\(\int x^2 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [397]

Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} b d^3 n x^3-\frac {b e^3 n x^{3 (1+r)}}{9 (1+r)^2}-\frac {3 b d^2 e n x^{3+r}}{(3+r)^2}-\frac {3 b d e^2 n x^{3+2 r}}{(3+2 r)^2}+\frac {1}{3} \left (d^3 x^3+\frac {e^3 x^{3 (1+r)}}{1+r}+\frac {9 d^2 e x^{3+r}}{3+r}+\frac {9 d e^2 x^{3+2 r}}{3+2 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \] Output:

-1/9*b*d^3*n*x^3-1/9*b*e^3*n*x^(3+3*r)/(1+r)^2-3*b*d^2*e*n*x^(3+r)/(3+r)^2 
-3*b*d*e^2*n*x^(3+2*r)/(3+2*r)^2+1/3*(d^3*x^3+e^3*x^(3+3*r)/(1+r)+9*d^2*e* 
x^(3+r)/(3+r)+9*d*e^2*x^(3+2*r)/(3+2*r))*(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.19 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{9} x^3 \left (b n \left (-d^3-\frac {27 d^2 e x^r}{(3+r)^2}-\frac {27 d e^2 x^{2 r}}{(3+2 r)^2}-\frac {e^3 x^{3 r}}{(1+r)^2}\right )+3 a \left (d^3+\frac {9 d^2 e x^r}{3+r}+\frac {9 d e^2 x^{2 r}}{3+2 r}+\frac {e^3 x^{3 r}}{1+r}\right )+3 b \left (d^3+\frac {9 d^2 e x^r}{3+r}+\frac {9 d e^2 x^{2 r}}{3+2 r}+\frac {e^3 x^{3 r}}{1+r}\right ) \log \left (c x^n\right )\right ) \] Input:

Integrate[x^2*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

(x^3*(b*n*(-d^3 - (27*d^2*e*x^r)/(3 + r)^2 - (27*d*e^2*x^(2*r))/(3 + 2*r)^ 
2 - (e^3*x^(3*r))/(1 + r)^2) + 3*a*(d^3 + (9*d^2*e*x^r)/(3 + r) + (9*d*e^2 
*x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r)) + 3*b*(d^3 + (9*d^2*e*x^r)/(3 
 + r) + (9*d*e^2*x^(2*r))/(3 + 2*r) + (e^3*x^(3*r))/(1 + r))*Log[c*x^n]))/ 
9
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2771, 27, 2010, 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.

\(\displaystyle \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2771

\(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{3} x^2 \left (\frac {9 d^2 e x^r}{r+3}+\frac {9 d e^2 x^{2 r}}{2 r+3}+\frac {e^3 x^{3 r}}{r+1}+d^3\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int x^2 \left (\frac {9 d^2 e x^r}{r+3}+\frac {9 d e^2 x^{2 r}}{2 r+3}+\frac {e^3 x^{3 r}}{r+1}+d^3\right )dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \int \left (\frac {9 d e^2 x^{2 (r+1)}}{2 r+3}+\frac {9 d^2 e x^{r+2}}{r+3}+\frac {e^3 x^{3 r+2}}{r+1}+d^3 x^2\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (d^3 x^3+\frac {9 d^2 e x^{r+3}}{r+3}+\frac {9 d e^2 x^{2 r+3}}{2 r+3}+\frac {e^3 x^{3 (r+1)}}{r+1}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{3} b n \left (\frac {d^3 x^3}{3}+\frac {9 d^2 e x^{r+3}}{(r+3)^2}+\frac {9 d e^2 x^{2 r+3}}{(2 r+3)^2}+\frac {e^3 x^{3 (r+1)}}{3 (r+1)^2}\right )\)

Input:

Int[x^2*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

-1/3*(b*n*((d^3*x^3)/3 + (e^3*x^(3*(1 + r)))/(3*(1 + r)^2) + (9*d^2*e*x^(3 
 + r))/(3 + r)^2 + (9*d*e^2*x^(3 + 2*r))/(3 + 2*r)^2)) + ((d^3*x^3 + (e^3* 
x^(3*(1 + r)))/(1 + r) + (9*d^2*e*x^(3 + r))/(3 + r) + (9*d*e^2*x^(3 + 2*r 
))/(3 + 2*r))*(a + b*Log[c*x^n]))/3
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

rule 2771

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1266\) vs. \(2(142)=284\).

Time = 10.99 (sec) , antiderivative size = 1267, normalized size of antiderivative = 8.56

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1267\)
risch	\(\text {Expression too large to display}\)	\(4027\)

Input:

int(x^2*(d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
 

Output:

-1/9*(-243*x^3*a*d^3-243*x^3*e^3*(x^r)^3*a-729*x^3*d*e^2*(x^r)^2*a-729*x^3 
*d^2*e*x^r*a-12*x^3*a*d^3*r^6-132*x^3*a*d^3*r^5-579*x^3*a*d^3*r^4-1296*x^3 
*a*d^3*r^3-1566*x^3*a*d^3*r^2-972*x^3*a*d^3*r-243*b*d^3*ln(c*x^n)*x^3-54*x 
^3*(x^r)^2*a*d*e^2*r^5-513*x^3*(x^r)^2*a*d*e^2*r^4-1836*x^3*(x^r)^2*a*d*e^ 
2*r^3-3078*x^3*(x^r)^2*a*d*e^2*r^2-2430*x^3*(x^r)^2*a*d*e^2*r+243*x^3*(x^r 
)^2*b*d*e^2*n-12*x^3*(x^r)^3*ln(c*x^n)*b*e^3*r^5-120*x^3*(x^r)^3*ln(c*x^n) 
*b*e^3*r^4-459*x^3*(x^r)^3*ln(c*x^n)*b*e^3*r^3-837*x^3*(x^r)^3*ln(c*x^n)*b 
*e^3*r^2-729*x^3*(x^r)^3*ln(c*x^n)*b*e^3*r+4*x^3*(x^r)^3*b*e^3*n*r^4+36*x^ 
3*(x^r)^3*b*e^3*n*r^3+117*x^3*(x^r)^3*b*e^3*n*r^2-12*x^3*ln(c*x^n)*b*d^3*r 
^6-132*x^3*ln(c*x^n)*b*d^3*r^5-579*x^3*ln(c*x^n)*b*d^3*r^4-1296*x^3*ln(c*x 
^n)*b*d^3*r^3-1566*x^3*ln(c*x^n)*b*d^3*r^2-972*x^3*ln(c*x^n)*b*d^3*r+4*x^3 
*b*d^3*n*r^6-243*b*e^3*ln(c*x^n)*(x^r)^3*x^3-108*x^3*x^r*ln(c*x^n)*b*d^2*e 
*r^5-864*x^3*x^r*ln(c*x^n)*b*d^2*e*r^4-2619*x^3*x^r*ln(c*x^n)*b*d^2*e*r^3- 
3807*x^3*x^r*ln(c*x^n)*b*d^2*e*r^2-2673*x^3*x^r*ln(c*x^n)*b*d^2*e*r-54*x^3 
*(x^r)^2*ln(c*x^n)*b*d*e^2*r^5-513*x^3*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-1836* 
x^3*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3-3078*x^3*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2-2 
430*x^3*(x^r)^2*ln(c*x^n)*b*d*e^2*r+27*x^3*(x^r)^2*b*d*e^2*n*r^4+216*x^3*( 
x^r)^2*b*d*e^2*n*r^3+594*x^3*(x^r)^2*b*d*e^2*n*r^2+648*x^3*(x^r)^2*b*d*e^2 
*n*r+108*x^3*x^r*b*d^2*e*n*r^4+540*x^3*x^r*b*d^2*e*n*r^3+999*x^3*x^r*b*d^2 
*e*n*r^2+810*x^3*x^r*b*d^2*e*n*r+162*x^3*(x^r)^3*b*e^3*n*r-108*x^3*x^r*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (142) = 284\).

Time = 0.09 (sec) , antiderivative size = 1022, normalized size of antiderivative = 6.91 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")
 

Output:

1/9*(3*(4*b*d^3*r^6 + 44*b*d^3*r^5 + 193*b*d^3*r^4 + 432*b*d^3*r^3 + 522*b 
*d^3*r^2 + 324*b*d^3*r + 81*b*d^3)*x^3*log(c) + 3*(4*b*d^3*n*r^6 + 44*b*d^ 
3*n*r^5 + 193*b*d^3*n*r^4 + 432*b*d^3*n*r^3 + 522*b*d^3*n*r^2 + 324*b*d^3* 
n*r + 81*b*d^3*n)*x^3*log(x) - (4*(b*d^3*n - 3*a*d^3)*r^6 + 44*(b*d^3*n - 
3*a*d^3)*r^5 + 81*b*d^3*n + 193*(b*d^3*n - 3*a*d^3)*r^4 - 243*a*d^3 + 432* 
(b*d^3*n - 3*a*d^3)*r^3 + 522*(b*d^3*n - 3*a*d^3)*r^2 + 324*(b*d^3*n - 3*a 
*d^3)*r)*x^3 + (3*(4*b*e^3*r^5 + 40*b*e^3*r^4 + 153*b*e^3*r^3 + 279*b*e^3* 
r^2 + 243*b*e^3*r + 81*b*e^3)*x^3*log(c) + 3*(4*b*e^3*n*r^5 + 40*b*e^3*n*r 
^4 + 153*b*e^3*n*r^3 + 279*b*e^3*n*r^2 + 243*b*e^3*n*r + 81*b*e^3*n)*x^3*l 
og(x) + (12*a*e^3*r^5 - 81*b*e^3*n - 4*(b*e^3*n - 30*a*e^3)*r^4 + 243*a*e^ 
3 - 9*(4*b*e^3*n - 51*a*e^3)*r^3 - 9*(13*b*e^3*n - 93*a*e^3)*r^2 - 81*(2*b 
*e^3*n - 9*a*e^3)*r)*x^3)*x^(3*r) + 27*((2*b*d*e^2*r^5 + 19*b*d*e^2*r^4 + 
68*b*d*e^2*r^3 + 114*b*d*e^2*r^2 + 90*b*d*e^2*r + 27*b*d*e^2)*x^3*log(c) + 
 (2*b*d*e^2*n*r^5 + 19*b*d*e^2*n*r^4 + 68*b*d*e^2*n*r^3 + 114*b*d*e^2*n*r^ 
2 + 90*b*d*e^2*n*r + 27*b*d*e^2*n)*x^3*log(x) + (2*a*d*e^2*r^5 - 9*b*d*e^2 
*n - (b*d*e^2*n - 19*a*d*e^2)*r^4 + 27*a*d*e^2 - 4*(2*b*d*e^2*n - 17*a*d*e 
^2)*r^3 - 2*(11*b*d*e^2*n - 57*a*d*e^2)*r^2 - 6*(4*b*d*e^2*n - 15*a*d*e^2) 
*r)*x^3)*x^(2*r) + 27*((4*b*d^2*e*r^5 + 32*b*d^2*e*r^4 + 97*b*d^2*e*r^3 + 
141*b*d^2*e*r^2 + 99*b*d^2*e*r + 27*b*d^2*e)*x^3*log(c) + (4*b*d^2*e*n*r^5 
 + 32*b*d^2*e*n*r^4 + 97*b*d^2*e*n*r^3 + 141*b*d^2*e*n*r^2 + 99*b*d^2*e...
 

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(x**2*(d+e*x**r)**3*(a+b*ln(c*x**n)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.51 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{9} \, b d^{3} n x^{3} + \frac {1}{3} \, b d^{3} x^{3} \log \left (c x^{n}\right ) + \frac {1}{3} \, a d^{3} x^{3} + \frac {b e^{3} x^{3 \, r + 3} \log \left (c x^{n}\right )}{3 \, {\left (r + 1\right )}} + \frac {3 \, b d e^{2} x^{2 \, r + 3} \log \left (c x^{n}\right )}{2 \, r + 3} + \frac {3 \, b d^{2} e x^{r + 3} \log \left (c x^{n}\right )}{r + 3} - \frac {b e^{3} n x^{3 \, r + 3}}{9 \, {\left (r + 1\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 3}}{3 \, {\left (r + 1\right )}} - \frac {3 \, b d e^{2} n x^{2 \, r + 3}}{{\left (2 \, r + 3\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 3}}{2 \, r + 3} - \frac {3 \, b d^{2} e n x^{r + 3}}{{\left (r + 3\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 3}}{r + 3} \] Input:

integrate(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (142) = 284\).

Time = 0.16 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.89 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")
 

Output:

1/9*(12*b*e^3*n*r^5*x^3*x^(3*r)*log(x) + 54*b*d*e^2*n*r^5*x^3*x^(2*r)*log( 
x) + 108*b*d^2*e*n*r^5*x^3*x^r*log(x) + 12*b*d^3*n*r^6*x^3*log(x) - 4*b*d^ 
3*n*r^6*x^3 + 12*b*e^3*r^5*x^3*x^(3*r)*log(c) + 54*b*d*e^2*r^5*x^3*x^(2*r) 
*log(c) + 108*b*d^2*e*r^5*x^3*x^r*log(c) + 12*b*d^3*r^6*x^3*log(c) + 120*b 
*e^3*n*r^4*x^3*x^(3*r)*log(x) + 513*b*d*e^2*n*r^4*x^3*x^(2*r)*log(x) + 864 
*b*d^2*e*n*r^4*x^3*x^r*log(x) + 132*b*d^3*n*r^5*x^3*log(x) - 4*b*e^3*n*r^4 
*x^3*x^(3*r) + 12*a*e^3*r^5*x^3*x^(3*r) - 27*b*d*e^2*n*r^4*x^3*x^(2*r) + 5 
4*a*d*e^2*r^5*x^3*x^(2*r) - 108*b*d^2*e*n*r^4*x^3*x^r + 108*a*d^2*e*r^5*x^ 
3*x^r - 44*b*d^3*n*r^5*x^3 + 12*a*d^3*r^6*x^3 + 120*b*e^3*r^4*x^3*x^(3*r)* 
log(c) + 513*b*d*e^2*r^4*x^3*x^(2*r)*log(c) + 864*b*d^2*e*r^4*x^3*x^r*log( 
c) + 132*b*d^3*r^5*x^3*log(c) + 459*b*e^3*n*r^3*x^3*x^(3*r)*log(x) + 1836* 
b*d*e^2*n*r^3*x^3*x^(2*r)*log(x) + 2619*b*d^2*e*n*r^3*x^3*x^r*log(x) + 579 
*b*d^3*n*r^4*x^3*log(x) - 36*b*e^3*n*r^3*x^3*x^(3*r) + 120*a*e^3*r^4*x^3*x 
^(3*r) - 216*b*d*e^2*n*r^3*x^3*x^(2*r) + 513*a*d*e^2*r^4*x^3*x^(2*r) - 540 
*b*d^2*e*n*r^3*x^3*x^r + 864*a*d^2*e*r^4*x^3*x^r - 193*b*d^3*n*r^4*x^3 + 1 
32*a*d^3*r^5*x^3 + 459*b*e^3*r^3*x^3*x^(3*r)*log(c) + 1836*b*d*e^2*r^3*x^3 
*x^(2*r)*log(c) + 2619*b*d^2*e*r^3*x^3*x^r*log(c) + 579*b*d^3*r^4*x^3*log( 
c) + 837*b*e^3*n*r^2*x^3*x^(3*r)*log(x) + 3078*b*d*e^2*n*r^2*x^3*x^(2*r)*l 
og(x) + 3807*b*d^2*e*n*r^2*x^3*x^r*log(x) + 1296*b*d^3*n*r^3*x^3*log(x) - 
117*b*e^3*n*r^2*x^3*x^(3*r) + 459*a*e^3*r^3*x^3*x^(3*r) - 594*b*d*e^2*n...
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

int(x^2*(d + e*x^r)^3*(a + b*log(c*x^n)),x)
 

Output:

int(x^2*(d + e*x^r)^3*(a + b*log(c*x^n)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1052, normalized size of antiderivative = 7.11 \[ \int x^2 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

int(x^2*(d+e*x^r)^3*(a+b*log(c*x^n)),x)
 

Output:

(x**3*(12*x**(3*r)*log(x**n*c)*b*e**3*r**5 + 120*x**(3*r)*log(x**n*c)*b*e* 
*3*r**4 + 459*x**(3*r)*log(x**n*c)*b*e**3*r**3 + 837*x**(3*r)*log(x**n*c)* 
b*e**3*r**2 + 729*x**(3*r)*log(x**n*c)*b*e**3*r + 243*x**(3*r)*log(x**n*c) 
*b*e**3 + 12*x**(3*r)*a*e**3*r**5 + 120*x**(3*r)*a*e**3*r**4 + 459*x**(3*r 
)*a*e**3*r**3 + 837*x**(3*r)*a*e**3*r**2 + 729*x**(3*r)*a*e**3*r + 243*x** 
(3*r)*a*e**3 - 4*x**(3*r)*b*e**3*n*r**4 - 36*x**(3*r)*b*e**3*n*r**3 - 117* 
x**(3*r)*b*e**3*n*r**2 - 162*x**(3*r)*b*e**3*n*r - 81*x**(3*r)*b*e**3*n + 
54*x**(2*r)*log(x**n*c)*b*d*e**2*r**5 + 513*x**(2*r)*log(x**n*c)*b*d*e**2* 
r**4 + 1836*x**(2*r)*log(x**n*c)*b*d*e**2*r**3 + 3078*x**(2*r)*log(x**n*c) 
*b*d*e**2*r**2 + 2430*x**(2*r)*log(x**n*c)*b*d*e**2*r + 729*x**(2*r)*log(x 
**n*c)*b*d*e**2 + 54*x**(2*r)*a*d*e**2*r**5 + 513*x**(2*r)*a*d*e**2*r**4 + 
 1836*x**(2*r)*a*d*e**2*r**3 + 3078*x**(2*r)*a*d*e**2*r**2 + 2430*x**(2*r) 
*a*d*e**2*r + 729*x**(2*r)*a*d*e**2 - 27*x**(2*r)*b*d*e**2*n*r**4 - 216*x* 
*(2*r)*b*d*e**2*n*r**3 - 594*x**(2*r)*b*d*e**2*n*r**2 - 648*x**(2*r)*b*d*e 
**2*n*r - 243*x**(2*r)*b*d*e**2*n + 108*x**r*log(x**n*c)*b*d**2*e*r**5 + 8 
64*x**r*log(x**n*c)*b*d**2*e*r**4 + 2619*x**r*log(x**n*c)*b*d**2*e*r**3 + 
3807*x**r*log(x**n*c)*b*d**2*e*r**2 + 2673*x**r*log(x**n*c)*b*d**2*e*r + 7 
29*x**r*log(x**n*c)*b*d**2*e + 108*x**r*a*d**2*e*r**5 + 864*x**r*a*d**2*e* 
r**4 + 2619*x**r*a*d**2*e*r**3 + 3807*x**r*a*d**2*e*r**2 + 2673*x**r*a*d** 
2*e*r + 729*x**r*a*d**2*e - 108*x**r*b*d**2*e*n*r**4 - 540*x**r*b*d**2*...