (c x^{6} + b x^{3}\right )}^{2} a^{12} + \frac {1}{3} \, {\left (c x^{6} + b x^{3}\right )} a^{13} \] Input:
integrate(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x, algorithm="giac")
Output:
1/42*(c*x^6 + b*x^3)^14 + 1/3*(c*x^6 + b*x^3)^13*a + 13/6*(c*x^6 + b*x^3)^ 12*a^2 + 26/3*(c*x^6 + b*x^3)^11*a^3 + 143/6*(c*x^6 + b*x^3)^10*a^4 + 143/ 3*(c*x^6 + b*x^3)^9*a^5 + 143/2*(c*x^6 + b*x^3)^8*a^6 + 572/7*(c*x^6 + b*x ^3)^7*a^7 + 143/2*(c*x^6 + b*x^3)^6*a^8 + 143/3*(c*x^6 + b*x^3)^5*a^9 + 14 3/6*(c*x^6 + b*x^3)^4*a^10 + 26/3*(c*x^6 + b*x^3)^3*a^11 + 13/6*(c*x^6 + b *x^3)^2*a^12 + 1/3*(c*x^6 + b*x^3)*a^13
Time = 23.37 (sec) , antiderivative size = 1210, normalized size of antiderivative = 67.22 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx=\text {Too large to display} \] Input:
int(x^2*(b + 2*c*x^3)*(a + b*x^3 + c*x^6)^13,x)
Output:
x^36*((13*a^2*b^12)/6 + (143*a^8*c^6)/2 + (286*a^3*b^10*c)/3 + (2145*a^4*b ^8*c^2)/2 + 4004*a^5*b^6*c^3 + 5005*a^6*b^4*c^4 + 1716*a^7*b^2*c^5) + x^48 *((143*a^6*c^8)/2 + (13*b^12*c^2)/6 + (286*a*b^10*c^3)/3 + (2145*a^2*b^8*c ^4)/2 + 4004*a^3*b^6*c^5 + 5005*a^4*b^4*c^6 + 1716*a^5*b^2*c^7) + x^39*((a *b^13)/3 + 26*a^2*b^11*c + 572*a^7*b*c^6 + (1430*a^3*b^9*c^2)/3 + 2860*a^4 *b^7*c^3 + 6006*a^5*b^5*c^4 + 4004*a^6*b^3*c^5) + x^45*((b^13*c)/3 + 26*a* b^11*c^2 + 572*a^6*b*c^7 + (1430*a^2*b^9*c^3)/3 + 2860*a^3*b^7*c^4 + 6006* a^4*b^5*c^5 + 4004*a^5*b^3*c^6) + x^18*((143*a^8*b^6)/2 + (26*a^11*c^3)/3 + (715*a^9*b^4*c)/3 + 143*a^10*b^2*c^2) + x^66*((26*a^3*c^11)/3 + (143*b^6 *c^8)/2 + (715*a*b^4*c^9)/3 + 143*a^2*b^2*c^10) + x^30*((143*a^4*b^10)/6 + (143*a^9*c^5)/3 + 429*a^5*b^8*c + 2002*a^6*b^6*c^2 + 2860*a^7*b^4*c^3 + ( 2145*a^8*b^2*c^4)/2) + x^54*((143*a^5*c^9)/3 + (143*b^10*c^4)/6 + 429*a*b^ 8*c^5 + 2002*a^2*b^6*c^6 + 2860*a^3*b^4*c^7 + (2145*a^4*b^2*c^8)/2) + x^42 *(b^14/42 + (572*a^7*c^7)/7 + 143*a^2*b^10*c^2 + 1430*a^3*b^8*c^3 + 5005*a ^4*b^6*c^4 + 6006*a^5*b^4*c^5 + 2002*a^6*b^2*c^6 + (13*a*b^12*c)/3) + x^24 *((143*a^6*b^8)/2 + (143*a^10*c^4)/6 + 572*a^7*b^6*c + (2145*a^8*b^4*c^2)/ 2 + (1430*a^9*b^2*c^3)/3) + x^60*((143*a^4*c^10)/6 + (143*b^8*c^6)/2 + 572 *a*b^6*c^7 + (2145*a^2*b^4*c^8)/2 + (1430*a^3*b^2*c^9)/3) + (c^14*x^84)/42 + x^6*((a^13*c)/3 + (13*a^12*b^2)/6) + (13*a^10*x^12*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/6 + (13*c^10*x^72*(11*b^4 + a^2*c^2 + 12*a*b^2*c))/6 + (a^...
Time = 0.15 (sec) , antiderivative size = 1454, normalized size of antiderivative = 80.78 \[ \int x^2 \left (b+2 c x^3\right ) \left (a+b x^3+c x^6\right )^{13} \, dx =\text {Too large to display} \] Input:
int(x^2*(2*c*x^3+b)*(c*x^6+b*x^3+a)^13,x)
Output:
(x**3*(14*a**13*b + 14*a**13*c*x**3 + 91*a**12*b**2*x**3 + 182*a**12*b*c*x **6 + 91*a**12*c**2*x**9 + 364*a**11*b**3*x**6 + 1092*a**11*b**2*c*x**9 + 1092*a**11*b*c**2*x**12 + 364*a**11*c**3*x**15 + 1001*a**10*b**4*x**9 + 40 04*a**10*b**3*c*x**12 + 6006*a**10*b**2*c**2*x**15 + 4004*a**10*b*c**3*x** 18 + 1001*a**10*c**4*x**21 + 2002*a**9*b**5*x**12 + 10010*a**9*b**4*c*x**1 5 + 20020*a**9*b**3*c**2*x**18 + 20020*a**9*b**2*c**3*x**21 + 10010*a**9*b *c**4*x**24 + 2002*a**9*c**5*x**27 + 3003*a**8*b**6*x**15 + 18018*a**8*b** 5*c*x**18 + 45045*a**8*b**4*c**2*x**21 + 60060*a**8*b**3*c**3*x**24 + 4504 5*a**8*b**2*c**4*x**27 + 18018*a**8*b*c**5*x**30 + 3003*a**8*c**6*x**33 + 3432*a**7*b**7*x**18 + 24024*a**7*b**6*c*x**21 + 72072*a**7*b**5*c**2*x**2 4 + 120120*a**7*b**4*c**3*x**27 + 120120*a**7*b**3*c**4*x**30 + 72072*a**7 *b**2*c**5*x**33 + 24024*a**7*b*c**6*x**36 + 3432*a**7*c**7*x**39 + 3003*a **6*b**8*x**21 + 24024*a**6*b**7*c*x**24 + 84084*a**6*b**6*c**2*x**27 + 16 8168*a**6*b**5*c**3*x**30 + 210210*a**6*b**4*c**4*x**33 + 168168*a**6*b**3 *c**5*x**36 + 84084*a**6*b**2*c**6*x**39 + 24024*a**6*b*c**7*x**42 + 3003* a**6*c**8*x**45 + 2002*a**5*b**9*x**24 + 18018*a**5*b**8*c*x**27 + 72072*a **5*b**7*c**2*x**30 + 168168*a**5*b**6*c**3*x**33 + 252252*a**5*b**5*c**4* x**36 + 252252*a**5*b**4*c**5*x**39 + 168168*a**5*b**3*c**6*x**42 + 72072* a**5*b**2*c**7*x**45 + 18018*a**5*b*c**8*x**48 + 2002*a**5*c**9*x**51 + 10 01*a**4*b**10*x**27 + 10010*a**4*b**9*c*x**30 + 45045*a**4*b**8*c**2*x*...