Integrand size = 26, antiderivative size = 20 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\frac {1}{28} \left (a-b x^2-c x^4\right )^{14} \] Output:
1/28*(-c*x^4-b*x^2+a)^14
Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(20)=40\).
Time = 0.22 (sec) , antiderivative size = 233, normalized size of antiderivative = 11.65 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\frac {1}{28} x^2 \left (b+c x^2\right ) \left (-14 a^{13}+91 a^{12} x^2 \left (b+c x^2\right )-364 a^{11} x^4 \left (b+c x^2\right )^2+1001 a^{10} x^6 \left (b+c x^2\right )^3-2002 a^9 x^8 \left (b+c x^2\right )^4+3003 a^8 x^{10} \left (b+c x^2\right )^5-3432 a^7 x^{12} \left (b+c x^2\right )^6+3003 a^6 x^{14} \left (b+c x^2\right )^7-2002 a^5 x^{16} \left (b+c x^2\right )^8+1001 a^4 x^{18} \left (b+c x^2\right )^9-364 a^3 x^{20} \left (b+c x^2\right )^{10}+91 a^2 x^{22} \left (b+c x^2\right )^{11}-14 a x^{24} \left (b+c x^2\right )^{12}+x^{26} \left (b+c x^2\right )^{13}\right ) \] Input:
Integrate[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^13,x]
Output:
(x^2*(b + c*x^2)*(-14*a^13 + 91*a^12*x^2*(b + c*x^2) - 364*a^11*x^4*(b + c *x^2)^2 + 1001*a^10*x^6*(b + c*x^2)^3 - 2002*a^9*x^8*(b + c*x^2)^4 + 3003* a^8*x^10*(b + c*x^2)^5 - 3432*a^7*x^12*(b + c*x^2)^6 + 3003*a^6*x^14*(b + c*x^2)^7 - 2002*a^5*x^16*(b + c*x^2)^8 + 1001*a^4*x^18*(b + c*x^2)^9 - 364 *a^3*x^20*(b + c*x^2)^10 + 91*a^2*x^22*(b + c*x^2)^11 - 14*a*x^24*(b + c*x ^2)^12 + x^26*(b + c*x^2)^13))/28
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1576, 25, 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.
\(\displaystyle \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{2} \int -\left (\left (2 c x^2+b\right ) \left (-c x^4-b x^2+a\right )^{13}\right )dx^2\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \left (2 c x^2+b\right ) \left (-c x^4-b x^2+a\right )^{13}dx^2\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {1}{28} \left (a-b x^2-c x^4\right )^{14}\) |
Input:
Int[x*(b + 2*c*x^2)*(-a + b*x^2 + c*x^4)^13,x]
Output:
(a - b*x^2 - c*x^4)^14/28
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_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\left (c \,x^{4}+b \,x^{2}-a \right )^{14}}{28}\) | \(19\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1455\) |
gosper | \(\text {Expression too large to display}\) | \(1457\) |
risch | \(\text {Expression too large to display}\) | \(1460\) |
orering | \(\text {Expression too large to display}\) | \(1489\) |
Input:
int(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^13,x,method=_RETURNVERBOSE)
Output:
1/28*(c*x^4+b*x^2-a)^14
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (18) = 36\).
Time = 0.07 (sec) , antiderivative size = 1242, normalized size of antiderivative = 62.10 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^13,x, algorithm="fricas")
Output:
1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 1/4*(13*b^2*c^12 - 2*a*c^13)*x^52 + 13/ 2*(2*b^3*c^11 - a*b*c^12)*x^50 + 13/4*(11*b^4*c^10 - 12*a*b^2*c^11 + a^2*c ^12)*x^48 + 13/2*(11*b^5*c^9 - 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^46 + 13/4*( 33*b^6*c^8 - 110*a*b^4*c^9 + 66*a^2*b^2*c^10 - 4*a^3*c^11)*x^44 + 143/14*( 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^42 + 143/4*( 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^4 0 + 143/2*(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^38 + 143/4*(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^36 + 13/2*(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 ^34 + 13/4*(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^32 + 1/2*(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^30 + 1/28*(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^28 - 1/2*(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^26 + 13/4*(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^ 24 - 13/2*(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. 1384 vs. \(2 (14) = 28\).
Time = 0.15 (sec) , antiderivative size = 1384, normalized size of antiderivative = 69.20 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x*(2*c*x**2+b)*(c*x**4+b*x**2-a)**13,x)
Output:
-a**13*b*x**2/2 + b*c**13*x**54/2 + c**14*x**56/28 + x**52*(-a*c**13/2 + 1 3*b**2*c**12/4) + x**50*(-13*a*b*c**12/2 + 13*b**3*c**11) + x**48*(13*a**2 *c**12/4 - 39*a*b**2*c**11 + 143*b**4*c**10/4) + x**46*(39*a**2*b*c**11 - 143*a*b**3*c**10 + 143*b**5*c**9/2) + x**44*(-13*a**3*c**11 + 429*a**2*b** 2*c**10/2 - 715*a*b**4*c**9/2 + 429*b**6*c**8/4) + x**42*(-143*a**3*b*c**1 0 + 715*a**2*b**3*c**9 - 1287*a*b**5*c**8/2 + 858*b**7*c**7/7) + x**40*(14 3*a**4*c**10/4 - 715*a**3*b**2*c**9 + 6435*a**2*b**4*c**8/4 - 858*a*b**6*c **7 + 429*b**8*c**6/4) + x**38*(715*a**4*b*c**9/2 - 2145*a**3*b**3*c**8 + 2574*a**2*b**5*c**7 - 858*a*b**7*c**6 + 143*b**9*c**5/2) + x**36*(-143*a** 5*c**9/2 + 6435*a**4*b**2*c**8/4 - 4290*a**3*b**4*c**7 + 3003*a**2*b**6*c* *6 - 1287*a*b**8*c**5/2 + 143*b**10*c**4/4) + x**34*(-1287*a**5*b*c**8/2 + 4290*a**4*b**3*c**7 - 6006*a**3*b**5*c**6 + 2574*a**2*b**7*c**5 - 715*a*b **9*c**4/2 + 13*b**11*c**3) + x**32*(429*a**6*c**8/4 - 2574*a**5*b**2*c**7 + 15015*a**4*b**4*c**6/2 - 6006*a**3*b**6*c**5 + 6435*a**2*b**8*c**4/4 - 143*a*b**10*c**3 + 13*b**12*c**2/4) + x**30*(858*a**6*b*c**7 - 6006*a**5*b **3*c**6 + 9009*a**4*b**5*c**5 - 4290*a**3*b**7*c**4 + 715*a**2*b**9*c**3 - 39*a*b**11*c**2 + b**13*c/2) + x**28*(-858*a**7*c**7/7 + 3003*a**6*b**2* c**6 - 9009*a**5*b**4*c**5 + 15015*a**4*b**6*c**4/2 - 2145*a**3*b**8*c**3 + 429*a**2*b**10*c**2/2 - 13*a*b**12*c/2 + b**14/28) + x**26*(-858*a**7*b* c**6 + 6006*a**6*b**3*c**5 - 9009*a**5*b**5*c**4 + 4290*a**4*b**7*c**3 ...
Leaf count of result is larger than twice the leaf count of optimal. 1242 vs. \(2 (18) = 36\).
Time = 0.04 (sec) , antiderivative size = 1242, normalized size of antiderivative = 62.10 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^13,x, algorithm="maxima")
Output:
1/28*c^14*x^56 + 1/2*b*c^13*x^54 + 1/4*(13*b^2*c^12 - 2*a*c^13)*x^52 + 13/ 2*(2*b^3*c^11 - a*b*c^12)*x^50 + 13/4*(11*b^4*c^10 - 12*a*b^2*c^11 + a^2*c ^12)*x^48 + 13/2*(11*b^5*c^9 - 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^46 + 13/4*( 33*b^6*c^8 - 110*a*b^4*c^9 + 66*a^2*b^2*c^10 - 4*a^3*c^11)*x^44 + 143/14*( 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^42 + 143/4*( 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^4 0 + 143/2*(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^38 + 143/4*(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^36 + 13/2*(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 ^34 + 13/4*(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^32 + 1/2*(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^30 + 1/28*(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^28 - 1/2*(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^26 + 13/4*(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^ 24 - 13/2*(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. \(2 (18) = 36\).
Time = 0.15 (sec) , antiderivative size = 246, normalized size of antiderivative = 12.30 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\frac {1}{28} \, {\left (c x^{4} + b x^{2}\right )}^{14} - \frac {1}{2} \, {\left (c x^{4} + b x^{2}\right )}^{13} a + \frac {13}{4} \, {\left (c x^{4} + b x^{2}\right )}^{12} a^{2} - 13 \, {\left (c x^{4} + b x^{2}\right )}^{11} a^{3} + \frac {143}{4} \, {\left (c x^{4} + b x^{2}\right )}^{10} a^{4} - \frac {143}{2} \, {\left (c x^{4} + b x^{2}\right )}^{9} a^{5} + \frac {429}{4} \, {\left (c x^{4} + b x^{2}\right )}^{8} a^{6} - \frac {858}{7} \, {\left (c x^{4} + b x^{2}\right )}^{7} a^{7} + \frac {429}{4} \, {\left (c x^{4} + b x^{2}\right )}^{6} a^{8} - \frac {143}{2} \, {\left (c x^{4} + b x^{2}\right )}^{5} a^{9} + \frac {143}{4} \, {\left (c x^{4} + b x^{2}\right )}^{4} a^{10} - 13 \, {\left (c x^{4} + b x^{2}\right )}^{3} a^{11} + \frac {13}{4} \, {\left (c x^{4} + b x^{2}\right )}^{2} a^{12} - \frac {1}{2} \, {\left (c x^{4} + b x^{2}\right )} a^{13} \] Input:
integrate(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^13,x, algorithm="giac")
Output:
1/28*(c*x^4 + b*x^2)^14 - 1/2*(c*x^4 + b*x^2)^13*a + 13/4*(c*x^4 + b*x^2)^ 12*a^2 - 13*(c*x^4 + b*x^2)^11*a^3 + 143/4*(c*x^4 + b*x^2)^10*a^4 - 143/2* (c*x^4 + b*x^2)^9*a^5 + 429/4*(c*x^4 + b*x^2)^8*a^6 - 858/7*(c*x^4 + b*x^2 )^7*a^7 + 429/4*(c*x^4 + b*x^2)^6*a^8 - 143/2*(c*x^4 + b*x^2)^5*a^9 + 143/ 4*(c*x^4 + b*x^2)^4*a^10 - 13*(c*x^4 + b*x^2)^3*a^11 + 13/4*(c*x^4 + b*x^2 )^2*a^12 - 1/2*(c*x^4 + b*x^2)*a^13
Time = 23.71 (sec) , antiderivative size = 1214, normalized size of antiderivative = 60.70 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx=\text {Too large to display} \] Input:
int(x*(b + 2*c*x^2)*(b*x^2 - a + c*x^4)^13,x)
Output:
x^24*((13*a^2*b^12)/4 + (429*a^8*c^6)/4 - 143*a^3*b^10*c + (6435*a^4*b^8*c ^2)/4 - 6006*a^5*b^6*c^3 + (15015*a^6*b^4*c^4)/2 - 2574*a^7*b^2*c^5) + x^3 2*((429*a^6*c^8)/4 + (13*b^12*c^2)/4 - 143*a*b^10*c^3 + (6435*a^2*b^8*c^4) /4 - 6006*a^3*b^6*c^5 + (15015*a^4*b^4*c^6)/2 - 2574*a^5*b^2*c^7) - x^26*( (a*b^13)/2 - 39*a^2*b^11*c + 858*a^7*b*c^6 + 715*a^3*b^9*c^2 - 4290*a^4*b^ 7*c^3 + 9009*a^5*b^5*c^4 - 6006*a^6*b^3*c^5) + x^30*((b^13*c)/2 - 39*a*b^1 1*c^2 + 858*a^6*b*c^7 + 715*a^2*b^9*c^3 - 4290*a^3*b^7*c^4 + 9009*a^4*b^5* c^5 - 6006*a^5*b^3*c^6) + x^12*((429*a^8*b^6)/4 - 13*a^11*c^3 - (715*a^9*b ^4*c)/2 + (429*a^10*b^2*c^2)/2) - x^44*(13*a^3*c^11 - (429*b^6*c^8)/4 + (7 15*a*b^4*c^9)/2 - (429*a^2*b^2*c^10)/2) + x^20*((143*a^4*b^10)/4 - (143*a^ 9*c^5)/2 - (1287*a^5*b^8*c)/2 + 3003*a^6*b^6*c^2 - 4290*a^7*b^4*c^3 + (643 5*a^8*b^2*c^4)/4) - x^36*((143*a^5*c^9)/2 - (143*b^10*c^4)/4 + (1287*a*b^8 *c^5)/2 - 3003*a^2*b^6*c^6 + 4290*a^3*b^4*c^7 - (6435*a^4*b^2*c^8)/4) + x^ 28*(b^14/28 - (858*a^7*c^7)/7 + (429*a^2*b^10*c^2)/2 - 2145*a^3*b^8*c^3 + (15015*a^4*b^6*c^4)/2 - 9009*a^5*b^4*c^5 + 3003*a^6*b^2*c^6 - (13*a*b^12*c )/2) + x^16*((429*a^6*b^8)/4 + (143*a^10*c^4)/4 - 858*a^7*b^6*c + (6435*a^ 8*b^4*c^2)/4 - 715*a^9*b^2*c^3) + x^40*((143*a^4*c^10)/4 + (429*b^8*c^6)/4 - 858*a*b^6*c^7 + (6435*a^2*b^4*c^8)/4 - 715*a^3*b^2*c^9) + (c^14*x^56)/2 8 - x^4*((a^13*c)/2 - (13*a^12*b^2)/4) + (13*a^10*x^8*(11*b^4 + a^2*c^2 - 12*a*b^2*c))/4 + (13*c^10*x^48*(11*b^4 + a^2*c^2 - 12*a*b^2*c))/4 - (a^...
Time = 0.15 (sec) , antiderivative size = 1454, normalized size of antiderivative = 72.70 \[ \int x \left (b+2 c x^2\right ) \left (-a+b x^2+c x^4\right )^{13} \, dx =\text {Too large to display} \] Input:
int(x*(2*c*x^2+b)*(c*x^4+b*x^2-a)^13,x)
Output:
(x**2*( - 14*a**13*b - 14*a**13*c*x**2 + 91*a**12*b**2*x**2 + 182*a**12*b* c*x**4 + 91*a**12*c**2*x**6 - 364*a**11*b**3*x**4 - 1092*a**11*b**2*c*x**6 - 1092*a**11*b*c**2*x**8 - 364*a**11*c**3*x**10 + 1001*a**10*b**4*x**6 + 4004*a**10*b**3*c*x**8 + 6006*a**10*b**2*c**2*x**10 + 4004*a**10*b*c**3*x* *12 + 1001*a**10*c**4*x**14 - 2002*a**9*b**5*x**8 - 10010*a**9*b**4*c*x**1 0 - 20020*a**9*b**3*c**2*x**12 - 20020*a**9*b**2*c**3*x**14 - 10010*a**9*b *c**4*x**16 - 2002*a**9*c**5*x**18 + 3003*a**8*b**6*x**10 + 18018*a**8*b** 5*c*x**12 + 45045*a**8*b**4*c**2*x**14 + 60060*a**8*b**3*c**3*x**16 + 4504 5*a**8*b**2*c**4*x**18 + 18018*a**8*b*c**5*x**20 + 3003*a**8*c**6*x**22 - 3432*a**7*b**7*x**12 - 24024*a**7*b**6*c*x**14 - 72072*a**7*b**5*c**2*x**1 6 - 120120*a**7*b**4*c**3*x**18 - 120120*a**7*b**3*c**4*x**20 - 72072*a**7 *b**2*c**5*x**22 - 24024*a**7*b*c**6*x**24 - 3432*a**7*c**7*x**26 + 3003*a **6*b**8*x**14 + 24024*a**6*b**7*c*x**16 + 84084*a**6*b**6*c**2*x**18 + 16 8168*a**6*b**5*c**3*x**20 + 210210*a**6*b**4*c**4*x**22 + 168168*a**6*b**3 *c**5*x**24 + 84084*a**6*b**2*c**6*x**26 + 24024*a**6*b*c**7*x**28 + 3003* a**6*c**8*x**30 - 2002*a**5*b**9*x**16 - 18018*a**5*b**8*c*x**18 - 72072*a **5*b**7*c**2*x**20 - 168168*a**5*b**6*c**3*x**22 - 252252*a**5*b**5*c**4* x**24 - 252252*a**5*b**4*c**5*x**26 - 168168*a**5*b**3*c**6*x**28 - 72072* a**5*b**2*c**7*x**30 - 18018*a**5*b*c**8*x**32 - 2002*a**5*c**9*x**34 + 10 01*a**4*b**10*x**18 + 10010*a**4*b**9*c*x**20 + 45045*a**4*b**8*c**2*x*...