Integrand size = 26, antiderivative size = 913 \[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {x \left (b^2-2 a c+b c x^n\right ) \left (d+e x^n\right )^2}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}-\frac {x \left (4 a^2 c \left (c d^2 (1-4 n)-a e^2 (1-2 n)\right )+4 a^2 b c d e (2-5 n)+b^4 d^2 (1-2 n)-2 a b^3 d e (1-n)-a b^2 \left (5 c d^2 (1-3 n)-a e^2 (1+n)\right )-\left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)-a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+a e^2 n\right )\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {c \left (b^3 d \left (2 a e-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d^2 \left (1-3 n+2 n^2\right )-8 a^2 c \left (c d^2 \left (1-6 n+8 n^2\right )-e \left (a e (1-2 n)-\sqrt {b^2-4 a c} d \left (1-4 n+3 n^2\right )\right )\right )+2 a b^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )+e \left (\sqrt {b^2-4 a c} d (1-n)-a e \left (1-2 n+3 n^2\right )\right )\right )-2 a b \left (3 a \sqrt {b^2-4 a c} e^2 (1-n) n+c d \left (4 a e \left (1-n-3 n^2\right )-\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )\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 )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) n^2}+\frac {c \left (b^3 d \left (2 a e+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d^2 \left (1-3 n+2 n^2\right )-8 a^2 c \left (c d^2 \left (1-6 n+8 n^2\right )-e \left (a e (1-2 n)+\sqrt {b^2-4 a c} d \left (1-4 n+3 n^2\right )\right )\right )+2 a b^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )-e \left (\sqrt {b^2-4 a c} d (1-n)+a e \left (1-2 n+3 n^2\right )\right )\right )+2 a b \left (3 a \sqrt {b^2-4 a c} e^2 (1-n) n-c d \left (4 a e \left (1-n-3 n^2\right )+\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )\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 )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) n^2} \] Output:
1/2*x*(b^2-2*a*c+b*c*x^n)*(d+e*x^n)^2/a/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n)) ^2-1/2*x*(4*a^2*c*(c*d^2*(1-4*n)-a*e^2*(1-2*n))+4*a^2*b*c*d*e*(2-5*n)+b^4* d^2*(1-2*n)-2*a*b^3*d*e*(1-n)-a*b^2*(5*c*d^2*(1-3*n)-a*e^2*(1+n))-(2*a*b^2 *c*d*e-8*a^2*c^2*d*e*(1-3*n)+2*a*b*c*(c*d^2*(2-7*n)-a*e^2*n)-b^3*(c*d^2*(1 -2*n)+a*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*c*(b^3 *d*(2*a*e-(-4*a*c+b^2)^(1/2)*d*(1-2*n))*(1-n)-b^4*d^2*(2*n^2-3*n+1)-8*a^2* c*(c*d^2*(8*n^2-6*n+1)-e*(a*e*(1-2*n)-(-4*a*c+b^2)^(1/2)*d*(3*n^2-4*n+1))) +2*a*b^2*(3*c*d^2*(3*n^2-4*n+1)+e*((-4*a*c+b^2)^(1/2)*d*(1-n)-a*e*(3*n^2-2 *n+1)))-2*a*b*(3*a*(-4*a*c+b^2)^(1/2)*e^2*(1-n)*n+c*d*(4*a*e*(-3*n^2-n+1)- (-4*a*c+b^2)^(1/2)*d*(7*n^2-9*n+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^2-4*a*c-b*(-4*a*c+b^2)^(1/ 2))/n^2+1/2*c*(b^3*d*(2*a*e+(-4*a*c+b^2)^(1/2)*d*(1-2*n))*(1-n)-b^4*d^2*(2 *n^2-3*n+1)-8*a^2*c*(c*d^2*(8*n^2-6*n+1)-e*(a*e*(1-2*n)+(-4*a*c+b^2)^(1/2) *d*(3*n^2-4*n+1)))+2*a*b^2*(3*c*d^2*(3*n^2-4*n+1)-e*((-4*a*c+b^2)^(1/2)*d* (1-n)+a*e*(3*n^2-2*n+1)))+2*a*b*(3*a*(-4*a*c+b^2)^(1/2)*e^2*(1-n)*n-c*d*(4 *a*e*(-3*n^2-n+1)+(-4*a*c+b^2)^(1/2)*d*(7*n^2-9*n+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)-4*a*c+b^2)/n^2
Leaf count is larger than twice the leaf count of optimal. \(10910\) vs. \(2(913)=1826\).
Time = 7.90 (sec) , antiderivative size = 10910, normalized size of antiderivative = 11.95 \[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^3,x]
Output:
Result too large to show
Time = 3.22 (sec) , antiderivative size = 1191, normalized size of antiderivative = 1.30, 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.
| \(\displaystyle \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx\) |
| \(\Big \downarrow \) 1766 |
| \(\displaystyle \int \left (\frac {-a e^2+x^n \left (2 c d e-b e^2\right )+c d^2}{c \left (a+b x^n+c x^{2 n}\right )^3}+\frac {e^2}{c \left (a+b x^n+c x^{2 n}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (-\left ((1-n) b^2\right )-\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 n)\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 \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) n}-\frac {\left (-\left ((1-n) b^2\right )+\sqrt {b^2-4 a c} (1-n) b+4 a c (1-2 n)\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 \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 (b c x^n+b^2-2 a c\right ) e^2}{a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac {\left ((1-n) \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )+\frac {-\left ((1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4\right )+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) n^2}-\frac {\left ((1-n) \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )-\frac {-\left ((1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4\right )+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-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 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) n^2}+\frac {x \left (c \left (-\left (\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3\right )+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right ) x^n+2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c (1-2 n) d^2+2 a e^2 n\right )\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (\left (b c d^2-4 a c e d+a b e^2\right ) x^n+b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )}{2 a \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)^2/(a + b*x^n + c*x^(2*n))^3,x]
Output:
(x*(b^2*d^2 - 2*a*b*d*e - 2*a*(c*d^2 - a*e^2) + (b*c*d^2 - 4*a*c*d*e + a*b *e^2)*x^n))/(2*a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2 - 2*a*c + b*c*x^n))/(a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n))) + (x*(2* a*b^3*c*d*e - a*b^2*c*(a*e^2*(1 - 9*n) - 5*c*d^2*(1 - 3*n)) - 4*a^2*c^2*(c *d^2 - a*e^2)*(1 - 4*n) - 4*a^2*b*c^2*d*e*(2 - 3*n) - b^4*(c*d^2*(1 - 2*n) + 2*a*e^2*n) + c*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^ 2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a*e^2*n))*x^n))/(2*a^2*c *(b^2 - 4*a*c)^2*n^2*(a + b*x^n + c*x^(2*n))) - (e^2*(4*a*c*(1 - 2*n) - b^ 2*(1 - n) - b*Sqrt[b^2 - 4*a*c]*(1 - n))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a *c - b*Sqrt[b^2 - 4*a*c])*n) - (((1 - n)*(2*a*b^2*c*d*e - 8*a^2*c^2*d*e*(1 - 3*n) + 2*a*b*c*(c*d^2*(2 - 7*n) + a*e^2*n) - b^3*(c*d^2*(1 - 2*n) + 2*a *e^2*n)) + (2*a*b^3*c*d*e*(1 - n) - b^4*(1 - n)*(c*d^2*(1 - 2*n) + 2*a*e^2 *n) - 8*a^2*b*c^2*d*e*(1 - n - 3*n^2) - 8*a^2*c^2*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 2*a*b^2*c*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(1 - 10*n + 15*n^ 2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x ^n)/(b - Sqrt[b^2 - 4*a*c])])/(2*a^2*(b^2 - 4*a*c)^2*(b - Sqrt[b^2 - 4*a*c ])*n^2) - (e^2*(4*a*c*(1 - 2*n) - b^2*(1 - n) + b*Sqrt[b^2 - 4*a*c]*(1 - n ))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4 *a*c])])/(a*(b^2 - 4*a*c)*(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])*n) - (((1...
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]))
\[\int \frac {\left (d +e \,x^{n}\right )^{2}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}d x\]
Input:
int((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
Output:
int((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
\[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="fricas")
Output:
integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)/(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)
Timed out. \[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Timed out} \] Input:
integrate((d+e*x**n)**2/(a+b*x**n+c*x**(2*n))**3,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="maxima")
Output:
1/2*((b^3*c^2*d^2*(2*n - 1) + 2*(4*c^3*d*e*(3*n - 1) - 3*b*c^2*e^2*n)*a^2 - 2*(b*c^3*d^2*(7*n - 2) - b^2*c^2*d*e)*a)*x*x^(3*n) + (2*b^4*c*d^2*(2*n - 1) + 4*a^3*c^2*e^2 - (b^2*c*e^2*(9*n + 1) - 4*b*c^2*d*e*(9*n - 4) - 4*c^3 *d^2*(4*n - 1))*a^2 - (b^2*c^2*d^2*(29*n - 9) - 4*b^3*c*d*e)*a)*x*x^(2*n) + (b^5*d^2*(2*n - 1) + 2*(4*c^2*d*e*(5*n - 1) - b*c*e^2*(5*n - 2))*a^3 + ( 2*b^2*c*d*e*(4*n - 3) - b^3*e^2*(2*n + 1) - 2*b*c^2*d^2*n)*a^2 - 2*(2*b^3* c*d^2*(3*n - 1) - b^4*d*e)*a)*x*x^n + (a*b^4*d^2*(3*n - 1) - 4*a^4*c*e^2*( 2*n - 1) + (4*c^2*d^2*(6*n - 1) + 4*b*c*d*e*(5*n - 2) - b^2*e^2*(n + 1))*a ^3 - (b^2*c*d^2*(21*n - 5) + 2*b^3*d*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^2 + 4*a^3*c*e^2*(2*n - 1) + (4*(8*n^2 - 6*n + 1 )*c^2*d^2 - 4*b*c*d*e*(5*n - 2) + b^2*e^2*(n + 1))*a^2 - ((16*n^2 - 21*n + 5)*b^2*c*d^2 - 2*b^3*d*e*(n - 1))*a + ((2*n^2 - 3*n + 1)*b^3*c*d^2 + 2*(4 *(3*n^2 - 4*n + 1)*c^2*d*e - 3*(n^2 - n)*b*c*e^2)*a^2 - 2*((7*n^2 - 9*n + 2)*b*c^2*d^2 - b^2*c*d*e*(n - 1))*a)*x^n)/(a^3*b^4*n^2 - 8*a^4*b^2*c*n^2 + 16*a^5*c^2*n^2 + (a^2*b^4*c*n^2 - 8*a^3*b^2*c^2*n^2 + 16*a^4*c^3*n^2)*x^( 2*n) + (a^2*b^5*n^2 - 8*a^3*b^3*c*n^2 + 16*a^4*b*c^2*n^2)*x^n), x)
\[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )}^{2}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \] Input:
integrate((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="giac")
Output:
integrate((e*x^n + d)^2/(c*x^(2*n) + b*x^n + a)^3, x)
Timed out. \[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int \frac {{\left (d+e\,x^n\right )}^2}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \] Input:
int((d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^3,x)
Output:
int((d + e*x^n)^2/(a + b*x^n + c*x^(2*n))^3, x)
\[ \int \frac {\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {too large to display} \] Input:
int((d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
Output:
(4*x**(4*n)*int(x**(2*n)/(2*x**(6*n)*c**3*n - x**(6*n)*c**3 + 6*x**(5*n)*b *c**2*n - 3*x**(5*n)*b*c**2 + 6*x**(4*n)*a*c**2*n - 3*x**(4*n)*a*c**2 + 6* x**(4*n)*b**2*c*n - 3*x**(4*n)*b**2*c + 12*x**(3*n)*a*b*c*n - 6*x**(3*n)*a *b*c + 2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2*n)*a**2*c*n - 3*x**(2*n )*a**2*c + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x **n*a**2*b + 2*a**3*n - a**3),x)*b*c**2*e**2*n**2 - 4*x**(4*n)*int(x**(2*n )/(2*x**(6*n)*c**3*n - x**(6*n)*c**3 + 6*x**(5*n)*b*c**2*n - 3*x**(5*n)*b* c**2 + 6*x**(4*n)*a*c**2*n - 3*x**(4*n)*a*c**2 + 6*x**(4*n)*b**2*c*n - 3*x **(4*n)*b**2*c + 12*x**(3*n)*a*b*c*n - 6*x**(3*n)*a*b*c + 2*x**(3*n)*b**3* n - x**(3*n)*b**3 + 6*x**(2*n)*a**2*c*n - 3*x**(2*n)*a**2*c + 6*x**(2*n)*a *b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*b*c**2*e**2*n + x**(4*n)*int(x**(2*n)/(2*x**(6*n)*c**3*n - x**(6 *n)*c**3 + 6*x**(5*n)*b*c**2*n - 3*x**(5*n)*b*c**2 + 6*x**(4*n)*a*c**2*n - 3*x**(4*n)*a*c**2 + 6*x**(4*n)*b**2*c*n - 3*x**(4*n)*b**2*c + 12*x**(3*n) *a*b*c*n - 6*x**(3*n)*a*b*c + 2*x**(3*n)*b**3*n - x**(3*n)*b**3 + 6*x**(2* n)*a**2*c*n - 3*x**(2*n)*a**2*c + 6*x**(2*n)*a*b**2*n - 3*x**(2*n)*a*b**2 + 6*x**n*a**2*b*n - 3*x**n*a**2*b + 2*a**3*n - a**3),x)*b*c**2*e**2 - 16*x **(4*n)*int(x**(2*n)/(2*x**(6*n)*c**3*n - x**(6*n)*c**3 + 6*x**(5*n)*b*c** 2*n - 3*x**(5*n)*b*c**2 + 6*x**(4*n)*a*c**2*n - 3*x**(4*n)*a*c**2 + 6*x**( 4*n)*b**2*c*n - 3*x**(4*n)*b**2*c + 12*x**(3*n)*a*b*c*n - 6*x**(3*n)*a*...