\(\int \frac {A+B x^n+C x^{2 n}+D x^{3 n}}{(a+b x^n+c x^{2 n})^3} \, dx\) [22]

Optimal result

Integrand size = 38, antiderivative size = 933 \[ \int \frac {A+B x^n+C x^{2 n}+D x^{3 n}}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {x \left (A c \left (b^2-2 a c\right )-a (b B c-2 a c C+a b D)+\left (A b c^2-a \left (2 B c^2-b c C+b^2 D-2 a c D\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}-\frac {x \left (A c \left (4 a^2 c^2 (1-4 n)-5 a b^2 c (1-3 n)+b^4 (1-2 n)\right )-a \left (4 a^2 c^2 C-a b^2 c C (1+3 n)-2 a b c (B c (2-3 n)-a D n)+b^3 (B c+a D n)\right )-c \left (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left (b^2 (B c-a D (1-2 n))-6 a b c C n-4 a c (B c (1-3 n)-a D (1+n))\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {\left ((1-n) \left (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left (b^2 (B c-a D (1-2 n))-6 a b c C n-4 a c (B c (1-3 n)-a D (1+n))\right )\right )-\frac {A c \left (b^4 \left (1-3 n+2 n^2\right )-6 a b^2 c \left (1-4 n+3 n^2\right )+8 a^2 c^2 \left (1-6 n+8 n^2\right )\right )-a \left (8 a^2 c^2 C (1-2 n)+b^3 (B c+a D (1-2 n)) (1-n)-2 a b^2 c C \left (1-2 n+3 n^2\right )-4 a b c \left (B c \left (1-n-3 n^2\right )+a D \left (1-3 n-n^2\right )\right )\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 (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left (b^2 (B c-a D (1-2 n))-6 a b c C n-4 a c (B c (1-3 n)-a D (1+n))\right )\right )+\frac {A c \left (b^4 \left (1-3 n+2 n^2\right )-6 a b^2 c \left (1-4 n+3 n^2\right )+8 a^2 c^2 \left (1-6 n+8 n^2\right )\right )-a \left (8 a^2 c^2 C (1-2 n)+b^3 (B c+a D (1-2 n)) (1-n)-2 a b^2 c C \left (1-2 n+3 n^2\right )-4 a b c \left (B c \left (1-n-3 n^2\right )+a D \left (1-3 n-n^2\right )\right )\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} \] Output:

1/2*x*(A*c*(-2*a*c+b^2)-a*(B*b*c-2*C*a*c+D*a*b)+(A*b*c^2-a*(2*B*c^2-C*b*c- 
2*D*a*c+D*b^2))*x^n)/a/c/(-4*a*c+b^2)/n/(a+b*x^n+c*x^(2*n))^2-1/2*x*(A*c*( 
4*a^2*c^2*(1-4*n)-5*a*b^2*c*(1-3*n)+b^4*(1-2*n))-a*(4*a^2*c^2*C-a*b^2*c*C* 
(1+3*n)-2*a*b*c*(B*c*(2-3*n)-a*D*n)+b^3*(D*a*n+B*c))-c*(A*b*c*(2*a*c*(2-7* 
n)-b^2*(1-2*n))+a*(b^2*(B*c-a*D*(1-2*n))-6*a*b*c*C*n-4*a*c*(B*c*(1-3*n)-a* 
D*(1+n))))*x^n)/a^2/c/(-4*a*c+b^2)^2/n^2/(a+b*x^n+c*x^(2*n))-1/2*((1-n)*(A 
*b*c*(2*a*c*(2-7*n)-b^2*(1-2*n))+a*(b^2*(B*c-a*D*(1-2*n))-6*a*b*c*C*n-4*a* 
c*(B*c*(1-3*n)-a*D*(1+n))))-(A*c*(b^4*(2*n^2-3*n+1)-6*a*b^2*c*(3*n^2-4*n+1 
)+8*a^2*c^2*(8*n^2-6*n+1))-a*(8*a^2*c^2*C*(1-2*n)+b^3*(B*c+a*D*(1-2*n))*(1 
-n)-2*a*b^2*c*C*(3*n^2-2*n+1)-4*a*b*c*(B*c*(-3*n^2-n+1)+a*D*(-n^2-3*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*c* 
(2*a*c*(2-7*n)-b^2*(1-2*n))+a*(b^2*(B*c-a*D*(1-2*n))-6*a*b*c*C*n-4*a*c*(B* 
c*(1-3*n)-a*D*(1+n))))+(A*c*(b^4*(2*n^2-3*n+1)-6*a*b^2*c*(3*n^2-4*n+1)+8*a 
^2*c^2*(8*n^2-6*n+1))-a*(8*a^2*c^2*C*(1-2*n)+b^3*(B*c+a*D*(1-2*n))*(1-n)-2 
*a*b^2*c*C*(3*n^2-2*n+1)-4*a*b*c*(B*c*(-3*n^2-n+1)+a*D*(-n^2-3*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. \(12750\) vs. \(2(933)=1866\).

Time = 9.71 (sec) , antiderivative size = 12750, normalized size of antiderivative = 13.67 \[ \int \frac {A+B x^n+C x^{2 n}+D x^{3 n}}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x^n + C*x^(2*n) + D*x^(3*n))/(a + b*x^n + c*x^(2*n))^3,x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.98 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2326, 1760, 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.

\(\displaystyle \int \frac {A+B x^n+C x^{2 n}+D x^{3 n}}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx\)

\(\Big \downarrow \) 2326

\(\displaystyle \frac {\int \frac {-\left (\left (-a D (1-n) b^2+c (A c+a C) (1-3 n) b+2 a c (a D (n+1)-B (c-3 c n))\right ) x^n\right )+a b (B c+a D)-2 a c (a C-A c (1-4 n))-A b^2 c (1-2 n)}{\left (b x^n+c x^{2 n}+a\right )^2}dx}{2 a c n \left (b^2-4 a c\right )}+\frac {x \left (x^n \left (b c (a C+A c)-a b^2 D-2 a c (B c-a D)\right )+A c \left (b^2-2 a c\right )-a (a b D-2 a c C+b B c)\right )}{2 a c n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\)

\(\Big \downarrow \) 1760

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

\(\Big \downarrow \) 1752

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

\(\Big \downarrow \) 778

\(\displaystyle \frac {x \left (\left (-a D b^2+c (A c+a C) b-2 a c (B c-a D)\right ) x^n+A c \left (b^2-2 a c\right )-a (b B c-2 a C c+a b D)\right )}{2 a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2}+\frac {-\frac {x \left (-c \left (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left ((B c-a D (1-2 n)) b^2-6 a c C n b-4 a c (B c (1-3 n)-a D (n+1))\right )\right ) x^n+A c \left ((1-2 n) b^4-5 a c (1-3 n) b^2+4 a^2 c^2 (1-4 n)\right )-a \left ((B c+a D n) b^3-a c C (3 n+1) b^2-2 a c (B c (2-3 n)-a D n) b+4 a^2 c^2 C\right )\right )}{a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac {\frac {c \left ((1-n) \left (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left ((B c-a D (1-2 n)) b^2-6 a c C n b-4 a c (B c (1-3 n)-a D (n+1))\right )\right )-\frac {A c \left (\left (2 n^2-3 n+1\right ) b^4-6 a c \left (3 n^2-4 n+1\right ) b^2+8 a^2 c^2 \left (8 n^2-6 n+1\right )\right )-a \left ((B c+a D (1-2 n)) (1-n) b^3-2 a c C \left (3 n^2-2 n+1\right ) b^2-4 a c \left (B c \left (-3 n^2-n+1\right )+a D \left (-n^2-3 n+1\right )\right ) b+8 a^2 c^2 C (1-2 n)\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 )}{b-\sqrt {b^2-4 a c}}+\frac {c \left ((1-n) \left (A b c \left (2 a c (2-7 n)-b^2 (1-2 n)\right )+a \left ((B c-a D (1-2 n)) b^2-6 a c C n b-4 a c (B c (1-3 n)-a D (n+1))\right )\right )+\frac {A c \left (\left (2 n^2-3 n+1\right ) b^4-6 a c \left (3 n^2-4 n+1\right ) b^2+8 a^2 c^2 \left (8 n^2-6 n+1\right )\right )-a \left ((B c+a D (1-2 n)) (1-n) b^3-2 a c C \left (3 n^2-2 n+1\right ) b^2-4 a c \left (B c \left (-3 n^2-n+1\right )+a D \left (-n^2-3 n+1\right )\right ) b+8 a^2 c^2 C (1-2 n)\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 )}{b+\sqrt {b^2-4 a c}}}{a \left (b^2-4 a c\right ) n}}{2 a c \left (b^2-4 a c\right ) n}\)

Input:

Int[(A + B*x^n + C*x^(2*n) + D*x^(3*n))/(a + b*x^n + c*x^(2*n))^3,x]
 

Output:

(x*(A*c*(b^2 - 2*a*c) - a*(b*B*c - 2*a*c*C + a*b*D) + (b*c*(A*c + a*C) - a 
*b^2*D - 2*a*c*(B*c - a*D))*x^n))/(2*a*c*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^ 
(2*n))^2) + (-((x*(A*c*(4*a^2*c^2*(1 - 4*n) - 5*a*b^2*c*(1 - 3*n) + b^4*(1 
 - 2*n)) - a*(4*a^2*c^2*C - a*b^2*c*C*(1 + 3*n) - 2*a*b*c*(B*c*(2 - 3*n) - 
 a*D*n) + b^3*(B*c + a*D*n)) - c*(A*b*c*(2*a*c*(2 - 7*n) - b^2*(1 - 2*n)) 
+ a*(b^2*(B*c - a*D*(1 - 2*n)) - 6*a*b*c*C*n - 4*a*c*(B*c*(1 - 3*n) - a*D* 
(1 + n))))*x^n))/(a*(b^2 - 4*a*c)*n*(a + b*x^n + c*x^(2*n)))) - ((c*((1 - 
n)*(A*b*c*(2*a*c*(2 - 7*n) - b^2*(1 - 2*n)) + a*(b^2*(B*c - a*D*(1 - 2*n)) 
 - 6*a*b*c*C*n - 4*a*c*(B*c*(1 - 3*n) - a*D*(1 + n)))) - (A*c*(b^4*(1 - 3* 
n + 2*n^2) - 6*a*b^2*c*(1 - 4*n + 3*n^2) + 8*a^2*c^2*(1 - 6*n + 8*n^2)) - 
a*(8*a^2*c^2*C*(1 - 2*n) + b^3*(B*c + a*D*(1 - 2*n))*(1 - n) - 2*a*b^2*c*C 
*(1 - 2*n + 3*n^2) - 4*a*b*c*(B*c*(1 - n - 3*n^2) + a*D*(1 - 3*n - 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])])/(b - Sqrt[b^2 - 4*a*c]) + (c*((1 - n)*(A*b*c*(2* 
a*c*(2 - 7*n) - b^2*(1 - 2*n)) + a*(b^2*(B*c - a*D*(1 - 2*n)) - 6*a*b*c*C* 
n - 4*a*c*(B*c*(1 - 3*n) - a*D*(1 + n)))) + (A*c*(b^4*(1 - 3*n + 2*n^2) - 
6*a*b^2*c*(1 - 4*n + 3*n^2) + 8*a^2*c^2*(1 - 6*n + 8*n^2)) - a*(8*a^2*c^2* 
C*(1 - 2*n) + b^3*(B*c + a*D*(1 - 2*n))*(1 - n) - 2*a*b^2*c*C*(1 - 2*n + 3 
*n^2) - 4*a*b*c*(B*c*(1 - n - 3*n^2) + a*D*(1 - 3*n - n^2))))/Sqrt[b^2 - 4 
*a*c])*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[...
 

Defintions of rubi rules used

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]
 

rule 2326

Int[(P3_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Wi 
th[{d = Coeff[P3, x^n, 0], e = Coeff[P3, x^n, 1], f = Coeff[P3, x^n, 2], g 
= Coeff[P3, x^n, 3]}, Simp[(-x)*(b^2*c*d - 2*a*c*(c*d - a*f) - a*b*(c*e + a 
*g) + (b*c*(c*d + a*f) - a*b^2*g - 2*a*c*(c*e - a*g))*x^n)*((a + b*x^n + c* 
x^(2*n))^(p + 1)/(a*c*n*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(a*c*n*(p + 1) 
*(b^2 - 4*a*c))   Int[(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[a*b*(c*e + a*g) 
- b^2*c*d*(n + n*p + 1) - 2*a*c*(a*f - c*d*(2*n*(p + 1) + 1)) + (a*b^2*g*(n 
*(p + 2) + 1) - b*c*(c*d + a*f)*(n*(2*p + 3) + 1) - 2*a*c*(a*g*(n + 1) - c* 
e*(n*(2*p + 3) + 1)))*x^n, x], x], x]] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 
 2*n] && PolyQ[P3, x^n, 3] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]
 

Maple [F]

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

Input:

int((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^3,x)
 

Output:

int((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^3,x)
 

Fricas [F]

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

integrate((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^3,x, algorithm 
="fricas")
 

Output:

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

integrate((A+B*x**n+C*x**(2*n)+D*x**(3*n))/(a+b*x**n+c*x**(2*n))**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^3,x, algorithm 
="maxima")
 

Output:

-1/2*((6*C*a^2*b*c^2*n + (2*a*b*c^3*(7*n - 2) - b^3*c^2*(2*n - 1))*A - (4* 
a^2*c^3*(3*n - 1) + a*b^2*c^2)*B - (a^2*b^2*c*(2*n - 1) + 4*a^3*c^2*(n + 1 
))*D)*x*x^(3*n) + ((a*b^2*c^2*(29*n - 9) - 4*a^2*c^3*(4*n - 1) - 2*b^4*c*( 
2*n - 1))*A - 2*(a^2*b*c^2*(9*n - 4) + a*b^3*c)*B + (a^2*b^2*c*(9*n + 1) - 
 4*a^3*c^2)*C - (2*a^3*b*c*(3*n + 2) + a^2*b^3*(3*n - 1))*D)*x*x^(2*n) + ( 
(4*a*b^3*c*(3*n - 1) - b^5*(2*n - 1) + 2*a^2*b*c^2*n)*A - (4*a^3*c^2*(5*n 
- 1) + a^2*b^2*c*(4*n - 3) + a*b^4)*B + (2*a^3*b*c*(5*n - 2) + a^2*b^3*(2* 
n + 1))*C - (a^3*b^2*(10*n - 1) - 4*a^4*c*(n - 1))*D)*x*x^n - (6*D*a^4*b*n 
 - (a^2*b^2*c*(21*n - 5) - 4*a^3*c^2*(6*n - 1) - a*b^4*(3*n - 1))*A + (2*a 
^3*b*c*(5*n - 2) - a^2*b^3*(n - 1))*B - (4*a^4*c*(2*n - 1) + a^3*b^2*(n + 
1))*C)*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*(6*D*a^3*b*n - ((2*n^2 - 3*n + 1)*b^4 - (16*n 
^2 - 21*n + 5)*a*b^2*c + 4*(8*n^2 - 6*n + 1)*a^2*c^2)*A + (2*a^2*b*c*(5*n 
- 2) - a*b^3*(n - 1))*B - (4*a^3*c*(2*n - 1) + a^2*b^2*(n + 1))*C + (6*(n^ 
2 - n)*C*a^2*b*c - ((2*n^2 - 3*n + 1)*b^3*c - 2*(7*n^2 - 9*n + 2)*a*b*c^2) 
*A - (4*(3*n^2 - 4*n + 1)*a^2*c^2 + a*b^2*c*(n - 1))*B - ((2*n^2 - 3*n + 1 
)*a^2*b^2 + 4*(n^2 - 1)*a^3*c)*D)*x^n)/(a^3*b^4*n^2 - 8*a^4*b^2*c*n^2 +...
 

Giac [F]

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

integrate((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(a+b*x^n+c*x^(2*n))^3,x, algorithm 
="giac")
 

Output:

integrate((D*x^(3*n) + C*x^(2*n) + B*x^n + A)/(c*x^(2*n) + b*x^n + a)^3, x 
)
 

Mupad [F(-1)]

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

int((A + C*x^(2*n) + x^(3*n)*D + B*x^n)/(a + b*x^n + c*x^(2*n))^3,x)
 

Output:

int((A + C*x^(2*n) + x^(3*n)*D + B*x^n)/(a + b*x^n + c*x^(2*n))^3, x)
 

Reduce [F]

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

int((A+B*x^n+C*x^(2*n)+D*x^(3*n))/(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*d*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*d*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)*b*c**...