Integrand size = 29, antiderivative size = 816 \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) f n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {(f x)^{1+m} \left (\left (b^2-2 a c\right ) \left (a b e (1+m)+2 a c d (1+m-4 n)-b^2 d (1+m-2 n)\right )+a b c (b d-2 a e) (1+m-3 n)+c \left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 f n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)+\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt {b^2-4 a c}\right ) f (1+m) n^2}-\frac {c \left (\left (a b^2 e (1+m)+2 a b c d (2+2 m-7 n)-4 a^2 c e (1+m-3 n)-b^3 d (1+m-2 n)\right ) (1+m-n)-\frac {a b^3 e (1+m) (1+m-n)-4 a^2 b c e \left (1+m^2+m (2-n)-n-3 n^2\right )-b^4 d \left (1+m^2+m (2-3 n)-3 n+2 n^2\right )+6 a b^2 c d \left (1+m^2+m (2-4 n)-4 n+3 n^2\right )-8 a^2 c^2 d \left (1+m^2+m (2-6 n)-6 n+8 n^2\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt {b^2-4 a c}\right ) f (1+m) n^2} \]
1/2*(f*x)^(1+m)*(b^2*d-2*a*c*d-a*b*e+c*(-2*a*e+b*d)*x^n)/a/(-4*a*c+b^2)/f/ n/(a+b*x^n+c*x^(2*n))^2+1/2*(f*x)^(1+m)*((-2*a*c+b^2)*(a*b*e*(1+m)+2*a*c*d *(1+m-4*n)-b^2*d*(1+m-2*n))+a*b*c*(-2*a*e+b*d)*(1+m-3*n)+c*(a*b^2*e*(1+m)+ 2*a*b*c*d*(2+2*m-7*n)-4*a^2*c*e*(1+m-3*n)-b^3*d*(1+m-2*n))*x^n)/a^2/(-4*a* c+b^2)^2/f/n^2/(a+b*x^n+c*x^(2*n))-1/2*c*(f*x)^(1+m)*hypergeom([1, (1+m)/n ],[(1+m+n)/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*((a*b^2*e*(1+m)+2*a*b*c*d*( 2+2*m-7*n)-4*a^2*c*e*(1+m-3*n)-b^3*d*(1+m-2*n))*(1+m-n)+(a*b^3*e*(1+m)*(1+ m-n)-4*a^2*b*c*e*(1+m^2+m*(2-n)-n-3*n^2)-b^4*d*(1+m^2+m*(2-3*n)-3*n+2*n^2) +6*a*b^2*c*d*(1+m^2+m*(2-4*n)-4*n+3*n^2)-8*a^2*c^2*d*(1+m^2+m*(2-6*n)-6*n+ 8*n^2))/(-4*a*c+b^2)^(1/2))/a^2/(-4*a*c+b^2)^2/f/(1+m)/n^2/(b-(-4*a*c+b^2) ^(1/2))-1/2*c*(f*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-2*c*x^n/(b+( -4*a*c+b^2)^(1/2)))*((a*b^2*e*(1+m)+2*a*b*c*d*(2+2*m-7*n)-4*a^2*c*e*(1+m-3 *n)-b^3*d*(1+m-2*n))*(1+m-n)+(-a*b^3*e*(1+m)*(1+m-n)+4*a^2*b*c*e*(1+m^2+m* (2-n)-n-3*n^2)+b^4*d*(1+m^2+m*(2-3*n)-3*n+2*n^2)-6*a*b^2*c*d*(1+m^2+m*(2-4 *n)-4*n+3*n^2)+8*a^2*c^2*d*(1+m^2+m*(2-6*n)-6*n+8*n^2))/(-4*a*c+b^2)^(1/2) )/a^2/(-4*a*c+b^2)^2/f/(1+m)/n^2/(b+(-4*a*c+b^2)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(20515\) vs. \(2(816)=1632\).
Time = 8.44 (sec) , antiderivative size = 20515, normalized size of antiderivative = 25.14 \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \]
Time = 3.48 (sec) , antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1882, 25, 1882, 25, 1884, 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.
| \(\displaystyle \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx\) |
| \(\Big \downarrow \) 1882 |
| \(\displaystyle \frac {(f x)^{m+1} \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}-\frac {\int -\frac {(f x)^m \left (-c (b d-2 a e) (m-3 n+1) x^n+a b e (m+1)+2 a c d (m-4 n+1)-b^2 d (m-2 n+1)\right )}{\left (b x^n+c x^{2 n}+a\right )^2}dx}{2 a n \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(f x)^m \left (-c (b d-2 a e) (m-3 n+1) x^n+a b e (m+1)+2 a c d (m-4 n+1)-b^2 d (m-2 n+1)\right )}{\left (b x^n+c x^{2 n}+a\right )^2}dx}{2 a n \left (b^2-4 a c\right )}+\frac {(f x)^{m+1} \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\) |
| \(\Big \downarrow \) 1882 |
| \(\displaystyle \frac {\frac {(f x)^{m+1} \left (c x^n \left (-4 a^2 c e (m-3 n+1)+a b^2 e (m+1)+2 a b c d (2 m-7 n+2)+b^3 (-d) (m-2 n+1)\right )+\left (b^2-2 a c\right ) \left (a b e (m+1)+2 a c d (m-4 n+1)+b^2 (-d) (m-2 n+1)\right )+a b c (m-3 n+1) (b d-2 a e)\right )}{a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac {\int -\frac {(f x)^m \left (-c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1) x^n+\left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right ) \left (2 a c (m-2 n+1)-b^2 (m-n+1)\right )-a b c (b d-2 a e) (m+1) (m-3 n+1)\right )}{b x^n+c x^{2 n}+a}dx}{a n \left (b^2-4 a c\right )}}{2 a n \left (b^2-4 a c\right )}+\frac {(f x)^{m+1} \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(f x)^m \left (-c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1) x^n+\left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right ) \left (2 a c (m-2 n+1)-b^2 (m-n+1)\right )-a b c (b d-2 a e) (m+1) (m-3 n+1)\right )}{b x^n+c x^{2 n}+a}dx}{a n \left (b^2-4 a c\right )}+\frac {(f x)^{m+1} \left (c x^n \left (-4 a^2 c e (m-3 n+1)+a b^2 e (m+1)+2 a b c d (2 m-7 n+2)+b^3 (-d) (m-2 n+1)\right )+\left (b^2-2 a c\right ) \left (a b e (m+1)+2 a c d (m-4 n+1)+b^2 (-d) (m-2 n+1)\right )+a b c (m-3 n+1) (b d-2 a e)\right )}{a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}}{2 a n \left (b^2-4 a c\right )}+\frac {(f x)^{m+1} \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2}\) |
| \(\Big \downarrow \) 1884 |
| \(\displaystyle \frac {\left (c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right ) (f x)^{m+1}}{2 a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )^2}+\frac {\frac {\left (c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) x^n+\left (b^2-2 a c\right ) \left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right )+a b c (b d-2 a e) (m-3 n+1)\right ) (f x)^{m+1}}{a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )}+\frac {\int \left (\frac {\left (\frac {c \left (d m^2 b^4+2 d n^2 b^4+d b^4+2 d m b^4-3 d n b^4-3 d m n b^4-a e m^2 b^3-a e b^3-2 a e m b^3+a e n b^3+a e m n b^3-6 a c d m^2 b^2-18 a c d n^2 b^2-6 a c d b^2-12 a c d m b^2+24 a c d n b^2+24 a c d m n b^2+4 a^2 c e m^2 b-12 a^2 c e n^2 b+4 a^2 c e b+8 a^2 c e m b-4 a^2 c e n b-4 a^2 c e m n b+8 a^2 c^2 d m^2+64 a^2 c^2 d n^2+8 a^2 c^2 d+16 a^2 c^2 d m-48 a^2 c^2 d n-48 a^2 c^2 d m n\right )}{\sqrt {b^2-4 a c}}-c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)\right ) (f x)^m}{2 c x^n+b-\sqrt {b^2-4 a c}}+\frac {\left (-c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)-\frac {c \left (d m^2 b^4+2 d n^2 b^4+d b^4+2 d m b^4-3 d n b^4-3 d m n b^4-a e m^2 b^3-a e b^3-2 a e m b^3+a e n b^3+a e m n b^3-6 a c d m^2 b^2-18 a c d n^2 b^2-6 a c d b^2-12 a c d m b^2+24 a c d n b^2+24 a c d m n b^2+4 a^2 c e m^2 b-12 a^2 c e n^2 b+4 a^2 c e b+8 a^2 c e m b-4 a^2 c e n b-4 a^2 c e m n b+8 a^2 c^2 d m^2+64 a^2 c^2 d n^2+8 a^2 c^2 d+16 a^2 c^2 d m-48 a^2 c^2 d n-48 a^2 c^2 d m n\right )}{\sqrt {b^2-4 a c}}\right ) (f x)^m}{2 c x^n+b+\sqrt {b^2-4 a c}}\right )dx}{a \left (b^2-4 a c\right ) n}}{2 a \left (b^2-4 a c\right ) n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (c (b d-2 a e) x^n+b^2 d-2 a c d-a b e\right ) (f x)^{m+1}}{2 a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )^2}+\frac {\frac {\left (c \left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) x^n+\left (b^2-2 a c\right ) \left (-d (m-2 n+1) b^2+a e (m+1) b+2 a c d (m-4 n+1)\right )+a b c (b d-2 a e) (m-3 n+1)\right ) (f x)^{m+1}}{a \left (b^2-4 a c\right ) f n \left (b x^n+c x^{2 n}+a\right )}+\frac {-\frac {c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)+\frac {-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right ) (f x)^{m+1}}{\left (b-\sqrt {b^2-4 a c}\right ) f (m+1)}-\frac {c \left (\left (-d (m-2 n+1) b^3+a e (m+1) b^2+2 a c d (2 m-7 n+2) b-4 a^2 c e (m-3 n+1)\right ) (m-n+1)-\frac {-d \left (m^2+(2-3 n) m+2 n^2-3 n+1\right ) b^4+a e (m+1) (m-n+1) b^3+6 a c d \left (m^2+(2-4 n) m+3 n^2-4 n+1\right ) b^2-4 a^2 c e \left (m^2+(2-n) m-3 n^2-n+1\right ) b-8 a^2 c^2 d \left (m^2+(2-6 n) m+8 n^2-6 n+1\right )}{\sqrt {b^2-4 a c}}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right ) (f x)^{m+1}}{\left (b+\sqrt {b^2-4 a c}\right ) f (m+1)}}{a \left (b^2-4 a c\right ) n}}{2 a \left (b^2-4 a c\right ) n}\) |
((f*x)^(1 + m)*(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)*f*n*(a + b*x^n + c*x^(2*n))^2) + (((f*x)^(1 + m)*((b^2 - 2*a*c)*( a*b*e*(1 + m) + 2*a*c*d*(1 + m - 4*n) - b^2*d*(1 + m - 2*n)) + a*b*c*(b*d - 2*a*e)*(1 + m - 3*n) + c*(a*b^2*e*(1 + m) + 2*a*b*c*d*(2 + 2*m - 7*n) - 4*a^2*c*e*(1 + m - 3*n) - b^3*d*(1 + m - 2*n))*x^n))/(a*(b^2 - 4*a*c)*f*n* (a + b*x^n + c*x^(2*n))) + (-((c*((a*b^2*e*(1 + m) + 2*a*b*c*d*(2 + 2*m - 7*n) - 4*a^2*c*e*(1 + m - 3*n) - b^3*d*(1 + m - 2*n))*(1 + m - n) + (a*b^3 *e*(1 + m)*(1 + m - n) - 4*a^2*b*c*e*(1 + m^2 + m*(2 - n) - n - 3*n^2) - b ^4*d*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*n^2) + 6*a*b^2*c*d*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) - 8*a^2*c^2*d*(1 + m^2 + m*(2 - 6*n) - 6*n + 8*n^2))/ Sqrt[b^2 - 4*a*c])*(f*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b - Sqrt[b^2 - 4*a*c])*f*(1 + m))) - (c*((a*b^2*e*(1 + m) + 2*a*b*c*d*(2 + 2*m - 7*n) - 4*a^2*c*e*(1 + m - 3*n) - b^3*d*(1 + m - 2*n))*(1 + m - n) - (a*b^3*e*(1 + m)*(1 + m - n) - 4*a^2*b*c*e*(1 + m^2 + m*(2 - n) - n - 3*n^2) - b^4*d*(1 + m^2 + m*(2 - 3*n) - 3*n + 2*n^2) + 6*a*b^2*c*d*(1 + m^2 + m*(2 - 4*n) - 4*n + 3*n^2) - 8*a^2*c^2*d*(1 + m^2 + m*(2 - 6*n) - 6*n + 8*n^2))/Sqrt[b^2 - 4*a*c])*(f* x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^2 - 4*a*c])*f*(1 + m)))/(a*(b^2 - 4*a*c) *n))/(2*a*(b^2 - 4*a*c)*n)
3.2.43.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + ( c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(a + b*x^n + c*x^ (2*n))^(p + 1)*((d*(b^2 - 2*a*c) - a*b*e + (b*d - 2*a*e)*c*x^n)/(a*f*n*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*n*(p + 1)*(b^2 - 4*a*c)) Int[(f*x)^m* (a + b*x^n + c*x^(2*n))^(p + 1)*Simp[d*(b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p + 1) + 1)) - a*b*e*(m + 1) + (m + n*(2*p + 3) + 1)*(b*d - 2*a*e)*c *x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[n2, 2*n] && N eQ[b^2 - 4*a*c, 0] && !RationalQ[n] && ILtQ[p + 1, 0]
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*( (d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && !RationalQ[n] && ( IGtQ[p, 0] || IGtQ[q, 0])
\[\int \frac {\left (f x \right )^{m} \left (d +e \,x^{n}\right )}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{3}}d x\]
\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
integral((e*x^n + d)*(f*x)^m/(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. \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
-1/2*((a*b^4*d*f^m*(m - 3*n + 1) + 2*(b*c*e*f^m*(2*m - 5*n + 2) + 2*c^2*d* f^m*(m - 6*n + 1))*a^3 - (b^2*c*d*f^m*(5*m - 21*n + 5) + b^3*e*f^m*(m - n + 1))*a^2)*x*x^m + (b^3*c^2*d*f^m*(m - 2*n + 1) + 4*a^2*c^3*e*f^m*(m - 3*n + 1) - (2*b*c^3*d*f^m*(2*m - 7*n + 2) + b^2*c^2*e*f^m*(m + 1))*a)*x*e^(m* log(x) + 3*n*log(x)) + (2*b^4*c*d*f^m*(m - 2*n + 1) + 2*(b*c^2*e*f^m*(4*m - 9*n + 4) + 2*c^3*d*f^m*(m - 4*n + 1))*a^2 - (b^2*c^2*d*f^m*(9*m - 29*n + 9) + 2*b^3*c*e*f^m*(m + 1))*a)*x*e^(m*log(x) + 2*n*log(x)) + (b^5*d*f^m*( m - 2*n + 1) + 4*a^3*c^2*e*f^m*(m - 5*n + 1) + (b^2*c*e*f^m*(3*m - 4*n + 3 ) + 2*b*c^2*d*f^m*n)*a^2 - (4*b^3*c*d*f^m*(m - 3*n + 1) + b^4*e*f^m*(m + 1 ))*a)*x*e^(m*log(x) + n*log(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*(((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^4*d*f^m + 2*(2*(m^2 - 2*m*(3*n - 1) + 8*n^2 - 6*n + 1) *c^2*d*f^m + (2*m^2 - m*(5*n - 4) - 5*n + 2)*b*c*e*f^m)*a^2 - ((5*m^2 - m* (21*n - 10) + 16*n^2 - 21*n + 5)*b^2*c*d*f^m + (m^2 - m*(n - 2) - n + 1)*b ^3*e*f^m)*a)*x^m + ((m^2 - m*(3*n - 2) + 2*n^2 - 3*n + 1)*b^3*c*d*f^m + 4* (m^2 - 2*m*(2*n - 1) + 3*n^2 - 4*n + 1)*a^2*c^2*e*f^m - (2*(2*m^2 - m*(9*n - 4) + 7*n^2 - 9*n + 2)*b*c^2*d*f^m + (m^2 - m*(n - 2) - n + 1)*b^2*c*...
\[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int { \frac {{\left (e x^{n} + d\right )} \left (f x\right )^{m}}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(f x)^m \left (d+e x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (d+e\,x^n\right )}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^3} \,d x \]