\(\int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx\) [58]

Optimal result

Integrand size = 20, antiderivative size = 227 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (1+c x)^2}+\frac {15 b}{8 c^5 d^3 (1+c x)}-\frac {19 b \text {arctanh}(c x)}{8 c^5 d^3}-\frac {3 b x \text {arctanh}(c x)}{c^4 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))}{2 c^3 d^3}-\frac {a+b \text {arctanh}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 (a+b \text {arctanh}(c x))}{c^5 d^3 (1+c x)}-\frac {6 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^5 d^3} \] Output:

-3*a*x/c^4/d^3+1/2*b*x/c^4/d^3-1/8*b/c^5/d^3/(c*x+1)^2+15/8*b/c^5/d^3/(c*x 
+1)-19/8*b*arctanh(c*x)/c^5/d^3-3*b*x*arctanh(c*x)/c^4/d^3+1/2*x^2*(a+b*ar 
ctanh(c*x))/c^3/d^3-1/2*(a+b*arctanh(c*x))/c^5/d^3/(c*x+1)^2+4*(a+b*arctan 
h(c*x))/c^5/d^3/(c*x+1)-6*(a+b*arctanh(c*x))*ln(2/(c*x+1))/c^5/d^3-3/2*b*l 
n(-c^2*x^2+1)/c^5/d^3+3*b*polylog(2,1-2/(c*x+1))/c^5/d^3
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\frac {-96 a c x+16 a c^2 x^2-\frac {16 a}{(1+c x)^2}+\frac {128 a}{1+c x}+192 a \log (1+c x)+b \left (16 c x+28 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1-c^2 x^2\right )+96 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-28 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (-4-24 c x+4 c^2 x^2+14 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-14 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )\right )}{32 c^5 d^3} \] Input:

Integrate[(x^4*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]
 

Output:

(-96*a*c*x + 16*a*c^2*x^2 - (16*a)/(1 + c*x)^2 + (128*a)/(1 + c*x) + 192*a 
*Log[1 + c*x] + b*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] 
 - 48*Log[1 - c^2*x^2] + 96*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 28*Sinh[2*A 
rcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-4 - 24*c*x + 4*c^2* 
x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*Ar 
cTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*c^5*d 
^3)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6502, 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.

\(\displaystyle \int \frac {x^4 (a+b \text {arctanh}(c x))}{(c d x+d)^3} \, dx\)

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (\frac {6 (a+b \text {arctanh}(c x))}{c^4 d^3 (c x+1)}-\frac {4 (a+b \text {arctanh}(c x))}{c^4 d^3 (c x+1)^2}-\frac {3 (a+b \text {arctanh}(c x))}{c^4 d^3}+\frac {a+b \text {arctanh}(c x)}{c^4 d^3 (c x+1)^3}+\frac {x (a+b \text {arctanh}(c x))}{c^3 d^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 (a+b \text {arctanh}(c x))}{c^5 d^3 (c x+1)}-\frac {a+b \text {arctanh}(c x)}{2 c^5 d^3 (c x+1)^2}-\frac {6 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^5 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))}{2 c^3 d^3}-\frac {3 a x}{c^4 d^3}-\frac {19 b \text {arctanh}(c x)}{8 c^5 d^3}-\frac {3 b x \text {arctanh}(c x)}{c^4 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{c^5 d^3}+\frac {15 b}{8 c^5 d^3 (c x+1)}-\frac {b}{8 c^5 d^3 (c x+1)^2}+\frac {b x}{2 c^4 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}\)

Input:

Int[(x^4*(a + b*ArcTanh[c*x]))/(d + c*d*x)^3,x]
 

Output:

(-3*a*x)/(c^4*d^3) + (b*x)/(2*c^4*d^3) - b/(8*c^5*d^3*(1 + c*x)^2) + (15*b 
)/(8*c^5*d^3*(1 + c*x)) - (19*b*ArcTanh[c*x])/(8*c^5*d^3) - (3*b*x*ArcTanh 
[c*x])/(c^4*d^3) + (x^2*(a + b*ArcTanh[c*x]))/(2*c^3*d^3) - (a + b*ArcTanh 
[c*x])/(2*c^5*d^3*(1 + c*x)^2) + (4*(a + b*ArcTanh[c*x]))/(c^5*d^3*(1 + c* 
x)) - (6*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^5*d^3) - (3*b*Log[1 - c 
^2*x^2])/(2*c^5*d^3) + (3*b*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-3 c x -\frac {1}{2 \left (c x +1\right )^{2}}+\frac {4}{c x +1}+6 \ln \left (c x +1\right )\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-3 \,\operatorname {arctanh}\left (c x \right ) c x -\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {c x}{2}+\frac {1}{2}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}-\frac {5 \ln \left (c x -1\right )}{16}\right )}{d^{3}}}{c^{5}}\)	\(197\)
default	\(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-3 c x -\frac {1}{2 \left (c x +1\right )^{2}}+\frac {4}{c x +1}+6 \ln \left (c x +1\right )\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-3 \,\operatorname {arctanh}\left (c x \right ) c x -\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {c x}{2}+\frac {1}{2}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}-\frac {5 \ln \left (c x -1\right )}{16}\right )}{d^{3}}}{c^{5}}\)	\(197\)
parts	\(\frac {a \left (\frac {\frac {1}{2} c \,x^{2}-3 x}{c^{4}}+\frac {4}{c^{5} \left (c x +1\right )}+\frac {6 \ln \left (c x +1\right )}{c^{5}}-\frac {1}{2 c^{5} \left (c x +1\right )^{2}}\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{2} x^{2}}{2}-3 \,\operatorname {arctanh}\left (c x \right ) c x -\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )+\frac {c x}{2}+\frac {1}{2}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}-\frac {5 \ln \left (c x -1\right )}{16}\right )}{d^{3} c^{5}}\)	\(207\)
risch	\(\frac {9 b}{8 c^{5} d^{3}}-\frac {7 b \ln \left (c x +1\right )}{4 c^{5} d^{3}}+\frac {b}{8 d^{3} c^{5} \left (-c x -1\right )}+\frac {a \,x^{2}}{2 d^{3} c^{3}}-\frac {4 a}{d^{3} c^{5} \left (-c x -1\right )}-\frac {a}{2 d^{3} c^{5} \left (-c x -1\right )^{2}}-\frac {5 b \ln \left (-c x +1\right )}{4 d^{3} c^{5}}+\frac {5 a}{2 d^{3} c^{5}}+\frac {3 b \ln \left (c x +1\right )^{2}}{2 c^{5} d^{3}}-\frac {15 b \ln \left (-c x -1\right )}{16 d^{3} c^{5}}+\frac {3 b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{3} c^{5}}+\frac {6 a \ln \left (-c x -1\right )}{d^{3} c^{5}}+\left (\frac {b \left (\frac {1}{2} c \,x^{2}-3 x \right )}{2 c^{4} d^{3}}+\frac {4 b \,d^{3} x +\frac {7 d^{3} b}{2 c}}{2 c^{4} d^{6} \left (c x +1\right )^{2}}\right ) \ln \left (c x +1\right )-\frac {b \ln \left (-c x +1\right ) x}{d^{3} c^{4} \left (-c x -1\right )}-\frac {b \ln \left (-c x +1\right ) x^{2}}{16 d^{3} c^{3} \left (-c x -1\right )^{2}}-\frac {b \ln \left (-c x +1\right ) x}{8 d^{3} c^{4} \left (-c x -1\right )^{2}}-\frac {b \ln \left (-c x +1\right ) x^{2}}{4 d^{3} c^{3}}+\frac {3 b \ln \left (-c x +1\right ) x}{2 d^{3} c^{4}}+\frac {b \ln \left (-c x +1\right )}{d^{3} c^{5} \left (-c x -1\right )}-\frac {3 b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{d^{3} c^{5}}+\frac {3 b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{d^{3} c^{5}}+\frac {3 b \ln \left (-c x +1\right )}{16 d^{3} c^{5} \left (-c x -1\right )^{2}}-\frac {3 a x}{c^{4} d^{3}}+\frac {b x}{2 c^{4} d^{3}}-\frac {b}{8 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {2 b}{c^{5} d^{3} \left (c x +1\right )}\)	\(496\)

Input:

int(x^4*(a+b*arctanh(c*x))/(c*d*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/c^5*(a/d^3*(1/2*c^2*x^2-3*c*x-1/2/(c*x+1)^2+4/(c*x+1)+6*ln(c*x+1))+b/d^3 
*(1/2*arctanh(c*x)*c^2*x^2-3*arctanh(c*x)*c*x-1/2/(c*x+1)^2*arctanh(c*x)+4 
/(c*x+1)*arctanh(c*x)+6*arctanh(c*x)*ln(c*x+1)-3/2*ln(c*x+1)^2+3*(ln(c*x+1 
)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1/2)-3*dilog(1/2*c*x+1/2)+1/2*c*x+1/2-1/8/( 
c*x+1)^2+15/8/(c*x+1)-43/16*ln(c*x+1)-5/16*ln(c*x-1)))
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \] Input:

integrate(x^4*(a+b*arctanh(c*x))/(c*d*x+d)^3,x, algorithm="fricas")
 

Output:

integral((b*x^4*arctanh(c*x) + a*x^4)/(c^3*d^3*x^3 + 3*c^2*d^3*x^2 + 3*c*d 
^3*x + d^3), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\frac {\int \frac {a x^{4}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \] Input:

integrate(x**4*(a+b*atanh(c*x))/(c*d*x+d)**3,x)
 

Output:

(Integral(a*x**4/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b*x* 
*4*atanh(c*x)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x))/d**3
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \] Input:

integrate(x^4*(a+b*arctanh(c*x))/(c*d*x+d)^3,x, algorithm="maxima")
 

Output:

1/32*(c^5*(2*(9*c*x + 8)/(c^12*d^3*x^2 + 2*c^11*d^3*x + c^10*d^3) + 4*(c*x 
^2 - 4*x)/(c^9*d^3) + 31*log(c*x + 1)/(c^10*d^3) + log(c*x - 1)/(c^10*d^3) 
) + 32*c^5*integrate(1/2*x^5*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2 
*c^5*d^3*x - c^4*d^3), x) + 3*c^4*(2*(7*c*x + 6)/(c^11*d^3*x^2 + 2*c^10*d^ 
3*x + c^9*d^3) - 8*x/(c^8*d^3) + 17*log(c*x + 1)/(c^9*d^3) - log(c*x - 1)/ 
(c^9*d^3)) - 32*c^4*integrate(1/2*x^4*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^ 
3*x^3 - 2*c^5*d^3*x - c^4*d^3), x) - 15*c^3*(2*(5*c*x + 4)/(c^10*d^3*x^2 + 
 2*c^9*d^3*x + c^8*d^3) + 7*log(c*x + 1)/(c^8*d^3) + log(c*x - 1)/(c^8*d^3 
)) + 192*c^3*integrate(1/2*x^3*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 
 2*c^5*d^3*x - c^4*d^3), x) + 9*c^2*(2*(3*c*x + 2)/(c^9*d^3*x^2 + 2*c^8*d^ 
3*x + c^7*d^3) + log(c*x + 1)/(c^7*d^3) - log(c*x - 1)/(c^7*d^3)) + 576*c^ 
2*integrate(1/2*x^2*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3* 
x - c^4*d^3), x) + 9*c*(2*x/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) - log(c* 
x + 1)/(c^6*d^3) + log(c*x - 1)/(c^6*d^3)) + 576*c*integrate(1/2*x*log(c*x 
 + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3*x - c^4*d^3), x) - 8*(c^4*x 
^4 - 4*c^3*x^3 - 11*c^2*x^2 + 2*c*x + 12*(c^2*x^2 + 2*c*x + 1)*log(c*x + 1 
) + 7)*log(-c*x + 1)/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) + 14*(c*x + 2)/ 
(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) - 7*log(c*x + 1)/(c^5*d^3) + 7*log(c 
*x - 1)/(c^5*d^3) + 192*integrate(1/2*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^ 
3*x^3 - 2*c^5*d^3*x - c^4*d^3), x))*b + 1/2*a*((8*c*x + 7)/(c^7*d^3*x^2...
 

Giac [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \] Input:

integrate(x^4*(a+b*arctanh(c*x))/(c*d*x+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)*x^4/(c*d*x + d)^3, x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^3} \,d x \] Input:

int((x^4*(a + b*atanh(c*x)))/(d + c*d*x)^3,x)
 

Output:

int((x^4*(a + b*atanh(c*x)))/(d + c*d*x)^3, x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\frac {2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b \,c^{7} x^{2}+4 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b \,c^{6} x +2 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{4}}{c^{3} x^{3}+3 c^{2} x^{2}+3 c x +1}d x \right ) b \,c^{5}+12 \,\mathrm {log}\left (c x +1\right ) a \,c^{2} x^{2}+24 \,\mathrm {log}\left (c x +1\right ) a c x +12 \,\mathrm {log}\left (c x +1\right ) a +a \,c^{4} x^{4}-4 a \,c^{3} x^{3}-12 a \,c^{2} x^{2}+6 a}{2 c^{5} d^{3} \left (c^{2} x^{2}+2 c x +1\right )} \] Input:

int(x^4*(a+b*atanh(c*x))/(c*d*x+d)^3,x)
 

Output:

(2*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1),x)*b*c**7*x 
**2 + 4*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1),x)*b*c 
**6*x + 2*int((atanh(c*x)*x**4)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1),x)*b 
*c**5 + 12*log(c*x + 1)*a*c**2*x**2 + 24*log(c*x + 1)*a*c*x + 12*log(c*x + 
 1)*a + a*c**4*x**4 - 4*a*c**3*x**3 - 12*a*c**2*x**2 + 6*a)/(2*c**5*d**3*( 
c**2*x**2 + 2*c*x + 1))