\(\int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} (1-c^2 x^2)^6} \, dx\) [166]

Optimal result

Integrand size = 41, antiderivative size = 139 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=-\frac {5 x}{11 (-1+c x)^{11/2} (1+c x)^{11/2}}+\frac {2 x}{33 (-1+c x)^{9/2} (1+c x)^{9/2}}-\frac {16 x}{231 (-1+c x)^{7/2} (1+c x)^{7/2}}+\frac {32 x}{385 (-1+c x)^{5/2} (1+c x)^{5/2}}-\frac {128 x}{1155 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {256 x}{1155 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:

-5/11*x/(c*x-1)^(11/2)/(c*x+1)^(11/2)+2/33*x/(c*x-1)^(9/2)/(c*x+1)^(9/2)-1 
6/231*x/(c*x-1)^(7/2)/(c*x+1)^(7/2)+32/385*x/(c*x-1)^(5/2)/(c*x+1)^(5/2)-1 
28/1155*x/(c*x-1)^(3/2)/(c*x+1)^(3/2)+256/1155*x/(c*x-1)^(1/2)/(c*x+1)^(1/ 
2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.47 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\frac {x \left (-1155+2310 c^2 x^2-3696 c^4 x^4+3168 c^6 x^6-1408 c^8 x^8+256 c^{10} x^{10}\right )}{1155 (-1+c x)^{11/2} (1+c x)^{11/2}} \] Input:

Integrate[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^6),x 
]
 

Output:

(x*(-1155 + 2310*c^2*x^2 - 3696*c^4*x^4 + 3168*c^6*x^6 - 1408*c^8*x^8 + 25 
6*c^10*x^10))/(1155*(-1 + c*x)^(11/2)*(1 + c*x)^(11/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {2003, 35, 645, 42, 42, 42, 42, 41}

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.

\(\displaystyle \int \frac {4 c^2 x^2+1}{\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^6} \, dx\)

\(\Big \downarrow \) 2003

\(\displaystyle \int \frac {4 c^2 x^2+1}{(-c x-1)^6 (c x-1)^{13/2} \sqrt {c x+1}}dx\)

\(\Big \downarrow \) 35

\(\displaystyle \int \frac {4 c^2 x^2+1}{(c x-1)^{13/2} (c x+1)^{13/2}}dx\)

\(\Big \downarrow \) 645

\(\displaystyle -\frac {6}{11} \int \frac {1}{(c x-1)^{11/2} (c x+1)^{11/2}}dx-\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle -\frac {6}{11} \left (-\frac {8}{9} \int \frac {1}{(c x-1)^{9/2} (c x+1)^{9/2}}dx-\frac {x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\right )-\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle -\frac {6}{11} \left (-\frac {8}{9} \left (-\frac {6}{7} \int \frac {1}{(c x-1)^{7/2} (c x+1)^{7/2}}dx-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )-\frac {x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\right )-\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle -\frac {6}{11} \left (-\frac {8}{9} \left (-\frac {6}{7} \left (-\frac {4}{5} \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )-\frac {x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\right )-\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}\)

\(\Big \downarrow \) 42

\(\displaystyle -\frac {6}{11} \left (-\frac {8}{9} \left (-\frac {6}{7} \left (-\frac {4}{5} \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\right )-\frac {x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}\right )-\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}\)

\(\Big \downarrow \) 41

\(\displaystyle -\frac {5 x}{11 (c x-1)^{11/2} (c x+1)^{11/2}}-\frac {6}{11} \left (-\frac {x}{9 (c x-1)^{9/2} (c x+1)^{9/2}}-\frac {8}{9} \left (-\frac {x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}-\frac {6}{7} \left (-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}-\frac {4}{5} \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )\right )\right )\)

Input:

Int[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^6),x]
 

Output:

(-5*x)/(11*(-1 + c*x)^(11/2)*(1 + c*x)^(11/2)) - (6*(-1/9*x/((-1 + c*x)^(9 
/2)*(1 + c*x)^(9/2)) - (8*(-1/7*x/((-1 + c*x)^(7/2)*(1 + c*x)^(7/2)) - (6* 
(-1/5*x/((-1 + c*x)^(5/2)*(1 + c*x)^(5/2)) - (4*(-1/3*x/((-1 + c*x)^(3/2)* 
(1 + c*x)^(3/2)) + (2*x)/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/5))/7))/9))/11
 

Defintions of rubi rules used

rule 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 41

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S 
imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq 
Q[b*c + a*d, 0]
 

rule 42

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- 
x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 
3)/(2*a*c*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
 

rule 645

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2), x_Symbol] :> Simp[(b*c*e - a*d*f)*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 
1)/(2*c*d*e*f*(m + 1))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(2*c*d*e*f*(m 
+ 1))   Int[(c + d*x)^(m + 1)*(e + f*x)^(n + 1), x], x] /; FreeQ[{a, b, c, 
d, e, f, m, n}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && LtQ[m, -1]
 

rule 2003

Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : 
> Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} 
, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && 
  !IntegerQ[n]))
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.51

method	result	size
gosper	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, x \left (256 c^{10} x^{10}-1408 c^{8} x^{8}+3168 c^{6} x^{6}-3696 c^{4} x^{4}+2310 c^{2} x^{2}-1155\right )}{1155 \left (c^{2} x^{2}-1\right )^{6}}\)	\(71\)
default	\(\frac {\left (256 c^{10} x^{10}-1408 c^{8} x^{8}+3168 c^{6} x^{6}-3696 c^{4} x^{4}+2310 c^{2} x^{2}-1155\right ) x}{1155 \left (c^{2} x^{2}-1\right )^{5} \sqrt {c x +1}\, \sqrt {c x -1}}\)	\(71\)
orering	\(\frac {x \left (256 c^{10} x^{10}-1408 c^{8} x^{8}+3168 c^{6} x^{6}-3696 c^{4} x^{4}+2310 c^{2} x^{2}-1155\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{1155 \left (-c^{2} x^{2}+1\right )^{6}}\)	\(72\)

Input:

int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^6,x,method=_RET 
URNVERBOSE)
 

Output:

1/1155*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*(256*c^10*x^10-1408*c^8*x^8+3168*c^6* 
x^6-3696*c^4*x^4+2310*c^2*x^2-1155)/(c^2*x^2-1)^6
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.17 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\frac {256 \, c^{12} x^{12} - 1536 \, c^{10} x^{10} + 3840 \, c^{8} x^{8} - 5120 \, c^{6} x^{6} + 3840 \, c^{4} x^{4} - 1536 \, c^{2} x^{2} + {\left (256 \, c^{11} x^{11} - 1408 \, c^{9} x^{9} + 3168 \, c^{7} x^{7} - 3696 \, c^{5} x^{5} + 2310 \, c^{3} x^{3} - 1155 \, c x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 256}{1155 \, {\left (c^{13} x^{12} - 6 \, c^{11} x^{10} + 15 \, c^{9} x^{8} - 20 \, c^{7} x^{6} + 15 \, c^{5} x^{4} - 6 \, c^{3} x^{2} + c\right )}} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^6,x, algo 
rithm="fricas")
 

Output:

1/1155*(256*c^12*x^12 - 1536*c^10*x^10 + 3840*c^8*x^8 - 5120*c^6*x^6 + 384 
0*c^4*x^4 - 1536*c^2*x^2 + (256*c^11*x^11 - 1408*c^9*x^9 + 3168*c^7*x^7 - 
3696*c^5*x^5 + 2310*c^3*x^3 - 1155*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + 256) 
/(c^13*x^12 - 6*c^11*x^10 + 15*c^9*x^8 - 20*c^7*x^6 + 15*c^5*x^4 - 6*c^3*x 
^2 + c)
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\text {Timed out} \] Input:

integrate((4*c**2*x**2+1)/(c*x-1)**(1/2)/(c*x+1)**(1/2)/(-c**2*x**2+1)**6, 
x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.61 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\frac {256 \, x}{1155 \, \sqrt {c^{2} x^{2} - 1}} - \frac {128 \, x}{1155 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {3}{2}}} + \frac {32 \, x}{385 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {5}{2}}} - \frac {16 \, x}{231 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {7}{2}}} + \frac {2 \, x}{33 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {9}{2}}} - \frac {5 \, x}{11 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {11}{2}}} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^6,x, algo 
rithm="maxima")
 

Output:

256/1155*x/sqrt(c^2*x^2 - 1) - 128/1155*x/(c^2*x^2 - 1)^(3/2) + 32/385*x/( 
c^2*x^2 - 1)^(5/2) - 16/231*x/(c^2*x^2 - 1)^(7/2) + 2/33*x/(c^2*x^2 - 1)^( 
9/2) - 5/11*x/(c^2*x^2 - 1)^(11/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (103) = 206\).

Time = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.31 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\frac {{\left ({\left ({\left ({\left (c x + 1\right )} {\left ({\left (c x + 1\right )} {\left (\frac {65536 \, {\left (c x + 1\right )}}{c} - \frac {680471}{c}\right )} + \frac {2839782}{c}\right )} - \frac {5963958}{c}\right )} {\left (c x + 1\right )} + \frac {6320160}{c}\right )} {\left (c x + 1\right )} - \frac {2716560}{c}\right )} \sqrt {c x + 1}}{591360 \, {\left (c x - 1\right )}^{\frac {11}{2}}} + \frac {40425 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{20} + 889350 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{18} + 8870400 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{16} + 52852800 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{14} + 207737376 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{12} + 553409472 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{10} + 961297920 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{8} + 1100267520 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{6} + 808910080 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 348401152 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 67108864}{147840 \, {\left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 2\right )}^{11} c} \] Input:

integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^6,x, algo 
rithm="giac")
 

Output:

1/591360*((((c*x + 1)*((c*x + 1)*(65536*(c*x + 1)/c - 680471/c) + 2839782/ 
c) - 5963958/c)*(c*x + 1) + 6320160/c)*(c*x + 1) - 2716560/c)*sqrt(c*x + 1 
)/(c*x - 1)^(11/2) + 1/147840*(40425*(sqrt(c*x + 1) - sqrt(c*x - 1))^20 + 
889350*(sqrt(c*x + 1) - sqrt(c*x - 1))^18 + 8870400*(sqrt(c*x + 1) - sqrt( 
c*x - 1))^16 + 52852800*(sqrt(c*x + 1) - sqrt(c*x - 1))^14 + 207737376*(sq 
rt(c*x + 1) - sqrt(c*x - 1))^12 + 553409472*(sqrt(c*x + 1) - sqrt(c*x - 1) 
)^10 + 961297920*(sqrt(c*x + 1) - sqrt(c*x - 1))^8 + 1100267520*(sqrt(c*x 
+ 1) - sqrt(c*x - 1))^6 + 808910080*(sqrt(c*x + 1) - sqrt(c*x - 1))^4 + 34 
8401152*(sqrt(c*x + 1) - sqrt(c*x - 1))^2 + 67108864)/(((sqrt(c*x + 1) - s 
qrt(c*x - 1))^2 + 2)^11*c)
 

Mupad [B] (verification not implemented)

Time = 22.82 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.02 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=-\frac {1155\,x\,\sqrt {c\,x-1}-2310\,c^2\,x^3\,\sqrt {c\,x-1}+3696\,c^4\,x^5\,\sqrt {c\,x-1}-3168\,c^6\,x^7\,\sqrt {c\,x-1}+1408\,c^8\,x^9\,\sqrt {c\,x-1}-256\,c^{10}\,x^{11}\,\sqrt {c\,x-1}}{{\left (c\,x-1\right )}^6\,\sqrt {c\,x+1}\,\left (\left (c\,x-1\right )\,\left (\left (\left (c\,x-1\right )\,\left (\left (c\,x-1\right )\,\left (1155\,c\,x+10395\right )+46200\right )+92400\right )\,\left (c\,x-1\right )+92400\right )+36960\right )} \] Input:

int((4*c^2*x^2 + 1)/((c^2*x^2 - 1)^6*(c*x - 1)^(1/2)*(c*x + 1)^(1/2)),x)
 

Output:

-(1155*x*(c*x - 1)^(1/2) - 2310*c^2*x^3*(c*x - 1)^(1/2) + 3696*c^4*x^5*(c* 
x - 1)^(1/2) - 3168*c^6*x^7*(c*x - 1)^(1/2) + 1408*c^8*x^9*(c*x - 1)^(1/2) 
 - 256*c^10*x^11*(c*x - 1)^(1/2))/((c*x - 1)^6*(c*x + 1)^(1/2)*((c*x - 1)* 
(((c*x - 1)*((c*x - 1)*(1155*c*x + 10395) + 46200) + 92400)*(c*x - 1) + 92 
400) + 36960))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.43 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^6} \, dx=\frac {-256 \sqrt {c x -1}\, c^{11} x^{11}-256 \sqrt {c x -1}\, c^{10} x^{10}+1280 \sqrt {c x -1}\, c^{9} x^{9}+1280 \sqrt {c x -1}\, c^{8} x^{8}-2560 \sqrt {c x -1}\, c^{7} x^{7}-2560 \sqrt {c x -1}\, c^{6} x^{6}+2560 \sqrt {c x -1}\, c^{5} x^{5}+2560 \sqrt {c x -1}\, c^{4} x^{4}-1280 \sqrt {c x -1}\, c^{3} x^{3}-1280 \sqrt {c x -1}\, c^{2} x^{2}+256 \sqrt {c x -1}\, c x +256 \sqrt {c x -1}+256 \sqrt {c x +1}\, c^{11} x^{11}-1408 \sqrt {c x +1}\, c^{9} x^{9}+3168 \sqrt {c x +1}\, c^{7} x^{7}-3696 \sqrt {c x +1}\, c^{5} x^{5}+2310 \sqrt {c x +1}\, c^{3} x^{3}-1155 \sqrt {c x +1}\, c x}{1155 \sqrt {c x -1}\, c \left (c^{11} x^{11}+c^{10} x^{10}-5 c^{9} x^{9}-5 c^{8} x^{8}+10 c^{7} x^{7}+10 c^{6} x^{6}-10 c^{5} x^{5}-10 c^{4} x^{4}+5 c^{3} x^{3}+5 c^{2} x^{2}-c x -1\right )} \] Input:

int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^6,x)
 

Output:

( - 256*sqrt(c*x - 1)*c**11*x**11 - 256*sqrt(c*x - 1)*c**10*x**10 + 1280*s 
qrt(c*x - 1)*c**9*x**9 + 1280*sqrt(c*x - 1)*c**8*x**8 - 2560*sqrt(c*x - 1) 
*c**7*x**7 - 2560*sqrt(c*x - 1)*c**6*x**6 + 2560*sqrt(c*x - 1)*c**5*x**5 + 
 2560*sqrt(c*x - 1)*c**4*x**4 - 1280*sqrt(c*x - 1)*c**3*x**3 - 1280*sqrt(c 
*x - 1)*c**2*x**2 + 256*sqrt(c*x - 1)*c*x + 256*sqrt(c*x - 1) + 256*sqrt(c 
*x + 1)*c**11*x**11 - 1408*sqrt(c*x + 1)*c**9*x**9 + 3168*sqrt(c*x + 1)*c* 
*7*x**7 - 3696*sqrt(c*x + 1)*c**5*x**5 + 2310*sqrt(c*x + 1)*c**3*x**3 - 11 
55*sqrt(c*x + 1)*c*x)/(1155*sqrt(c*x - 1)*c*(c**11*x**11 + c**10*x**10 - 5 
*c**9*x**9 - 5*c**8*x**8 + 10*c**7*x**7 + 10*c**6*x**6 - 10*c**5*x**5 - 10 
*c**4*x**4 + 5*c**3*x**3 + 5*c**2*x**2 - c*x - 1))