\(\int x^4 (d+e x^r)^3 (a+b \log (c x^n)) \, dx\) [396]

Optimal result

Integrand size = 23, antiderivative size = 151 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} b d^3 n x^5-\frac {3 b d^2 e n x^{5+r}}{(5+r)^2}-\frac {3 b d e^2 n x^{5+2 r}}{(5+2 r)^2}-\frac {b e^3 n x^{5+3 r}}{(5+3 r)^2}+\frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{5+r}}{5+r}+\frac {15 d e^2 x^{5+2 r}}{5+2 r}+\frac {5 e^3 x^{5+3 r}}{5+3 r}\right ) \left (a+b \log \left (c x^n\right )\right ) \] Output:

-1/25*b*d^3*n*x^5-3*b*d^2*e*n*x^(5+r)/(5+r)^2-3*b*d*e^2*n*x^(5+2*r)/(5+2*r 
)^2-b*e^3*n*x^(5+3*r)/(5+3*r)^2+1/5*(d^3*x^5+15*d^2*e*x^(5+r)/(5+r)+15*d*e 
^2*x^(5+2*r)/(5+2*r)+5*e^3*x^(5+3*r)/(5+3*r))*(a+b*ln(c*x^n))
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.22 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {1}{25} x^5 \left (b n \left (-d^3-\frac {75 d^2 e x^r}{(5+r)^2}-\frac {75 d e^2 x^{2 r}}{(5+2 r)^2}-\frac {25 e^3 x^{3 r}}{(5+3 r)^2}\right )+5 a \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right )+5 b \left (d^3+\frac {15 d^2 e x^r}{5+r}+\frac {15 d e^2 x^{2 r}}{5+2 r}+\frac {5 e^3 x^{3 r}}{5+3 r}\right ) \log \left (c x^n\right )\right ) \] Input:

Integrate[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

(x^5*(b*n*(-d^3 - (75*d^2*e*x^r)/(5 + r)^2 - (75*d*e^2*x^(2*r))/(5 + 2*r)^ 
2 - (25*e^3*x^(3*r))/(5 + 3*r)^2) + 5*a*(d^3 + (15*d^2*e*x^r)/(5 + r) + (1 
5*d*e^2*x^(2*r))/(5 + 2*r) + (5*e^3*x^(3*r))/(5 + 3*r)) + 5*b*(d^3 + (15*d 
^2*e*x^r)/(5 + r) + (15*d*e^2*x^(2*r))/(5 + 2*r) + (5*e^3*x^(3*r))/(5 + 3* 
r))*Log[c*x^n]))/25
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 150, 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.

\(\displaystyle \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2771

\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {1}{5} x^4 \left (\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}+d^3\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int x^4 \left (\frac {15 d^2 e x^r}{r+5}+\frac {15 d e^2 x^{2 r}}{2 r+5}+\frac {5 e^3 x^{3 r}}{3 r+5}+d^3\right )dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \int \left (\frac {15 d e^2 x^{2 (r+2)}}{2 r+5}+\frac {15 d^2 e x^{r+4}}{r+5}+\frac {5 e^3 x^{3 r+4}}{3 r+5}+d^3 x^4\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} \left (d^3 x^5+\frac {15 d^2 e x^{r+5}}{r+5}+\frac {15 d e^2 x^{2 r+5}}{2 r+5}+\frac {5 e^3 x^{3 r+5}}{3 r+5}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{5} b n \left (\frac {d^3 x^5}{5}+\frac {15 d^2 e x^{r+5}}{(r+5)^2}+\frac {15 d e^2 x^{2 r+5}}{(2 r+5)^2}+\frac {5 e^3 x^{3 r+5}}{(3 r+5)^2}\right )\)

Input:

Int[x^4*(d + e*x^r)^3*(a + b*Log[c*x^n]),x]
 

Output:

-1/5*(b*n*((d^3*x^5)/5 + (15*d^2*e*x^(5 + r))/(5 + r)^2 + (15*d*e^2*x^(5 + 
 2*r))/(5 + 2*r)^2 + (5*e^3*x^(5 + 3*r))/(5 + 3*r)^2)) + ((d^3*x^5 + (15*d 
^2*e*x^(5 + r))/(5 + r) + (15*d*e^2*x^(5 + 2*r))/(5 + 2*r) + (5*e^3*x^(5 + 
 3*r))/(5 + 3*r))*(a + b*Log[c*x^n]))/5
 

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. \(1268\) vs. \(2(147)=294\).

Time = 29.89 (sec) , antiderivative size = 1269, normalized size of antiderivative = 8.40

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1269\)
risch	\(\text {Expression too large to display}\)	\(4031\)

Input:

int(x^4*(d+e*x^r)^3*(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
 

Output:

-1/25*(-78125*x^5*e^3*(x^r)^3*a-234375*x^5*d*e^2*(x^r)^2*b*ln(c*x^n)-23437 
5*x^5*d^2*e*x^r*b*ln(c*x^n)-78125*x^5*a*d^3-234375*x^5*d^2*e*x^r*a-234375* 
x^5*d*e^2*(x^r)^2*a-78125*x^5*e^3*(x^r)^3*b*ln(c*x^n)-78125*x^5*b*ln(c*x^n 
)*d^3+15625*b*d^3*n*x^5+36*x^5*b*d^3*n*r^6+660*x^5*b*d^3*n*r^5+4825*x^5*b* 
d^3*n*r^4+9000*x^5*(x^r)^2*b*d*e^2*n*r^3+41250*x^5*(x^r)^2*b*d*e^2*n*r^2+7 
5000*x^5*(x^r)^2*b*d*e^2*n*r-2700*x^5*x^r*ln(c*x^n)*b*d^2*e*r^5-36000*x^5* 
x^r*ln(c*x^n)*b*d^2*e*r^4-181875*x^5*x^r*ln(c*x^n)*b*d^2*e*r^3-440625*x^5* 
x^r*ln(c*x^n)*b*d^2*e*r^2-515625*x^5*x^r*ln(c*x^n)*b*d^2*e*r-1350*x^5*(x^r 
)^2*ln(c*x^n)*b*d*e^2*r^5-21375*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r^4-127500*x 
^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r^3-356250*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r^2- 
468750*x^5*(x^r)^2*ln(c*x^n)*b*d*e^2*r+2700*x^5*x^r*b*d^2*e*n*r^4+22500*x^ 
5*x^r*b*d^2*e*n*r^3+69375*x^5*x^r*b*d^2*e*n*r^2+93750*x^5*x^r*b*d^2*e*n*r+ 
675*x^5*(x^r)^2*b*d*e^2*n*r^4-180*x^5*a*d^3*r^6-3300*x^5*a*d^3*r^5-24125*x 
^5*a*d^3*r^4-90000*x^5*a*d^3*r^3-181250*x^5*a*d^3*r^2-187500*x^5*a*d^3*r-3 
56250*x^5*(x^r)^2*a*d*e^2*r^2-468750*x^5*(x^r)^2*a*d*e^2*r+46875*x^5*(x^r) 
^2*b*d*e^2*n-300*x^5*(x^r)^3*ln(c*x^n)*b*e^3*r^5-5000*x^5*(x^r)^3*ln(c*x^n 
)*b*e^3*r^4-31875*x^5*(x^r)^3*ln(c*x^n)*b*e^3*r^3-96875*x^5*(x^r)^3*ln(c*x 
^n)*b*e^3*r^2-140625*x^5*(x^r)^3*ln(c*x^n)*b*e^3*r+100*x^5*(x^r)^3*b*e^3*n 
*r^4+1500*x^5*(x^r)^3*b*e^3*n*r^3+8125*x^5*(x^r)^3*b*e^3*n*r^2+18750*x^5*( 
x^r)^3*b*e^3*n*r-2700*x^5*x^r*a*d^2*e*r^5-36000*x^5*x^r*a*d^2*e*r^4-181...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (147) = 294\).

Time = 0.10 (sec) , antiderivative size = 1023, normalized size of antiderivative = 6.77 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="fricas")
 

Output:

1/25*(5*(36*b*d^3*r^6 + 660*b*d^3*r^5 + 4825*b*d^3*r^4 + 18000*b*d^3*r^3 + 
 36250*b*d^3*r^2 + 37500*b*d^3*r + 15625*b*d^3)*x^5*log(c) + 5*(36*b*d^3*n 
*r^6 + 660*b*d^3*n*r^5 + 4825*b*d^3*n*r^4 + 18000*b*d^3*n*r^3 + 36250*b*d^ 
3*n*r^2 + 37500*b*d^3*n*r + 15625*b*d^3*n)*x^5*log(x) - (36*(b*d^3*n - 5*a 
*d^3)*r^6 + 660*(b*d^3*n - 5*a*d^3)*r^5 + 15625*b*d^3*n + 4825*(b*d^3*n - 
5*a*d^3)*r^4 - 78125*a*d^3 + 18000*(b*d^3*n - 5*a*d^3)*r^3 + 36250*(b*d^3* 
n - 5*a*d^3)*r^2 + 37500*(b*d^3*n - 5*a*d^3)*r)*x^5 + 25*((12*b*e^3*r^5 + 
200*b*e^3*r^4 + 1275*b*e^3*r^3 + 3875*b*e^3*r^2 + 5625*b*e^3*r + 3125*b*e^ 
3)*x^5*log(c) + (12*b*e^3*n*r^5 + 200*b*e^3*n*r^4 + 1275*b*e^3*n*r^3 + 387 
5*b*e^3*n*r^2 + 5625*b*e^3*n*r + 3125*b*e^3*n)*x^5*log(x) + (12*a*e^3*r^5 
- 625*b*e^3*n - 4*(b*e^3*n - 50*a*e^3)*r^4 + 3125*a*e^3 - 15*(4*b*e^3*n - 
85*a*e^3)*r^3 - 25*(13*b*e^3*n - 155*a*e^3)*r^2 - 375*(2*b*e^3*n - 15*a*e^ 
3)*r)*x^5)*x^(3*r) + 75*((18*b*d*e^2*r^5 + 285*b*d*e^2*r^4 + 1700*b*d*e^2* 
r^3 + 4750*b*d*e^2*r^2 + 6250*b*d*e^2*r + 3125*b*d*e^2)*x^5*log(c) + (18*b 
*d*e^2*n*r^5 + 285*b*d*e^2*n*r^4 + 1700*b*d*e^2*n*r^3 + 4750*b*d*e^2*n*r^2 
 + 6250*b*d*e^2*n*r + 3125*b*d*e^2*n)*x^5*log(x) + (18*a*d*e^2*r^5 - 625*b 
*d*e^2*n - 3*(3*b*d*e^2*n - 95*a*d*e^2)*r^4 + 3125*a*d*e^2 - 20*(6*b*d*e^2 
*n - 85*a*d*e^2)*r^3 - 50*(11*b*d*e^2*n - 95*a*d*e^2)*r^2 - 250*(4*b*d*e^2 
*n - 25*a*d*e^2)*r)*x^5)*x^(2*r) + 75*((36*b*d^2*e*r^5 + 480*b*d^2*e*r^4 + 
 2425*b*d^2*e*r^3 + 5875*b*d^2*e*r^2 + 6875*b*d^2*e*r + 3125*b*d^2*e)*x...
 

Sympy [F(-1)]

Timed out. \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(x**4*(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 = 228, normalized size of antiderivative = 1.51 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {1}{25} \, b d^{3} n x^{5} + \frac {1}{5} \, b d^{3} x^{5} \log \left (c x^{n}\right ) + \frac {1}{5} \, a d^{3} x^{5} + \frac {b e^{3} x^{3 \, r + 5} \log \left (c x^{n}\right )}{3 \, r + 5} + \frac {3 \, b d e^{2} x^{2 \, r + 5} \log \left (c x^{n}\right )}{2 \, r + 5} + \frac {3 \, b d^{2} e x^{r + 5} \log \left (c x^{n}\right )}{r + 5} - \frac {b e^{3} n x^{3 \, r + 5}}{{\left (3 \, r + 5\right )}^{2}} + \frac {a e^{3} x^{3 \, r + 5}}{3 \, r + 5} - \frac {3 \, b d e^{2} n x^{2 \, r + 5}}{{\left (2 \, r + 5\right )}^{2}} + \frac {3 \, a d e^{2} x^{2 \, r + 5}}{2 \, r + 5} - \frac {3 \, b d^{2} e n x^{r + 5}}{{\left (r + 5\right )}^{2}} + \frac {3 \, a d^{2} e x^{r + 5}}{r + 5} \] Input:

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="maxima")
 

Output:

-1/25*b*d^3*n*x^5 + 1/5*b*d^3*x^5*log(c*x^n) + 1/5*a*d^3*x^5 + b*e^3*x^(3* 
r + 5)*log(c*x^n)/(3*r + 5) + 3*b*d*e^2*x^(2*r + 5)*log(c*x^n)/(2*r + 5) + 
 3*b*d^2*e*x^(r + 5)*log(c*x^n)/(r + 5) - b*e^3*n*x^(3*r + 5)/(3*r + 5)^2 
+ a*e^3*x^(3*r + 5)/(3*r + 5) - 3*b*d*e^2*n*x^(2*r + 5)/(2*r + 5)^2 + 3*a* 
d*e^2*x^(2*r + 5)/(2*r + 5) - 3*b*d^2*e*n*x^(r + 5)/(r + 5)^2 + 3*a*d^2*e* 
x^(r + 5)/(r + 5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (147) = 294\).

Time = 0.15 (sec) , antiderivative size = 1611, normalized size of antiderivative = 10.67 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x, algorithm="giac")
 

Output:

1/25*(300*b*e^3*n*r^5*x^5*x^(3*r)*log(x) + 1350*b*d*e^2*n*r^5*x^5*x^(2*r)* 
log(x) + 2700*b*d^2*e*n*r^5*x^5*x^r*log(x) + 180*b*d^3*n*r^6*x^5*log(x) - 
36*b*d^3*n*r^6*x^5 + 300*b*e^3*r^5*x^5*x^(3*r)*log(c) + 1350*b*d*e^2*r^5*x 
^5*x^(2*r)*log(c) + 2700*b*d^2*e*r^5*x^5*x^r*log(c) + 180*b*d^3*r^6*x^5*lo 
g(c) + 5000*b*e^3*n*r^4*x^5*x^(3*r)*log(x) + 21375*b*d*e^2*n*r^4*x^5*x^(2* 
r)*log(x) + 36000*b*d^2*e*n*r^4*x^5*x^r*log(x) + 3300*b*d^3*n*r^5*x^5*log( 
x) - 100*b*e^3*n*r^4*x^5*x^(3*r) + 300*a*e^3*r^5*x^5*x^(3*r) - 675*b*d*e^2 
*n*r^4*x^5*x^(2*r) + 1350*a*d*e^2*r^5*x^5*x^(2*r) - 2700*b*d^2*e*n*r^4*x^5 
*x^r + 2700*a*d^2*e*r^5*x^5*x^r - 660*b*d^3*n*r^5*x^5 + 180*a*d^3*r^6*x^5 
+ 5000*b*e^3*r^4*x^5*x^(3*r)*log(c) + 21375*b*d*e^2*r^4*x^5*x^(2*r)*log(c) 
 + 36000*b*d^2*e*r^4*x^5*x^r*log(c) + 3300*b*d^3*r^5*x^5*log(c) + 31875*b* 
e^3*n*r^3*x^5*x^(3*r)*log(x) + 127500*b*d*e^2*n*r^3*x^5*x^(2*r)*log(x) + 1 
81875*b*d^2*e*n*r^3*x^5*x^r*log(x) + 24125*b*d^3*n*r^4*x^5*log(x) - 1500*b 
*e^3*n*r^3*x^5*x^(3*r) + 5000*a*e^3*r^4*x^5*x^(3*r) - 9000*b*d*e^2*n*r^3*x 
^5*x^(2*r) + 21375*a*d*e^2*r^4*x^5*x^(2*r) - 22500*b*d^2*e*n*r^3*x^5*x^r + 
 36000*a*d^2*e*r^4*x^5*x^r - 4825*b*d^3*n*r^4*x^5 + 3300*a*d^3*r^5*x^5 + 3 
1875*b*e^3*r^3*x^5*x^(3*r)*log(c) + 127500*b*d*e^2*r^3*x^5*x^(2*r)*log(c) 
+ 181875*b*d^2*e*r^3*x^5*x^r*log(c) + 24125*b*d^3*r^4*x^5*log(c) + 96875*b 
*e^3*n*r^2*x^5*x^(3*r)*log(x) + 356250*b*d*e^2*n*r^2*x^5*x^(2*r)*log(x) + 
440625*b*d^2*e*n*r^2*x^5*x^r*log(x) + 90000*b*d^3*n*r^3*x^5*log(x) - 81...
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^4\,{\left (d+e\,x^r\right )}^3\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

int(x^4*(d + e*x^r)^3*(a + b*log(c*x^n)),x)
 

Output:

int(x^4*(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 = 6.97 \[ \int x^4 \left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:

int(x^4*(d+e*x^r)^3*(a+b*log(c*x^n)),x)
 

Output:

(x**5*(300*x**(3*r)*log(x**n*c)*b*e**3*r**5 + 5000*x**(3*r)*log(x**n*c)*b* 
e**3*r**4 + 31875*x**(3*r)*log(x**n*c)*b*e**3*r**3 + 96875*x**(3*r)*log(x* 
*n*c)*b*e**3*r**2 + 140625*x**(3*r)*log(x**n*c)*b*e**3*r + 78125*x**(3*r)* 
log(x**n*c)*b*e**3 + 300*x**(3*r)*a*e**3*r**5 + 5000*x**(3*r)*a*e**3*r**4 
+ 31875*x**(3*r)*a*e**3*r**3 + 96875*x**(3*r)*a*e**3*r**2 + 140625*x**(3*r 
)*a*e**3*r + 78125*x**(3*r)*a*e**3 - 100*x**(3*r)*b*e**3*n*r**4 - 1500*x** 
(3*r)*b*e**3*n*r**3 - 8125*x**(3*r)*b*e**3*n*r**2 - 18750*x**(3*r)*b*e**3* 
n*r - 15625*x**(3*r)*b*e**3*n + 1350*x**(2*r)*log(x**n*c)*b*d*e**2*r**5 + 
21375*x**(2*r)*log(x**n*c)*b*d*e**2*r**4 + 127500*x**(2*r)*log(x**n*c)*b*d 
*e**2*r**3 + 356250*x**(2*r)*log(x**n*c)*b*d*e**2*r**2 + 468750*x**(2*r)*l 
og(x**n*c)*b*d*e**2*r + 234375*x**(2*r)*log(x**n*c)*b*d*e**2 + 1350*x**(2* 
r)*a*d*e**2*r**5 + 21375*x**(2*r)*a*d*e**2*r**4 + 127500*x**(2*r)*a*d*e**2 
*r**3 + 356250*x**(2*r)*a*d*e**2*r**2 + 468750*x**(2*r)*a*d*e**2*r + 23437 
5*x**(2*r)*a*d*e**2 - 675*x**(2*r)*b*d*e**2*n*r**4 - 9000*x**(2*r)*b*d*e** 
2*n*r**3 - 41250*x**(2*r)*b*d*e**2*n*r**2 - 75000*x**(2*r)*b*d*e**2*n*r - 
46875*x**(2*r)*b*d*e**2*n + 2700*x**r*log(x**n*c)*b*d**2*e*r**5 + 36000*x* 
*r*log(x**n*c)*b*d**2*e*r**4 + 181875*x**r*log(x**n*c)*b*d**2*e*r**3 + 440 
625*x**r*log(x**n*c)*b*d**2*e*r**2 + 515625*x**r*log(x**n*c)*b*d**2*e*r + 
234375*x**r*log(x**n*c)*b*d**2*e + 2700*x**r*a*d**2*e*r**5 + 36000*x**r*a* 
d**2*e*r**4 + 181875*x**r*a*d**2*e*r**3 + 440625*x**r*a*d**2*e*r**2 + 5...