\(\int (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [398]

Optimal result

Integrand size = 20, antiderivative size = 169 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^3 n x-\frac {3 b d^2 e n x^{1+r}}{(1+r)^2}-\frac {3 b d e^2 n x^{1+2 r}}{(1+2 r)^2}-\frac {b e^3 n x^{1+3 r}}{(1+3 r)^2}+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{1+r} \left (a+b \log \left (c x^n\right )\right )}{1+r}+\frac {3 d e^2 x^{1+2 r} \left (a+b \log \left (c x^n\right )\right )}{1+2 r}+\frac {e^3 x^{1+3 r} \left (a+b \log \left (c x^n\right )\right )}{1+3 r} \] Output:

-b*d^3*n*x-3*b*d^2*e*n*x^(1+r)/(1+r)^2-3*b*d*e^2*n*x^(1+2*r)/(1+2*r)^2-b*e 
^3*n*x^(1+3*r)/(1+3*r)^2+d^3*x*(a+b*ln(c*x^n))+3*d^2*e*x^(1+r)*(a+b*ln(c*x 
^n))/(1+r)+3*d*e^2*x^(1+2*r)*(a+b*ln(c*x^n))/(1+2*r)+e^3*x^(1+3*r)*(a+b*ln 
(c*x^n))/(1+3*r)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=x \left (a d^3-b d^3 n-\frac {3 b d^2 e n x^r}{(1+r)^2}-\frac {3 b d e^2 n x^{2 r}}{(1+2 r)^2}-\frac {b e^3 n x^{3 r}}{(1+3 r)^2}+b d^3 \log \left (c x^n\right )+\frac {3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )}{1+r}+\frac {3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{1+2 r}+\frac {e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{1+3 r}\right ) \] Input:

Integrate[(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

x*(a*d^3 - b*d^3*n - (3*b*d^2*e*n*x^r)/(1 + r)^2 - (3*b*d*e^2*n*x^(2*r))/( 
1 + 2*r)^2 - (b*e^3*n*x^(3*r))/(1 + 3*r)^2 + b*d^3*Log[c*x^n] + (3*d^2*e*x 
^r*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n]))/(1 + 
 2*r) + (e^3*x^(3*r)*(a + b*Log[c*x^n]))/(1 + 3*r))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2750, 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.

\(\displaystyle \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2750

\(\displaystyle -b n \int \left (\frac {3 d^2 e x^r}{r+1}+\frac {3 d e^2 x^{2 r}}{2 r+1}+\frac {e^3 x^{3 r}}{3 r+1}+d^3\right )dx+d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac {3 d e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac {e^3 x^{3 r+1} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}\)

\(\Big \downarrow \) 2009

\(\displaystyle d^3 x \left (a+b \log \left (c x^n\right )\right )+\frac {3 d^2 e x^{r+1} \left (a+b \log \left (c x^n\right )\right )}{r+1}+\frac {3 d e^2 x^{2 r+1} \left (a+b \log \left (c x^n\right )\right )}{2 r+1}+\frac {e^3 x^{3 r+1} \left (a+b \log \left (c x^n\right )\right )}{3 r+1}-b n \left (d^3 x+\frac {3 d^2 e x^{r+1}}{(r+1)^2}+\frac {3 d e^2 x^{2 r+1}}{(2 r+1)^2}+\frac {e^3 x^{3 r+1}}{(3 r+1)^2}\right )\)

Input:

Int[(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

-(b*n*(d^3*x + (3*d^2*e*x^(1 + r))/(1 + r)^2 + (3*d*e^2*x^(1 + 2*r))/(1 + 
2*r)^2 + (e^3*x^(1 + 3*r))/(1 + 3*r)^2)) + d^3*x*(a + b*Log[c*x^n]) + (3*d 
^2*e*x^(1 + r)*(a + b*Log[c*x^n]))/(1 + r) + (3*d*e^2*x^(1 + 2*r)*(a + b*L 
og[c*x^n]))/(1 + 2*r) + (e^3*x^(1 + 3*r)*(a + b*Log[c*x^n]))/(1 + 3*r)
 

Defintions of rubi rules used

rule 2009

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

rule 2750

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), 
x_Symbol] :> With[{u = IntHide[(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) 
  u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, 
 c, d, e, n, r}, x] && IGtQ[q, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1112\) vs. \(2(169)=338\).

Time = 3.32 (sec) , antiderivative size = 1113, normalized size of antiderivative = 6.59

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1113\)
risch	\(\text {Expression too large to display}\)	\(4023\)

Input:

int((d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
 

Output:

-(-x*a*d^3-x*e^3*(x^r)^3*a-36*x*a*d^3*r^6-132*x*a*d^3*r^5-193*x*a*d^3*r^4- 
144*x*a*d^3*r^3-58*x*a*d^3*r^2-12*x*a*d^3*r-x*ln(c*x^n)*b*d^3-3*x*d^2*e*x^ 
r*a-3*x*d*e^2*(x^r)^2*a-12*x*(x^r)^3*a*e^3*r^5-40*x*(x^r)^3*a*e^3*r^4-36*x 
*ln(c*x^n)*b*d^3*r^6+36*x*b*d^3*n*r^6-51*x*(x^r)^3*a*e^3*r^3-132*x*ln(c*x^ 
n)*b*d^3*r^5+132*x*b*d^3*n*r^5-31*x*(x^r)^3*a*e^3*r^2-193*x*ln(c*x^n)*b*d^ 
3*r^4+193*x*b*d^3*n*r^4-9*x*(x^r)^3*a*e^3*r+x*(x^r)^3*b*e^3*n-144*x*ln(c*x 
^n)*b*d^3*r^3+144*x*b*d^3*n*r^3-58*x*ln(c*x^n)*b*d^3*r^2+58*x*b*d^3*n*r^2- 
12*x*ln(c*x^n)*b*d^3*r+12*x*b*d^3*n*r-x*(x^r)^3*ln(c*x^n)*b*e^3-3*x*(x^r)^ 
2*ln(c*x^n)*b*d*e^2-3*x*x^r*ln(c*x^n)*b*d^2*e-12*x*(x^r)^3*ln(c*x^n)*b*e^3 
*r^5-40*x*(x^r)^3*ln(c*x^n)*b*e^3*r^4+4*x*(x^r)^3*b*e^3*n*r^4-51*x*(x^r)^3 
*ln(c*x^n)*b*e^3*r^3+12*x*(x^r)^3*b*e^3*n*r^3-54*x*(x^r)^2*a*d*e^2*r^5-31* 
x*(x^r)^3*ln(c*x^n)*b*e^3*r^2+13*x*(x^r)^3*b*e^3*n*r^2-171*x*(x^r)^2*a*d*e 
^2*r^4-108*x*x^r*a*d^2*e*r^5-9*x*(x^r)^3*ln(c*x^n)*b*e^3*r+6*x*(x^r)^3*b*e 
^3*n*r-204*x*(x^r)^2*a*d*e^2*r^3-288*x*x^r*a*d^2*e*r^4-114*x*(x^r)^2*a*d*e 
^2*r^2-291*x*x^r*a*d^2*e*r^3-30*x*(x^r)^2*a*d*e^2*r+3*x*(x^r)^2*b*d*e^2*n- 
141*x*x^r*a*d^2*e*r^2-33*x*x^r*a*d^2*e*r+3*x*x^r*b*d^2*e*n-141*x*x^r*ln(c* 
x^n)*b*d^2*e*r^2+111*x*x^r*b*d^2*e*n*r^2-33*x*x^r*ln(c*x^n)*b*d^2*e*r+30*x 
*x^r*b*d^2*e*n*r-54*x*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5-171*x*(x^r)^2*ln(c*x^n 
)*b*d*e^2*r^4+27*x*(x^r)^2*b*d*e^2*n*r^4-108*x*x^r*ln(c*x^n)*b*d^2*e*r^5-2 
04*x*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3+72*x*(x^r)^2*b*d*e^2*n*r^3-288*x*x^r...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (169) = 338\).

Time = 0.10 (sec) , antiderivative size = 983, normalized size of antiderivative = 5.82 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")
 

Output:

((36*b*d^3*r^6 + 132*b*d^3*r^5 + 193*b*d^3*r^4 + 144*b*d^3*r^3 + 58*b*d^3* 
r^2 + 12*b*d^3*r + b*d^3)*x*log(c) + (36*b*d^3*n*r^6 + 132*b*d^3*n*r^5 + 1 
93*b*d^3*n*r^4 + 144*b*d^3*n*r^3 + 58*b*d^3*n*r^2 + 12*b*d^3*n*r + b*d^3*n 
)*x*log(x) - (36*(b*d^3*n - a*d^3)*r^6 + 132*(b*d^3*n - a*d^3)*r^5 + b*d^3 
*n + 193*(b*d^3*n - a*d^3)*r^4 - a*d^3 + 144*(b*d^3*n - a*d^3)*r^3 + 58*(b 
*d^3*n - a*d^3)*r^2 + 12*(b*d^3*n - a*d^3)*r)*x + ((12*b*e^3*r^5 + 40*b*e^ 
3*r^4 + 51*b*e^3*r^3 + 31*b*e^3*r^2 + 9*b*e^3*r + b*e^3)*x*log(c) + (12*b* 
e^3*n*r^5 + 40*b*e^3*n*r^4 + 51*b*e^3*n*r^3 + 31*b*e^3*n*r^2 + 9*b*e^3*n*r 
 + b*e^3*n)*x*log(x) + (12*a*e^3*r^5 - b*e^3*n - 4*(b*e^3*n - 10*a*e^3)*r^ 
4 + a*e^3 - 3*(4*b*e^3*n - 17*a*e^3)*r^3 - (13*b*e^3*n - 31*a*e^3)*r^2 - 3 
*(2*b*e^3*n - 3*a*e^3)*r)*x)*x^(3*r) + 3*((18*b*d*e^2*r^5 + 57*b*d*e^2*r^4 
 + 68*b*d*e^2*r^3 + 38*b*d*e^2*r^2 + 10*b*d*e^2*r + b*d*e^2)*x*log(c) + (1 
8*b*d*e^2*n*r^5 + 57*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 + 38*b*d*e^2*n*r^2 + 
 10*b*d*e^2*n*r + b*d*e^2*n)*x*log(x) + (18*a*d*e^2*r^5 - b*d*e^2*n - 3*(3 
*b*d*e^2*n - 19*a*d*e^2)*r^4 + a*d*e^2 - 4*(6*b*d*e^2*n - 17*a*d*e^2)*r^3 
- 2*(11*b*d*e^2*n - 19*a*d*e^2)*r^2 - 2*(4*b*d*e^2*n - 5*a*d*e^2)*r)*x)*x^ 
(2*r) + 3*((36*b*d^2*e*r^5 + 96*b*d^2*e*r^4 + 97*b*d^2*e*r^3 + 47*b*d^2*e* 
r^2 + 11*b*d^2*e*r + b*d^2*e)*x*log(c) + (36*b*d^2*e*n*r^5 + 96*b*d^2*e*n* 
r^4 + 97*b*d^2*e*n*r^3 + 47*b*d^2*e*n*r^2 + 11*b*d^2*e*n*r + b*d^2*e*n)*x* 
log(x) + (36*a*d^2*e*r^5 - b*d^2*e*n - 12*(3*b*d^2*e*n - 8*a*d^2*e)*r^4...
 

Sympy [A] (verification not implemented)

Time = 3.64 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.92 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((d+e*x**r)**3*(a+b*ln(c*x**n)),x)
 

Output:

a*d**3*x + 3*a*d**2*e*Piecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), 
True)) + 3*a*d*e**2*Piecewise((x**(2*r + 1)/(2*r + 1), Ne(r, -1/2)), (log( 
x), True)) + a*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(r, -1/3)), (log( 
x), True)) - b*d**3*n*x + b*d**3*x*log(c*x**n) - 3*b*d**2*e*n*Piecewise((P 
iecewise((x**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True))/(r + 1), (r > -o 
o) & (r < oo) & Ne(r, -1)), (log(x)**2/2, True)) + 3*b*d**2*e*Piecewise((x 
**(r + 1)/(r + 1), Ne(r, -1)), (log(x), True))*log(c*x**n) - 3*b*d*e**2*n* 
Piecewise((Piecewise((x**(2*r + 1)/(2*r + 1), Ne(r, -1/2)), (log(x), True) 
)/(2*r + 1), (r > -oo) & (r < oo) & Ne(r, -1/2)), (log(x)**2/2, True)) + 3 
*b*d*e**2*Piecewise((x**(2*r + 1)/(2*r + 1), Ne(r, -1/2)), (log(x), True)) 
*log(c*x**n) - b*e**3*n*Piecewise((Piecewise((x**(3*r + 1)/(3*r + 1), Ne(r 
, -1/3)), (log(x), True))/(3*r + 1), (r > -oo) & (r < oo) & Ne(r, -1/3)), 
(log(x)**2/2, True)) + b*e**3*Piecewise((x**(3*r + 1)/(3*r + 1), Ne(r, -1/ 
3)), (log(x), True))*log(c*x**n)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-b d^{3} n x + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x + \frac {b e^{3} x^{3 \, r + 1} \log \left (c x^{n}\right )}{3 \, r + 1} + \frac {3 \, b d e^{2} x^{2 \, r + 1} \log \left (c x^{n}\right )}{2 \, r + 1} + \frac {3 \, b d^{2} e x^{r + 1} \log \left (c x^{n}\right )}{r + 1} - \frac {b e^{3} n x^{3 \, r + 1}}{{\left (3 \, r + 1\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 1}}{3 \, r + 1} - \frac {3 \, b d e^{2} n x^{2 \, r + 1}}{{\left (2 \, r + 1\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 1}}{2 \, r + 1} - \frac {3 \, b d^{2} e n x^{r + 1}}{{\left (r + 1\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 1}}{r + 1} \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")
 

Output:

-b*d^3*n*x + b*d^3*x*log(c*x^n) + a*d^3*x + b*e^3*x^(3*r + 1)*log(c*x^n)/( 
3*r + 1) + 3*b*d*e^2*x^(2*r + 1)*log(c*x^n)/(2*r + 1) + 3*b*d^2*e*x^(r + 1 
)*log(c*x^n)/(r + 1) - b*e^3*n*x^(3*r + 1)/(3*r + 1)^2 + a*e^3*x^(3*r + 1) 
/(3*r + 1) - 3*b*d*e^2*n*x^(2*r + 1)/(2*r + 1)^2 + 3*a*d*e^2*x^(2*r + 1)/( 
2*r + 1) - 3*b*d^2*e*n*x^(r + 1)/(r + 1)^2 + 3*a*d^2*e*x^(r + 1)/(r + 1)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (169) = 338\).

Time = 0.13 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.24 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3 \, b e^{3} n r x x^{3 \, r} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} + \frac {6 \, b d e^{2} n r x x^{2 \, r} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac {3 \, b d^{2} e n r x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} + b d^{3} n x \log \left (x\right ) + \frac {b e^{3} n x x^{3 \, r} \log \left (x\right )}{9 \, r^{2} + 6 \, r + 1} + \frac {3 \, b d e^{2} n x x^{2 \, r} \log \left (x\right )}{4 \, r^{2} + 4 \, r + 1} + \frac {3 \, b d^{2} e n x x^{r} \log \left (x\right )}{r^{2} + 2 \, r + 1} - b d^{3} n x - \frac {b e^{3} n x x^{3 \, r}}{9 \, r^{2} + 6 \, r + 1} - \frac {3 \, b d e^{2} n x x^{2 \, r}}{4 \, r^{2} + 4 \, r + 1} - \frac {3 \, b d^{2} e n x x^{r}}{r^{2} + 2 \, r + 1} + b d^{3} x \log \left (c\right ) + \frac {b e^{3} x x^{3 \, r} \log \left (c\right )}{3 \, r + 1} + \frac {3 \, b d e^{2} x x^{2 \, r} \log \left (c\right )}{2 \, r + 1} + \frac {3 \, b d^{2} e x x^{r} \log \left (c\right )}{r + 1} + a d^{3} x + \frac {a e^{3} x x^{3 \, r}}{3 \, r + 1} + \frac {3 \, a d e^{2} x x^{2 \, r}}{2 \, r + 1} + \frac {3 \, a d^{2} e x x^{r}}{r + 1} \] Input:

integrate((d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

3*b*e^3*n*r*x*x^(3*r)*log(x)/(9*r^2 + 6*r + 1) + 6*b*d*e^2*n*r*x*x^(2*r)*l 
og(x)/(4*r^2 + 4*r + 1) + 3*b*d^2*e*n*r*x*x^r*log(x)/(r^2 + 2*r + 1) + b*d 
^3*n*x*log(x) + b*e^3*n*x*x^(3*r)*log(x)/(9*r^2 + 6*r + 1) + 3*b*d*e^2*n*x 
*x^(2*r)*log(x)/(4*r^2 + 4*r + 1) + 3*b*d^2*e*n*x*x^r*log(x)/(r^2 + 2*r + 
1) - b*d^3*n*x - b*e^3*n*x*x^(3*r)/(9*r^2 + 6*r + 1) - 3*b*d*e^2*n*x*x^(2* 
r)/(4*r^2 + 4*r + 1) - 3*b*d^2*e*n*x*x^r/(r^2 + 2*r + 1) + b*d^3*x*log(c) 
+ b*e^3*x*x^(3*r)*log(c)/(3*r + 1) + 3*b*d*e^2*x*x^(2*r)*log(c)/(2*r + 1) 
+ 3*b*d^2*e*x*x^r*log(c)/(r + 1) + a*d^3*x + a*e^3*x*x^(3*r)/(3*r + 1) + 3 
*a*d*e^2*x*x^(2*r)/(2*r + 1) + 3*a*d^2*e*x*x^r/(r + 1)
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

int((d + e*x^r)^3*(a + b*log(c*x^n)),x)
 

Output:

int((d + e*x^r)^3*(a + b*log(c*x^n)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1046, normalized size of antiderivative = 6.19 \[ \int \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

int((d+e*x^r)^3*(a+b*log(c*x^n)),x)
 

Output:

(x*(12*x**(3*r)*log(x**n*c)*b*e**3*r**5 + 40*x**(3*r)*log(x**n*c)*b*e**3*r 
**4 + 51*x**(3*r)*log(x**n*c)*b*e**3*r**3 + 31*x**(3*r)*log(x**n*c)*b*e**3 
*r**2 + 9*x**(3*r)*log(x**n*c)*b*e**3*r + x**(3*r)*log(x**n*c)*b*e**3 + 12 
*x**(3*r)*a*e**3*r**5 + 40*x**(3*r)*a*e**3*r**4 + 51*x**(3*r)*a*e**3*r**3 
+ 31*x**(3*r)*a*e**3*r**2 + 9*x**(3*r)*a*e**3*r + x**(3*r)*a*e**3 - 4*x**( 
3*r)*b*e**3*n*r**4 - 12*x**(3*r)*b*e**3*n*r**3 - 13*x**(3*r)*b*e**3*n*r**2 
 - 6*x**(3*r)*b*e**3*n*r - x**(3*r)*b*e**3*n + 54*x**(2*r)*log(x**n*c)*b*d 
*e**2*r**5 + 171*x**(2*r)*log(x**n*c)*b*d*e**2*r**4 + 204*x**(2*r)*log(x** 
n*c)*b*d*e**2*r**3 + 114*x**(2*r)*log(x**n*c)*b*d*e**2*r**2 + 30*x**(2*r)* 
log(x**n*c)*b*d*e**2*r + 3*x**(2*r)*log(x**n*c)*b*d*e**2 + 54*x**(2*r)*a*d 
*e**2*r**5 + 171*x**(2*r)*a*d*e**2*r**4 + 204*x**(2*r)*a*d*e**2*r**3 + 114 
*x**(2*r)*a*d*e**2*r**2 + 30*x**(2*r)*a*d*e**2*r + 3*x**(2*r)*a*d*e**2 - 2 
7*x**(2*r)*b*d*e**2*n*r**4 - 72*x**(2*r)*b*d*e**2*n*r**3 - 66*x**(2*r)*b*d 
*e**2*n*r**2 - 24*x**(2*r)*b*d*e**2*n*r - 3*x**(2*r)*b*d*e**2*n + 108*x**r 
*log(x**n*c)*b*d**2*e*r**5 + 288*x**r*log(x**n*c)*b*d**2*e*r**4 + 291*x**r 
*log(x**n*c)*b*d**2*e*r**3 + 141*x**r*log(x**n*c)*b*d**2*e*r**2 + 33*x**r* 
log(x**n*c)*b*d**2*e*r + 3*x**r*log(x**n*c)*b*d**2*e + 108*x**r*a*d**2*e*r 
**5 + 288*x**r*a*d**2*e*r**4 + 291*x**r*a*d**2*e*r**3 + 141*x**r*a*d**2*e* 
r**2 + 33*x**r*a*d**2*e*r + 3*x**r*a*d**2*e - 108*x**r*b*d**2*e*n*r**4 - 1 
80*x**r*b*d**2*e*n*r**3 - 111*x**r*b*d**2*e*n*r**2 - 30*x**r*b*d**2*e*n...