\(\int \frac {d+e x^n}{(a+b x^n+c x^{2 n})^3} \, dx\) [76]

Optimal result

Integrand size = 24, antiderivative size = 708 \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {x \left (\left (b^2-2 a c\right ) \left (a b e+2 a c d (1-4 n)-b^2 d (1-2 n)\right )+a b c (b d-2 a e) (1-3 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) n^2}-\frac {c \left (a b^2 \left (\sqrt {b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt {b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) n^2} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(8593\) vs. \(2(708)=1416\).

Time = 7.04 (sec) , antiderivative size = 8593, normalized size of antiderivative = 12.14 \[ \int \frac {d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.95 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1760, 25, 1760, 25, 1752, 778}

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.

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

\(\Big \downarrow \) 1760

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1760

\(\displaystyle \frac {\frac {x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac {\int -\frac {-c \left (-d (1-2 n) b^3+a e b^2+2 a c d (2-7 n) b-4 a^2 c e (1-3 n)\right ) (1-n) x^n+2 a^2 b c e (2-5 n)-a b^3 e (1-n)+b^4 d \left (2 n^2-3 n+1\right )+4 a^2 c^2 d \left (8 n^2-6 n+1\right )-a b^2 c d \left (16 n^2-21 n+5\right )}{b x^n+c x^{2 n}+a}dx}{a n \left (b^2-4 a c\right )}}{2 a n \left (b^2-4 a c\right )}+\frac {x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {-c \left (-d (1-2 n) b^3+a e b^2+2 a c d (2-7 n) b-4 a^2 c e (1-3 n)\right ) (1-n) x^n+2 a^2 b c e (2-5 n)-a b^3 e (1-n)+b^4 d \left (2 n^2-3 n+1\right )+4 a^2 c^2 d \left (8 n^2-6 n+1\right )-a b^2 c d \left (16 n^2-21 n+5\right )}{b x^n+c x^{2 n}+a}dx}{a n \left (b^2-4 a c\right )}+\frac {x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}}{2 a n \left (b^2-4 a c\right )}+\frac {x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\)

\(\Big \downarrow \) 1752

\(\displaystyle \frac {\frac {-\frac {c \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt {b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt {b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt {b^2-4 a c}+6 c d (1-3 n)\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt {b^2-4 a c}\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \int \frac {1}{c x^n+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}-\frac {c \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt {b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt {b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )+a b^2 (1-n) \left (e \sqrt {b^2-4 a c}-6 c d (1-3 n)\right )-b^3 (1-n) \left (d (1-2 n) \sqrt {b^2-4 a c}+a e\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \int \frac {1}{c x^n+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 \sqrt {b^2-4 a c}}}{a n \left (b^2-4 a c\right )}+\frac {x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+a b^3 e+5 a b^2 c d (1-3 n)-b^4 d (1-2 n)\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}}{2 a n \left (b^2-4 a c\right )}+\frac {x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\)

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 1752

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x 
_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) 
   Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   I 
nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 
, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 
 - 4*a*c] ||  !IGtQ[n/2, 0])
 

rule 1760

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Simp[(-x)*(d*b^2 - a*b*e - 2*a*c*d + (b*d - 2*a*e)*c*x^n)* 
((a + b*x^n + c*x^(2*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/ 
(a*n*(p + 1)*(b^2 - 4*a*c))   Int[Simp[(n*p + n + 1)*d*b^2 - a*b*e - 2*a*c* 
d*(2*n*p + 2*n + 1) + (2*n*p + 3*n + 1)*(d*b - 2*a*e)*c*x^n, x]*(a + b*x^n 
+ c*x^(2*n))^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n 
] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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