\(\int \frac {(d+e x^n)^3}{(a+b x^n+c x^{2 n})^3} \, dx\) [74]

Optimal result

Integrand size = 26, antiderivative size = 1048 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx =\text {Too large to display} \] Output:

1/2*x*(b^2-2*a*c+b*c*x^n)*(d+e*x^n)^3/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n)) 
^2-1/2*b*e^3*x^(1+2*n)/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))-1/2*x*(4*a^2*c 
*d*(c*d^2*(1-4*n)-3*a*e^2*(1-2*n))+b^4*d^3*(1-2*n)-3*a*b^3*d^2*e*(1-n)+6*a 
^2*b*e*(c*d^2*(2-5*n)-a*e^2*n)-a*b^2*d*(5*c*d^2*(1-3*n)-3*a*e^2*(1+n))-(a* 
b^2*e*(3*c*d^2-a*e^2*(1-4*n))-4*a^2*c*e*(3*c*d^2*(1-3*n)-a*e^2*(1-n))+2*a* 
b*c*d*(c*d^2*(2-7*n)-3*a*e^2*n)-b^3*(c*d^3*(1-2*n)+3*a*d*e^2*n))*x^n)/a^2/ 
(-4*a*c+b^2)^2/n^2/(a+b*x^n+c*x^(2*n))-1/2*((1-n)*(a*b^2*e*(3*c*d^2-a*e^2* 
(1-2*n))-b^3*c*d^3*(1-2*n)+2*a*b*c*d*(c*d^2*(2-7*n)-9*a*e^2*n)-4*a^2*c*e*( 
3*c*d^2*(1-3*n)-a*e^2*(1+n)))+(8*a^2*c^2*d*(3*a*e^2-c*d^2*(1-4*n))*(1-2*n) 
+a*b^3*e*(3*c*d^2+a*e^2*(1-2*n))*(1-n)-b^4*c*d^3*(2*n^2-3*n+1)-4*a^2*b*c*e 
*(3*c*d^2*(-3*n^2-n+1)+a*e^2*(-n^2-3*n+1))+6*a*b^2*c*d*(c*d^2*(3*n^2-4*n+1 
)-a*e^2*(3*n^2-2*n+1)))/(-4*a*c+b^2)^(1/2))*x*hypergeom([1, 1/n],[1+1/n],- 
2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2))/ 
n^2-1/2*((1-n)*(a*b^2*e*(3*c*d^2-a*e^2*(1-2*n))-b^3*c*d^3*(1-2*n)+2*a*b*c* 
d*(c*d^2*(2-7*n)-9*a*e^2*n)-4*a^2*c*e*(3*c*d^2*(1-3*n)-a*e^2*(1+n)))-(8*a^ 
2*c^2*d*(3*a*e^2-c*d^2*(1-4*n))*(1-2*n)+a*b^3*e*(3*c*d^2+a*e^2*(1-2*n))*(1 
-n)-b^4*c*d^3*(2*n^2-3*n+1)-4*a^2*b*c*e*(3*c*d^2*(-3*n^2-n+1)+a*e^2*(-n^2- 
3*n+1))+6*a*b^2*c*d*(c*d^2*(3*n^2-4*n+1)-a*e^2*(3*n^2-2*n+1)))/(-4*a*c+b^2 
)^(1/2))*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/a^2 
/(-4*a*c+b^2)^2/(b+(-4*a*c+b^2)^(1/2))/n^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(13018\) vs. \(2(1048)=2096\).

Time = 8.29 (sec) , antiderivative size = 13018, normalized size of antiderivative = 12.42 \[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 4.49 (sec) , antiderivative size = 1707, normalized size of antiderivative = 1.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1766, 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.

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

\(\Big \downarrow \) 1766

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (-e (1-n) b^3+\left (3 c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-5 n)+3 \sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (6 c d (1-2 n)+\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) e^2}{a c \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) n}+\frac {\left (-e (1-n) b^3+\left (3 c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-5 n)-3 \sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (6 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) e^2}{a c \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) n}+\frac {x \left (c \left (-e b^2+3 c d b-2 a c e\right ) x^n-6 a c^2 d+3 b^2 c d-b^3 e+a b c e\right ) e^2}{a c^2 \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}+\frac {\left ((1-n) \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right )-\frac {2 a e^3 (1-n) n b^5-c d (1-n) \left (c (1-2 n) d^2+6 a e^2 n\right ) b^4+a c e \left (3 c (1-n) d^2+a e^2 \left (30 n^2-19 n+1\right )\right ) b^3+6 a c^2 d \left (c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-4 a^2 c^2 e \left (3 c \left (-3 n^2-n+1\right ) d^2+a e^2 \left (19 n^2-11 n+1\right )\right ) b-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}+\frac {\left ((1-n) \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right )+\frac {2 a e^3 (1-n) n b^5-c d (1-n) \left (c (1-2 n) d^2+6 a e^2 n\right ) b^4+a c e \left (3 c (1-n) d^2+a e^2 \left (30 n^2-19 n+1\right )\right ) b^3+6 a c^2 d \left (c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-4 a^2 c^2 e \left (3 c \left (-3 n^2-n+1\right ) d^2+a e^2 \left (19 n^2-11 n+1\right )\right ) b-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 c \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}-\frac {x \left (c \left (-2 a e^3 n b^4+c d \left (c (1-2 n) d^2+6 a e^2 n\right ) b^3-a c e \left (3 c d^2-a e^2 (2 n+1)\right ) b^2-2 a c^2 d \left (c (2-7 n) d^2+3 a e^2 n\right ) b+4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)\right ) x^n+a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c (1-2 n) d^2+6 a e^2 n\right )\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (-\left (\left (a b^2 e^3+2 a c \left (3 c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )+b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )}{2 a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2}\)

Input:

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

Output:

(x*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2) - (a*b 
^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^n))/(2*a*c 
*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(3*b^2*c*d - 6*a*c^2* 
d - b^3*e + a*b*c*e + c*(3*b*c*d - b^2*e - 2*a*c*e)*x^n))/(a*c^2*(b^2 - 4* 
a*c)*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*d*(3*a*e^2*(1 - 9*n) - 5*c 
*d^2*(1 - 3*n)) + 4*a^2*c^3*d*(c*d^2 - 3*a*e^2)*(1 - 4*n) - 2*a*b^5*e^3*n 
+ 2*a^2*b*c^2*e*(3*c*d^2*(2 - 3*n) - 5*a*e^2*n) - 3*a*b^3*c*e*(c*d^2 - 3*a 
*e^2*n) + b^4*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) + c*(4*a^2*c^2*e*(3*c*d^2 
- a*e^2)*(1 - 3*n) - 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^ 
2*n) + b^3*c*d*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2* 
(1 + 2*n)))*x^n))/(2*a^2*c^2*(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) 
+ (e^2*(b*c*(2*a*e*(2 - 5*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(6*c 
*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(3*c*d - 
 Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), 
(-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*c*(b^2 - 4*a*c)*(b^2 - 4*a*c - b*S 
qrt[b^2 - 4*a*c])*n) + (((1 - n)*(4*a^2*c^2*e*(3*c*d^2 - a*e^2)*(1 - 3*n) 
- 2*a*b^4*e^3*n - 2*a*b*c^2*d*(c*d^2*(2 - 7*n) + 3*a*e^2*n) + b^3*c*d*(c*d 
^2*(1 - 2*n) + 6*a*e^2*n) - a*b^2*c*e*(3*c*d^2 - a*e^2*(1 + 2*n))) - (2*a* 
b^5*e^3*(1 - n)*n - b^4*c*d*(1 - n)*(c*d^2*(1 - 2*n) + 6*a*e^2*n) - 8*a^2* 
c^3*d*(c*d^2 - 3*a*e^2)*(1 - 6*n + 8*n^2) + 6*a*b^2*c^2*d*(c*d^2*(1 - 4...
 

Defintions of rubi rules used

rule 1766

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
))^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + b*x^n + c*x^(2 
*n))^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ 
[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ((IntegersQ[p, q] && 
!IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]))
 

rule 2009

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

Maple [F]

\[\int \frac {\left (d +e \,x^{n}\right )^{3}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:

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

Output:

integral((e^3*x^(3*n) + 3*d*e^2*x^(2*n) + 3*d^2*e*x^n + d^3)/(c^3*x^(6*n) 
+ b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3 + 3*(b*c^2*x^n + a*c^2 
)*x^(4*n) + 3*(b^2*c*x^(2*n) + 2*a*b*c*x^n + a^2*c)*x^(2*n)), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:

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

Output:

1/2*((b^3*c^2*d^3*(2*n - 1) + 4*a^3*c^2*e^3*(n + 1) + (12*c^3*d^2*e*(3*n - 
 1) + b^2*c*e^3*(2*n - 1) - 18*b*c^2*d*e^2*n)*a^2 - (2*b*c^3*d^3*(7*n - 2) 
 - 3*b^2*c^2*d^2*e)*a)*x*x^(3*n) + (2*b^4*c*d^3*(2*n - 1) + 2*(b*c*e^3*(3* 
n + 2) + 6*c^2*d*e^2)*a^3 - (3*b^2*c*d*e^2*(9*n + 1) - 6*b*c^2*d^2*e*(9*n 
- 4) - 4*c^3*d^3*(4*n - 1) - b^3*e^3*(3*n - 1))*a^2 - (b^2*c^2*d^3*(29*n - 
 9) - 6*b^3*c*d^2*e)*a)*x*x^(2*n) + (b^5*d^3*(2*n - 1) - 4*a^4*c*e^3*(n - 
1) + (b^2*e^3*(10*n - 1) + 12*c^2*d^2*e*(5*n - 1) - 6*b*c*d*e^2*(5*n - 2)) 
*a^3 + (3*b^2*c*d^2*e*(4*n - 3) - 3*b^3*d*e^2*(2*n + 1) - 2*b*c^2*d^3*n)*a 
^2 - (4*b^3*c*d^3*(3*n - 1) - 3*b^4*d^2*e)*a)*x*x^n + (a*b^4*d^3*(3*n - 1) 
 - 6*(2*c*d*e^2*(2*n - 1) - b*e^3*n)*a^4 + (4*c^2*d^3*(6*n - 1) + 6*b*c*d^ 
2*e*(5*n - 2) - 3*b^2*d*e^2*(n + 1))*a^3 - (b^2*c*d^3*(21*n - 5) + 3*b^3*d 
^2*e*(n - 1))*a^2)*x)/(a^4*b^4*n^2 - 8*a^5*b^2*c*n^2 + 16*a^6*c^2*n^2 + (a 
^2*b^4*c^2*n^2 - 8*a^3*b^2*c^3*n^2 + 16*a^4*c^4*n^2)*x^(4*n) + 2*(a^2*b^5* 
c*n^2 - 8*a^3*b^3*c^2*n^2 + 16*a^4*b*c^3*n^2)*x^(3*n) + (a^2*b^6*n^2 - 6*a 
^3*b^4*c*n^2 + 32*a^5*c^3*n^2)*x^(2*n) + 2*(a^3*b^5*n^2 - 8*a^4*b^3*c*n^2 
+ 16*a^5*b*c^2*n^2)*x^n) + integrate(1/2*((2*n^2 - 3*n + 1)*b^4*d^3 + 6*(2 
*c*d*e^2*(2*n - 1) - b*e^3*n)*a^3 + (4*(8*n^2 - 6*n + 1)*c^2*d^3 - 6*b*c*d 
^2*e*(5*n - 2) + 3*b^2*d*e^2*(n + 1))*a^2 - ((16*n^2 - 21*n + 5)*b^2*c*d^3 
 - 3*b^3*d^2*e*(n - 1))*a + ((2*n^2 - 3*n + 1)*b^3*c*d^3 + 4*(n^2 - 1)*a^3 
*c*e^3 + (12*(3*n^2 - 4*n + 1)*c^2*d^2*e - 18*(n^2 - n)*b*c*d*e^2 + (2*...
 

Giac [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{3}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:

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

Output:

integrate((e*x^n + d)^3/(c*x^(2*n) + b*x^n + a)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {too large to display} \] Input:

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

Output:

( - 24*x**(4*n)*int(x**(2*n)/(6*x**(6*n)*c**3*n**2 - 5*x**(6*n)*c**3*n + x 
**(6*n)*c**3 + 18*x**(5*n)*b*c**2*n**2 - 15*x**(5*n)*b*c**2*n + 3*x**(5*n) 
*b*c**2 + 18*x**(4*n)*a*c**2*n**2 - 15*x**(4*n)*a*c**2*n + 3*x**(4*n)*a*c* 
*2 + 18*x**(4*n)*b**2*c*n**2 - 15*x**(4*n)*b**2*c*n + 3*x**(4*n)*b**2*c + 
36*x**(3*n)*a*b*c*n**2 - 30*x**(3*n)*a*b*c*n + 6*x**(3*n)*a*b*c + 6*x**(3* 
n)*b**3*n**2 - 5*x**(3*n)*b**3*n + x**(3*n)*b**3 + 18*x**(2*n)*a**2*c*n**2 
 - 15*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + 18*x**(2*n)*a*b**2*n**2 - 15 
*x**(2*n)*a*b**2*n + 3*x**(2*n)*a*b**2 + 18*x**n*a**2*b*n**2 - 15*x**n*a** 
2*b*n + 3*x**n*a**2*b + 6*a**3*n**2 - 5*a**3*n + a**3),x)*a*c**3*e**3*n**4 
 + 2*x**(4*n)*int(x**(2*n)/(6*x**(6*n)*c**3*n**2 - 5*x**(6*n)*c**3*n + x** 
(6*n)*c**3 + 18*x**(5*n)*b*c**2*n**2 - 15*x**(5*n)*b*c**2*n + 3*x**(5*n)*b 
*c**2 + 18*x**(4*n)*a*c**2*n**2 - 15*x**(4*n)*a*c**2*n + 3*x**(4*n)*a*c**2 
 + 18*x**(4*n)*b**2*c*n**2 - 15*x**(4*n)*b**2*c*n + 3*x**(4*n)*b**2*c + 36 
*x**(3*n)*a*b*c*n**2 - 30*x**(3*n)*a*b*c*n + 6*x**(3*n)*a*b*c + 6*x**(3*n) 
*b**3*n**2 - 5*x**(3*n)*b**3*n + x**(3*n)*b**3 + 18*x**(2*n)*a**2*c*n**2 - 
 15*x**(2*n)*a**2*c*n + 3*x**(2*n)*a**2*c + 18*x**(2*n)*a*b**2*n**2 - 15*x 
**(2*n)*a*b**2*n + 3*x**(2*n)*a*b**2 + 18*x**n*a**2*b*n**2 - 15*x**n*a**2* 
b*n + 3*x**n*a**2*b + 6*a**3*n**2 - 5*a**3*n + a**3),x)*a*c**3*e**3*n**3 + 
 17*x**(4*n)*int(x**(2*n)/(6*x**(6*n)*c**3*n**2 - 5*x**(6*n)*c**3*n + x**( 
6*n)*c**3 + 18*x**(5*n)*b*c**2*n**2 - 15*x**(5*n)*b*c**2*n + 3*x**(5*n)...