\(\int \frac {1}{(d+e x^n)^2 (a+b x^n+c x^{2 n})^3} \, dx\) [78]

Optimal result

Integrand size = 26, antiderivative size = 3487 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx =\text {Too large to display} \] Output:

1/2*x*(c*(-2*a*c+b^2)*d-b*(-3*a*c+b^2)*e+c*(2*a*c*e-b^2*e+b*c*d)*x^n)/a/(- 
4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/n/(d+e*x^n)/(a+b*x^n+c*x^(2*n))^2+1/2*x*(4* 
a^2*b*c^3*d*e*(a*e^2*(3-28*n)+c*d^2*(5-16*n))-a*b^2*c^2*(a^2*e^4*(13-85*n) 
+2*a*c*d^2*e^2*(12-23*n)-5*c^2*d^4*(1-3*n))-b^5*c*d*e*(a*e^2*(5-6*n)-3*c*d 
^2*(1-2*n))-b^6*e^2*(a*e^2*(1-5*n)+3*c*d^2*(1-2*n))+b^4*c*(a^2*e^4*(7-40*n 
)+6*a*c*d^2*e^2*(3-7*n)-c^2*d^4*(1-2*n))+b^7*d*e^3*(1-2*n)+4*a^2*c^3*(a^2* 
e^4*(1-8*n)-c^2*d^4*(1-4*n)+8*a*c*d^2*e^2*n)-a*b^3*c^2*d*e*(c*d^2*(17-46*n 
)-a*e^2*(1+42*n))-c*(2*a*b*c^2*(a^2*e^4*(4-27*n)-c^2*d^4*(2-7*n)+2*a*c*d^2 
*e^2*(3-4*n))-8*a^2*c^3*d*e*(a*e^2*(1-9*n)+c*d^2*(1-3*n))+2*a*b^2*c^2*d*e* 
(a*e^2*(1-23*n)+7*c*d^2*(1-3*n))+b^5*e^2*(a*e^2*(1-5*n)+3*c*d^2*(1-2*n))-b 
^3*c*(a*c*d^2*e^2*(15-37*n)+a^2*e^4*(6-35*n)-c^2*d^4*(1-2*n))-b^4*c*d*e*(3 
*c*d^2*(1-2*n)-4*a*e^2*(1-n))-b^6*d*e^3*(1-2*n))*x^n)/a^2/(-4*a*c+b^2)^2/( 
a*e^2-b*d*e+c*d^2)^3/n^2/(a+b*x^n+c*x^(2*n))+1/2*c*e*(b^4*e^3*(e*(a*e*(1-7 
*n)-(-4*a*c+b^2)^(1/2)*d*(1-n))+c*d^2*(5-7*n))-b^3*e^2*(c*d*e*(a*e*(1-23*n 
)-3*(-4*a*c+b^2)^(1/2)*d*(1-n))-a*(-4*a*c+b^2)^(1/2)*e^3*(1-7*n)+3*c^2*d^3 
*(3-5*n))+b*c*(2*a*c*d*e^3*(a*e*(5-48*n)-3*(-4*a*c+b^2)^(1/2)*d*(1-n))+c^2 
*d^3*e*(4*a*e*(4-7*n)+(-4*a*c+b^2)^(1/2)*d*(1-n))-3*a^2*(-4*a*c+b^2)^(1/2) 
*e^5*(1-9*n)-2*c^3*d^5*(1-2*n))-b^5*d*e^4*(1-n)-4*a*c^2*e*(c*d^2*e*(2*a*e* 
(1-12*n)-(-4*a*c+b^2)^(1/2)*d*(1-n))+2*c^2*d^4*(1-2*n)-a*e^3*((-4*a*c+b^2) 
^(1/2)*d*(1-13*n)-4*a*e*n))+b^2*c*e*(c^2*d^4*(7-13*n)-c*d^2*e*(3*(-4*a*...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(56566\) vs. \(2(3487)=6974\).

Time = 10.83 (sec) , antiderivative size = 56566, normalized size of antiderivative = 16.22 \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 6.70 (sec) , antiderivative size = 2446, normalized size of antiderivative = 0.70, 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.

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

\(\Big \downarrow \) 1766

\(\displaystyle \int \left (\frac {e^2 \left (-a c e^2+2 b^2 e^2+x^n \left (2 b c e^2-4 c^2 d e\right )-5 b c d e+3 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^3 \left (a+b x^n+c x^{2 n}\right )^2}+\frac {-a c e^2+b^2 e^2-\left (x^n \left (2 c^2 d e-b c e^2\right )\right )-2 b c d e+c^2 d^2}{\left (a e^2-b d e+c d^2\right )^2 \left (a+b x^n+c x^{2 n}\right )^3}+\frac {e^4 \left (-a c e^2+3 b^2 e^2+x^n \left (3 b c e^2-6 c^2 d e\right )-8 b c d e+5 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4 \left (a+b x^n+c x^{2 n}\right )}-\frac {3 e^6 (b e-2 c d)}{\left (d+e x^n\right ) \left (a e^2-b d e+c d^2\right )^4}+\frac {e^6}{\left (d+e x^n\right )^2 \left (a e^2-b d e+c d^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3 (2 c d-b e) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right ) e^6}{d \left (c d^2-b e d+a e^2\right )^4}+\frac {x \operatorname {Hypergeometric2F1}\left (2,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right ) e^6}{d^2 \left (c d^2-b e d+a e^2\right )^3}-\frac {c \left (10 c^2 d^2+3 b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (5 b d+3 \sqrt {b^2-4 a c} d+a e\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^4}{\left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^4}-\frac {c \left (10 c^2 d^2+3 b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (5 b d-3 \sqrt {b^2-4 a c} d+a e\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^4}{\left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^4}+\frac {c \left (2 e^2 (1-n) b^4-e \left (5 c d-2 \sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (9-13 n)+5 \sqrt {b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right ) b^2+c \left (c d \left (4 a e (5-8 n)+3 \sqrt {b^2-4 a c} d (1-n)\right )-5 a \sqrt {b^2-4 a c} e^2 (1-n)\right ) b+4 a c^2 \left (e \left (a e (1-2 n)+2 \sqrt {b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 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 \left (b^2-4 a c\right ) \left (b^2-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3 n}+\frac {c \left (2 e^2 (1-n) b^4-e \left (5 c d+2 \sqrt {b^2-4 a c} e\right ) (1-n) b^3-c \left (e \left (a e (9-13 n)-5 \sqrt {b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right ) b^2+c \left (5 a \sqrt {b^2-4 a c} (1-n) e^2+c d \left (4 a e (5-8 n)-3 \sqrt {b^2-4 a c} d (1-n)\right )\right ) b+4 a c^2 \left (e \left (a e (1-2 n)-2 \sqrt {b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 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 \left (b^2-4 a c\right ) \left (b^2+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3 n}-\frac {x \left (c \left (-2 e^2 b^3+5 c d e b^2-c \left (3 c d^2-5 a e^2\right ) b-8 a c^2 d e\right ) x^n-2 b^4 e^2-14 a b c^2 d e+5 b^3 c d e-b^2 c \left (3 c d^2-7 a e^2\right )+2 a c^2 \left (3 c d^2-a e^2\right )\right ) e^2}{a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^3 n \left (b x^n+c x^{2 n}+a\right )}+\frac {c \left (\left (e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right ) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right ) b-8 a^2 c^3 d e (1-3 n)\right ) (1-n)-\frac {-e^2 \left (2 n^2-3 n+1\right ) b^6+2 c d e \left (2 n^2-3 n+1\right ) b^5+c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n) b^4-2 a c^2 d e \left (18 n^2-25 n+7\right ) b^3+2 a c^2 \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (35 n^2-38 n+9\right )\right ) b^2+8 a^2 c^3 d e \left (13 n^2-13 n+3\right ) b-8 a^2 c^3 \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 ) \left (c d^2-b e d+a e^2\right )^2 n^2}+\frac {c \left (\left (e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right ) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right ) b-8 a^2 c^3 d e (1-3 n)\right ) (1-n)+\frac {-e^2 \left (2 n^2-3 n+1\right ) b^6+2 c d e \left (2 n^2-3 n+1\right ) b^5+c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n) b^4-2 a c^2 d e \left (18 n^2-25 n+7\right ) b^3+2 a c^2 \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (35 n^2-38 n+9\right )\right ) b^2+8 a^2 c^3 d e \left (13 n^2-13 n+3\right ) b-8 a^2 c^3 \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 ) \left (c d^2-b e d+a e^2\right )^2 n^2}-\frac {x \left (c \left (e^2 (1-2 n) b^5-2 c d e (1-2 n) b^4-c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right ) b^3+2 a c^2 d e (5-14 n) b^2+2 a c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right ) b-8 a^2 c^3 d e (1-3 n)\right ) x^n+a b^2 c^2 \left (a e^2 (13-37 n)-5 c d^2 (1-3 n)\right )-b^4 c \left (a e^2 (7-17 n)-c d^2 (1-2 n)\right )-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left (c d^2-a e^2\right ) (1-4 n)+b^6 e^2 (1-2 n)-2 b^5 c d e (1-2 n)\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}-\frac {x \left (c \left (-e^2 b^3+2 c d e b^2-c \left (c d^2-3 a e^2\right ) b-4 a c^2 d e\right ) x^n-b^4 e^2-6 a b c^2 d e+2 b^3 c d e-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )^2 n \left (b x^n+c x^{2 n}+a\right )^2}\)

Input:

Int[1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3),x]
 

Output:

-1/2*(x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 
 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c 
*d^2 - 3*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*n*(a + b 
*x^n + c*x^(2*n))^2) - (e^2*x*(5*b^3*c*d*e - 14*a*b*c^2*d*e - 2*b^4*e^2 - 
b^2*c*(3*c*d^2 - 7*a*e^2) + 2*a*c^2*(3*c*d^2 - a*e^2) + c*(5*b^2*c*d*e - 8 
*a*c^2*d*e - 2*b^3*e^2 - b*c*(3*c*d^2 - 5*a*e^2))*x^n))/(a*(b^2 - 4*a*c)*( 
c*d^2 - b*d*e + a*e^2)^3*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*(a*e^2 
*(13 - 37*n) - 5*c*d^2*(1 - 3*n)) - b^4*c*(a*e^2*(7 - 17*n) - c*d^2*(1 - 2 
*n)) - 4*a^2*b*c^3*d*e*(4 - 11*n) + 6*a*b^3*c^2*d*e*(2 - 5*n) + 4*a^2*c^3* 
(c*d^2 - a*e^2)*(1 - 4*n) - 2*b^5*c*d*e*(1 - 2*n) + b^6*e^2*(1 - 2*n) + c* 
(2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) 
 - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) 
 - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2 
*(c*d^2 - b*d*e + a*e^2)^2*n^2*(a + b*x^n + c*x^(2*n))) - (c*e^4*(10*c^2*d 
^2 + 3*b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d + 3*Sqrt[b^2 - 4*a*c]* 
d + a*e))*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b - Sqrt[ 
b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^ 
2)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) + 2*Sqrt[b^2 - 4*a*c]*d*(1 - n)) 
 - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) + 5*Sqrt[b^2 - 4*a*c]*d*( 
1 - n)) - 3*c*d^2*(1 - n)) + b*c*(c*d*(4*a*e*(5 - 8*n) + 3*Sqrt[b^2 - 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]

\[\int \frac {1}{\left (d +e \,x^{n}\right )^{2} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}d x\]

Input:

int(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
 

Output:

int(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
 

Fricas [F]

\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3} {\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="fricas")
 

Output:

integral(1/(b^3*e^2*x^(5*n) + a^3*d^2 + (c^3*e^2*x^(2*n) + 2*c^3*d*e*x^n + 
 c^3*d^2)*x^(6*n) + 3*(b*c^2*e^2*x^(3*n) + a*c^2*d^2 + (2*b*c^2*d*e + a*c^ 
2*e^2)*x^(2*n) + (b*c^2*d^2 + 2*a*c^2*d*e)*x^n)*x^(4*n) + (2*b^3*d*e + 3*a 
*b^2*e^2)*x^(4*n) + (b^3*d^2 + 6*a*b^2*d*e + 3*a^2*b*e^2)*x^(3*n) + 3*(b^2 
*c*e^2*x^(4*n) + a^2*c*d^2 + 2*(b^2*c*d*e + a*b*c*e^2)*x^(3*n) + (b^2*c*d^ 
2 + 4*a*b*c*d*e + a^2*c*e^2)*x^(2*n) + 2*(a*b*c*d^2 + a^2*c*d*e)*x^n)*x^(2 
*n) + (3*a*b^2*d^2 + 6*a^2*b*d*e + a^3*e^2)*x^(2*n) + (3*a^2*b*d^2 + 2*a^3 
*d*e)*x^n), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(d+e*x**n)**2/(a+b*x**n+c*x**(2*n))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3} {\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="maxima")
 

Output:

(c*d^2*e^6*(7*n - 1) - b*d*e^7*(4*n - 1) + a*e^8*(n - 1))*integrate(1/(c^4 
*d^10*n - 4*b*c^3*d^9*e*n + 6*b^2*c^2*d^8*e^2*n - 4*b^3*c*d^7*e^3*n + b^4* 
d^6*e^4*n + a^4*d^2*e^8*n + 4*(c*d^4*e^6*n - b*d^3*e^7*n)*a^3 + 6*(c^2*d^6 
*e^4*n - 2*b*c*d^5*e^5*n + b^2*d^4*e^6*n)*a^2 + 4*(c^3*d^8*e^2*n - 3*b*c^2 
*d^7*e^3*n + 3*b^2*c*d^6*e^4*n - b^3*d^5*e^5*n)*a + (c^4*d^9*e*n - 4*b*c^3 
*d^8*e^2*n + 6*b^2*c^2*d^7*e^3*n - 4*b^3*c*d^6*e^4*n + b^4*d^5*e^5*n + a^4 
*d*e^9*n + 4*(c*d^3*e^7*n - b*d^2*e^8*n)*a^3 + 6*(c^2*d^5*e^5*n - 2*b*c*d^ 
4*e^6*n + b^2*d^3*e^7*n)*a^2 + 4*(c^3*d^7*e^3*n - 3*b*c^2*d^6*e^4*n + 3*b^ 
2*c*d^5*e^5*n - b^3*d^4*e^6*n)*a)*x^n), x) + 1/2*((b^3*c^5*d^5*e*(2*n - 1) 
 - 3*b^4*c^4*d^4*e^2*(2*n - 1) + 3*b^5*c^3*d^3*e^3*(2*n - 1) - b^6*c^2*d^2 
*e^4*(2*n - 1) + 32*a^4*c^4*e^6*n + 2*(b*c^4*d*e^5*(33*n - 4) - 4*c^5*d^2* 
e^4*(11*n - 1) - 8*b^2*c^3*e^6*n)*a^3 + 2*(b^2*c^4*d^2*e^4*(29*n - 1) - 3* 
b^3*c^3*d*e^5*(7*n - 1) - 4*c^6*d^4*e^2*(3*n - 1) + 6*b*c^5*d^3*e^3*(n - 1 
) + b^4*c^2*e^6*n)*a^2 - (3*b^3*c^4*d^3*e^3*(12*n - 5) + 2*b*c^6*d^5*e*(7* 
n - 2) - b^5*c^2*d*e^5*(6*n - 1) - 14*b^2*c^5*d^4*e^2*(3*n - 1) - 2*b^4*c^ 
3*d^2*e^4*(n - 2))*a)*x*x^(4*n) + (b^3*c^5*d^6*(2*n - 1) - b^4*c^4*d^5*e*( 
2*n - 1) - 3*b^5*c^3*d^4*e^2*(2*n - 1) + 5*b^6*c^2*d^3*e^3*(2*n - 1) - 2*b 
^7*c*d^2*e^4*(2*n - 1) - 4*(c^4*d*e^5*(8*n - 1) - 16*b*c^3*e^6*n)*a^4 + (b 
^2*c^3*d*e^5*(163*n - 21) - 6*b*c^4*d^2*e^4*(27*n - 2) - 8*c^5*d^3*e^3*(5* 
n - 1) - 32*b^3*c^2*e^6*n)*a^3 - (b^4*c^2*d*e^5*(89*n - 13) - b^3*c^3*d...
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3} {\left (e x^{n} + d\right )}^{2}} \,d x } \] Input:

integrate(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="giac")
 

Output:

integrate(1/((c*x^(2*n) + b*x^n + a)^3*(e*x^n + d)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int \frac {1}{{\left (d+e\,x^n\right )}^2\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \] Input:

int(1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3),x)
 

Output:

int(1/((d + e*x^n)^2*(a + b*x^n + c*x^(2*n))^3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int \frac {1}{x^{5 n} b^{3} e^{2}+x^{3 n} b^{3} d^{2}+3 x^{7 n} b \,c^{2} e^{2}+3 x^{6 n} b^{2} c \,e^{2}+3 x^{5 n} b \,c^{2} d^{2}+3 x^{4 n} a \,b^{2} e^{2}+2 x^{4 n} b^{3} d e +3 x^{4 n} b^{2} c \,d^{2}+3 x^{3 n} a^{2} b \,e^{2}+3 x^{2 n} a \,b^{2} d^{2}+3 x^{n} a^{2} b \,d^{2}+a^{3} d^{2}+x^{8 n} c^{3} e^{2}+x^{2 n} a^{3} e^{2}+x^{6 n} c^{3} d^{2}+2 x^{7 n} c^{3} d e +3 x^{6 n} a \,c^{2} e^{2}+3 x^{4 n} a^{2} c \,e^{2}+3 x^{4 n} a \,c^{2} d^{2}+3 x^{2 n} a^{2} c \,d^{2}+2 x^{n} a^{3} d e +6 x^{5 n} a \,c^{2} d e +6 x^{3 n} a^{2} c d e +6 x^{6 n} b \,c^{2} d e +6 x^{5 n} a b c \,e^{2}+6 x^{5 n} b^{2} c d e +6 x^{3 n} a \,b^{2} d e +6 x^{3 n} a b c \,d^{2}+6 x^{2 n} a^{2} b d e +12 x^{4 n} a b c d e}d x \] Input:

int(1/(d+e*x^n)^2/(a+b*x^n+c*x^(2*n))^3,x)
 

Output:

int(1/(x**(8*n)*c**3*e**2 + 3*x**(7*n)*b*c**2*e**2 + 2*x**(7*n)*c**3*d*e + 
 3*x**(6*n)*a*c**2*e**2 + 3*x**(6*n)*b**2*c*e**2 + 6*x**(6*n)*b*c**2*d*e + 
 x**(6*n)*c**3*d**2 + 6*x**(5*n)*a*b*c*e**2 + 6*x**(5*n)*a*c**2*d*e + x**( 
5*n)*b**3*e**2 + 6*x**(5*n)*b**2*c*d*e + 3*x**(5*n)*b*c**2*d**2 + 3*x**(4* 
n)*a**2*c*e**2 + 3*x**(4*n)*a*b**2*e**2 + 12*x**(4*n)*a*b*c*d*e + 3*x**(4* 
n)*a*c**2*d**2 + 2*x**(4*n)*b**3*d*e + 3*x**(4*n)*b**2*c*d**2 + 3*x**(3*n) 
*a**2*b*e**2 + 6*x**(3*n)*a**2*c*d*e + 6*x**(3*n)*a*b**2*d*e + 6*x**(3*n)* 
a*b*c*d**2 + x**(3*n)*b**3*d**2 + x**(2*n)*a**3*e**2 + 6*x**(2*n)*a**2*b*d 
*e + 3*x**(2*n)*a**2*c*d**2 + 3*x**(2*n)*a*b**2*d**2 + 2*x**n*a**3*d*e + 3 
*x**n*a**2*b*d**2 + a**3*d**2),x)