\(\int \frac {1}{(d+e x^n) (a+b x^n+c x^{2 n})^3} \, dx\) [83]

Optimal result

Integrand size = 26, antiderivative size = 1708 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)-c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c e^4 \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\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 )}{\left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {c e^2 \left (b c \left (2 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (2 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-n)\right )-b^3 e (1-n)+b^2 \left (c d-\sqrt {b^2-4 a c} e\right ) (1-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 )}{a \left (b^2-4 a c\right ) \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right ) (1-n)+b^3 c \left (a e (7-18 n)+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)-b^5 e \left (1-3 n+2 n^2\right )+b^4 \left (c d-\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \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 )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )+2 a e \left (3-13 n+13 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 ) \left (c d^2-b d e+a e^2\right ) n^2}-\frac {c e^4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\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 )}{\left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {c e^2 \left (b c \left (2 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right )-2 a c \left (2 c d (1-2 n)+\sqrt {b^2-4 a c} e (1-n)\right )-b^3 e (1-n)+b^2 \left (c d+\sqrt {b^2-4 a c} e\right ) (1-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 )}{a \left (b^2-4 a c\right ) \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {c \left (a b^2 c \left (\sqrt {b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right ) (1-n)-b^3 c \left (a e (7-18 n)-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n)+b^5 e \left (1-3 n+2 n^2\right )-b^4 \left (c d+\sqrt {b^2-4 a c} e\right ) \left (1-3 n+2 n^2\right )-4 a^2 c^2 \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 )-2 a b c^2 \left (\sqrt {b^2-4 a c} d \left (2-9 n+7 n^2\right )-2 a e \left (3-13 n+13 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 ) \left (c d^2-b d e+a e^2\right ) n^2}+\frac {e^6 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3} \]

[Out]

1/2*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*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/(a+
b*x^n+c*x^(2*n))^2+e^2*x*(b^2*c*d-2*a*c^2*d-b^3*e+3*a*b*c*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)^2/n/(a+b*x^n+c*x^(2*n))+1/2*x*(2*a^2*b*c^2*e*(4-11*n)-3*a*b^3*c*e*(2-5*n)-4*a^2*c^3*d*(1-4*n)+5*
a*b^2*c^2*d*(1-3*n)-b^4*c*d*(1-2*n)+b^5*(-2*e*n+e)-c*(a*b^2*c*e*(5-14*n)-2*a*b*c^2*d*(2-7*n)-4*a^2*c^2*e*(1-3*
n)+b^3*c*d*(1-2*n)-b^4*e*(1-2*n))*x^n)/a^2/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/n^2/(a+b*x^n+c*x^(2*n))+e^6*x*hy
pergeom([1, 1/n],[1+1/n],-e*x^n/d)/d/(a*e^2-b*d*e+c*d^2)^3-c*e^4*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*
a*c+b^2)^(1/2)))*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^3/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))-c*e^4
*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c
*d^2)^3/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))+c*e^2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(
-b^3*e*(1-n)+b^2*(1-n)*(c*d-e*(-4*a*c+b^2)^(1/2))+b*c*(2*a*e*(2-3*n)+d*(1-n)*(-4*a*c+b^2)^(1/2))-2*a*c*(2*c*d*
(1-2*n)-e*(1-n)*(-4*a*c+b^2)^(1/2)))/a/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/n/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))+c
*e^2*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))*(-b^3*e*(1-n)+b^2*(1-n)*(c*d+e*(-4*a*c+b^2)
^(1/2))+b*c*(2*a*e*(2-3*n)-d*(1-n)*(-4*a*c+b^2)^(1/2))-2*a*c*(2*c*d*(1-2*n)+e*(1-n)*(-4*a*c+b^2)^(1/2)))/a/(-4
*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/n/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))+1/2*c*x*hypergeom([1, 1/n],[1+1/n],-2*c*x^n
/(b+(-4*a*c+b^2)^(1/2)))*(b^5*e*(2*n^2-3*n+1)-b^4*(2*n^2-3*n+1)*(c*d+e*(-4*a*c+b^2)^(1/2))+a*b^2*c*(1-n)*(6*c*
d*(1-3*n)+e*(5-14*n)*(-4*a*c+b^2)^(1/2))-b^3*c*(1-n)*(a*e*(7-18*n)-d*(1-2*n)*(-4*a*c+b^2)^(1/2))-4*a^2*c^2*(2*
c*d*(8*n^2-6*n+1)+e*(3*n^2-4*n+1)*(-4*a*c+b^2)^(1/2))-2*a*b*c^2*(-2*a*e*(13*n^2-13*n+3)+d*(7*n^2-9*n+2)*(-4*a*
c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/n^2/(b^2-4*a*c+b*(-4*a*c+b^2)^(1/2))-1/2*c*x*hypergeom([
1, 1/n],[1+1/n],-2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))*(-b^5*e*(2*n^2-3*n+1)+b^4*(2*n^2-3*n+1)*(c*d-e*(-4*a*c+b^2)^(
1/2))+a*b^2*c*(1-n)*(-6*c*d*(1-3*n)+e*(5-14*n)*(-4*a*c+b^2)^(1/2))+b^3*c*(1-n)*(a*e*(7-18*n)+d*(1-2*n)*(-4*a*c
+b^2)^(1/2))-4*a^2*c^2*(-2*c*d*(8*n^2-6*n+1)+e*(3*n^2-4*n+1)*(-4*a*c+b^2)^(1/2))-2*a*b*c^2*(2*a*e*(13*n^2-13*n
+3)+d*(7*n^2-9*n+2)*(-4*a*c+b^2)^(1/2)))/a^2/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/n^2/(b^2-4*a*c-b*(-4*a*c+b^2)^
(1/2))

Rubi [A] (verified)

Time = 3.23 (sec) , antiderivative size = 1708, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1450, 251, 1444, 1436} \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\frac {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 )^3}-\frac {c \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\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 )^3}-\frac {c \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\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 )^3}+\frac {c \left (-e (1-n) b^3+\left (c d-\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-3 n)+\sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (2 c d (1-2 n)-\sqrt {b^2-4 a c} e (1-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 )^2 n}+\frac {c \left (-e (1-n) b^3+\left (c d+\sqrt {b^2-4 a c} e\right ) (1-n) b^2+c \left (2 a e (2-3 n)-\sqrt {b^2-4 a c} d (1-n)\right ) b-2 a c \left (2 c d (1-2 n)+\sqrt {b^2-4 a c} e (1-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 )^2 n}+\frac {x \left (c \left (-e b^2+c d b+2 a c e\right ) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right ) e^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 )}-\frac {c \left (-e \left (2 n^2-3 n+1\right ) b^5+\left (c d-\sqrt {b^2-4 a c} e\right ) \left (2 n^2-3 n+1\right ) b^4+c \left (a e (7-18 n)+\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n) b^3+a c \left (\sqrt {b^2-4 a c} e (5-14 n)-6 c d (1-3 n)\right ) (1-n) b^2-2 a c^2 \left (\sqrt {b^2-4 a c} d \left (7 n^2-9 n+2\right )+2 a e \left (13 n^2-13 n+3\right )\right ) b-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (3 n^2-4 n+1\right )-2 c d \left (8 n^2-6 n+1\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-\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n^2}+\frac {c \left (e \left (2 n^2-3 n+1\right ) b^5-\left (c d+\sqrt {b^2-4 a c} e\right ) \left (2 n^2-3 n+1\right ) b^4-c \left (a e (7-18 n)-\sqrt {b^2-4 a c} d (1-2 n)\right ) (1-n) b^3+a c \left (\sqrt {b^2-4 a c} e (5-14 n)+6 c d (1-3 n)\right ) (1-n) b^2-2 a c^2 \left (\sqrt {b^2-4 a c} d \left (7 n^2-9 n+2\right )-2 a e \left (13 n^2-13 n+3\right )\right ) b-4 a^2 c^2 \left (\sqrt {b^2-4 a c} e \left (3 n^2-4 n+1\right )+2 c d \left (8 n^2-6 n+1\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+\sqrt {b^2-4 a c} b-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n^2}+\frac {x \left (-c \left (-e (1-2 n) b^4+c d (1-2 n) b^3+a c e (5-14 n) b^2-2 a c^2 d (2-7 n) b-4 a^2 c^2 e (1-3 n)\right ) x^n+2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b e d+a e^2\right ) n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac {x \left (c \left (-e b^2+c d b+2 a c e\right ) x^n-2 a c^2 d+b^2 c d-b^3 e+3 a b c e\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]

[In]

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

[Out]

(x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e + 2*a*c*e)*x^n))/(2*a*(b^2 - 4*a*c)*(c*d^2 - b*
d*e + a*e^2)*n*(a + b*x^n + c*x^(2*n))^2) + (e^2*x*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e + c*(b*c*d - b^2*e
+ 2*a*c*e)*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))) + (x*(2*a^2*b*c^2*e*(4
- 11*n) - 3*a*b^3*c*e*(2 - 5*n) - 4*a^2*c^3*d*(1 - 4*n) + 5*a*b^2*c^2*d*(1 - 3*n) - b^4*c*d*(1 - 2*n) + b^5*(
e - 2*e*n) - c*(a*b^2*c*e*(5 - 14*n) - 2*a*b*c^2*d*(2 - 7*n) - 4*a^2*c^2*e*(1 - 3*n) + b^3*c*d*(1 - 2*n) - b^4
*e*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*n^2*(a + b*x^n + c*x^(2*n))) - (c*e^4*(2*c*
d - (b + Sqrt[b^2 - 4*a*c])*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)^3) + (c*e^2*(b*c*(2*a*e*(2 - 3*n) + Sqrt[b^2 - 4
*a*c]*d*(1 - n)) - 2*a*c*(2*c*d*(1 - 2*n) - Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(c*d - Sqrt[b^2
- 4*a*c]*e)*(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])*(c*d^2 - b*d*e + a*e^2)^2*n) - (c*(a*b^2*c*(Sqrt[b^2 - 4*a*c]*e*(
5 - 14*n) - 6*c*d*(1 - 3*n))*(1 - n) + b^3*c*(a*e*(7 - 18*n) + Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) - b^5*e*
(1 - 3*n + 2*n^2) + b^4*(c*d - Sqrt[b^2 - 4*a*c]*e)*(1 - 3*n + 2*n^2) - 4*a^2*c^2*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*
n + 3*n^2) - 2*c*d*(1 - 6*n + 8*n^2)) - 2*a*b*c^2*(Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2) + 2*a*e*(3 - 13*n + 1
3*n^2)))*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^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)*n^2) - (c*e^4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*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)^3) + (c*e^2*(b*c*(2*a*e*(2 - 3*n) - Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 2*a*c*(2*c
*d*(1 - 2*n) + Sqrt[b^2 - 4*a*c]*e*(1 - n)) - b^3*e*(1 - n) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e)*(1 - n))*x*Hyper
geometric2F1[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*Sqr
t[b^2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n) + (c*(a*b^2*c*(Sqrt[b^2 - 4*a*c]*e*(5 - 14*n) + 6*c*d*(1 - 3*n))*
(1 - n) - b^3*c*(a*e*(7 - 18*n) - Sqrt[b^2 - 4*a*c]*d*(1 - 2*n))*(1 - n) + b^5*e*(1 - 3*n + 2*n^2) - b^4*(c*d
+ Sqrt[b^2 - 4*a*c]*e)*(1 - 3*n + 2*n^2) - 4*a^2*c^2*(Sqrt[b^2 - 4*a*c]*e*(1 - 4*n + 3*n^2) + 2*c*d*(1 - 6*n +
8*n^2)) - 2*a*b*c^2*(Sqrt[b^2 - 4*a*c]*d*(2 - 9*n + 7*n^2) - 2*a*e*(3 - 13*n + 13*n^2)))*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^2 - 4*a*c + b*Sqrt[b^2 -
4*a*c])*(c*d^2 - b*d*e + a*e^2)*n^2) + (e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d
^2 - b*d*e + a*e^2)^3)

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-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 1436

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*
c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^n), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), In
t[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 1444

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] + Di
st[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] && Eq
Q[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && ILtQ[p, -1]

Rule 1450

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandInt
egrand[(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]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^6}{\left (c d^2-b d e+a e^2\right )^3 \left (d+e x^n\right )}+\frac {c d-b e-c e x^n}{\left (c d^2-b d e+a e^2\right ) \left (a+b x^n+c x^{2 n}\right )^3}-\frac {e^2 \left (-c d+b e+c e x^n\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^n+c x^{2 n}\right )^2}-\frac {e^4 \left (-c d+b e+c e x^n\right )}{\left (c d^2-b d e+a e^2\right )^3 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx \\ & = -\frac {e^4 \int \frac {-c d+b e+c e x^n}{a+b x^n+c x^{2 n}} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac {e^6 \int \frac {1}{d+e x^n} \, dx}{\left (c d^2-b d e+a e^2\right )^3}-\frac {e^2 \int \frac {-c d+b e+c e x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\int \frac {c d-b e-c e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c d^2-b d e+a e^2} \\ & = \frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}-\frac {\left (c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \left (c d^2-b d e+a e^2\right )^3}+\frac {e^2 \int \frac {-a b c e (3-4 n)+2 a c^2 d (1-2 n)-b^2 c d (1-n)+b^3 (e-e n)-c \left (b c d-b^2 e+2 a c e\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\frac {\int \frac {a b c e-2 a c (c d-b e) (1-4 n)+b^2 (c d-b e) (1-2 n)+c \left (b c d-b^2 e+2 a c e\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n} \\ & = \frac {x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac {e^2 x \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e+c \left (b c d-b^2 e+2 a c e\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n \left (a+b x^n+c x^{2 n}\right )}+\frac {x \left (2 a^2 b c^2 e (4-11 n)-3 a b^3 c e (2-5 n)-4 a^2 c^3 d (1-4 n)+5 a b^2 c^2 d (1-3 n)-b^4 c d (1-2 n)+b^5 (e-2 e n)-c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac {c e^4 \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}-\frac {c e^4 \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^3}+\frac {e^6 x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^3}+\frac {\int \frac {b^4 c d \left (1-3 n+2 n^2\right )-b^5 e \left (1-3 n+2 n^2\right )+2 a b^3 c e \left (3-11 n+8 n^2\right )+4 a^2 c^3 d \left (1-6 n+8 n^2\right )-a b^2 c^2 d \left (5-21 n+16 n^2\right )-2 a^2 b c^2 e \left (4-17 n+16 n^2\right )+c \left (a b^2 c e (5-14 n)-2 a b c^2 d (2-7 n)-4 a^2 c^2 e (1-3 n)+b^3 c d (1-2 n)-b^4 e (1-2 n)\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) n^2}-\frac {\left (c e^2 \left (\left (b c d-b^2 e+2 a c e\right ) (1-n)-\frac {2 a b c e (2-3 n)-4 a c^2 d (1-2 n)+b^2 c d (1-n)-b^3 (e-e n)}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n}+\frac {\left (e^2 \left (-\frac {1}{2} c \left (b c d-b^2 e+2 a c e\right ) (1-n)+\frac {b c \left (b c d-b^2 e+2 a c e\right ) (1-n)+2 c \left (-a b c e (3-4 n)+2 a c^2 d (1-2 n)-b^2 c d (1-n)+b^3 (e-e n)\right )}{2 \sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 n} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(43535\) vs. \(2(1708)=3416\).

Time = 7.87 (sec) , antiderivative size = 43535, normalized size of antiderivative = 25.49 \[ \int \frac {1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

e^6*integrate(1/(c^3*d^7 - 3*b*c^2*d^6*e + 3*b^2*c*d^5*e^2 - b^3*d^4*e^3 + a^3*d*e^6 + 3*(c*d^3*e^4 - b*d^2*e^
5)*a^2 + 3*(c^2*d^5*e^2 - 2*b*c*d^4*e^3 + b^2*d^3*e^4)*a + (c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^
3*d^3*e^4 + a^3*e^7 + 3*(c*d^2*e^5 - b*d*e^6)*a^2 + 3*(c^2*d^4*e^3 - 2*b*c*d^3*e^4 + b^2*d^2*e^5)*a)*x^n), x)
- 1/2*((4*a^3*c^4*e^3*(7*n - 1) - b^3*c^4*d^3*(2*n - 1) + 2*b^4*c^3*d^2*e*(2*n - 1) - b^5*c^2*d*e^2*(2*n - 1)
- (b^2*c^3*e^3*(26*n - 5) - 4*c^5*d^2*e*(3*n - 1) - 10*b*c^4*d*e^2*n)*a^2 - (b^2*c^4*d^2*e*(28*n - 9) - 2*b*c^
5*d^3*(7*n - 2) - 2*b^3*c^3*d*e^2*(5*n - 2) - b^4*c^2*e^3*(4*n - 1))*a)*x*x^(3*n) - (2*b^4*c^3*d^3*(2*n - 1) -
4*b^5*c^2*d^2*e*(2*n - 1) + 2*b^6*c*d*e^2*(2*n - 1) - 2*(b*c^3*e^3*(37*n - 6) - 2*c^4*d*e^2*(8*n - 1))*a^3 -
(2*b*c^4*d^2*e*(25*n - 8) + 3*b^2*c^3*d*e^2*(5*n + 1) - 11*b^3*c^2*e^3*(5*n - 1) - 4*c^5*d^3*(4*n - 1))*a^2 -
(b^2*c^4*d^3*(29*n - 9) - 2*b^3*c^3*d^2*e*(29*n - 10) + 3*b^4*c^2*d*e^2*(7*n - 3) + 2*b^5*c*e^3*(4*n - 1))*a)*
x*x^(2*n) + (4*a^4*c^3*e^3*(9*n - 1) - b^5*c^2*d^3*(2*n - 1) + 2*b^6*c*d^2*e*(2*n - 1) - b^7*d*e^2*(2*n - 1) +
(b^2*c^2*e^3*(14*n - 3) - 2*b*c^3*d*e^2*(13*n - 2) + 4*c^4*d^2*e*(5*n - 1))*a^3 - (b^4*c*e^3*(24*n - 5) - b^3
*c^2*d*e^2*(20*n - 1) - 2*b*c^4*d^3*n + 3*b^2*c^3*d^2*e)*a^2 - (3*b^4*c^2*d^2*e*(8*n - 3) - b^6*e^3*(4*n - 1)
- 4*b^3*c^3*d^3*(3*n - 1) - 4*b^5*c*d*e^2*(2*n - 1))*a)*x*x^n + (2*(b*c^2*e^3*(29*n - 4) - 2*c^3*d*e^2*(10*n -
1))*a^4 + (2*b*c^3*d^2*e*(29*n - 6) - 4*c^4*d^3*(6*n - 1) - 6*b^3*c*e^3*(6*n - 1) - b^2*c^2*d*e^2*(n - 3))*a^
3 - (b^3*c^2*d^2*e*(43*n - 11) - b^2*c^3*d^3*(21*n - 5) - b^4*c*d*e^2*(17*n - 5) - b^5*e^3*(5*n - 1))*a^2 - (b
^4*c^2*d^3*(3*n - 1) - 2*b^5*c*d^2*e*(3*n - 1) + b^6*d*e^2*(3*n - 1))*a)*x)/(16*a^8*c^2*e^4*n^2 + 8*(4*c^3*d^2
*e^2*n^2 - 4*b*c^2*d*e^3*n^2 - b^2*c*e^4*n^2)*a^7 + (16*c^4*d^4*n^2 - 32*b*c^3*d^3*e*n^2 + 16*b^3*c*d*e^3*n^2
+ b^4*e^4*n^2)*a^6 - 2*(4*b^2*c^3*d^4*n^2 - 8*b^3*c^2*d^3*e*n^2 + 3*b^4*c*d^2*e^2*n^2 + b^5*d*e^3*n^2)*a^5 + (
b^4*c^2*d^4*n^2 - 2*b^5*c*d^3*e*n^2 + b^6*d^2*e^2*n^2)*a^4 + (16*a^6*c^4*e^4*n^2 + 8*(4*c^5*d^2*e^2*n^2 - 4*b*
c^4*d*e^3*n^2 - b^2*c^3*e^4*n^2)*a^5 + (16*c^6*d^4*n^2 - 32*b*c^5*d^3*e*n^2 + 16*b^3*c^3*d*e^3*n^2 + b^4*c^2*e
^4*n^2)*a^4 - 2*(4*b^2*c^5*d^4*n^2 - 8*b^3*c^4*d^3*e*n^2 + 3*b^4*c^3*d^2*e^2*n^2 + b^5*c^2*d*e^3*n^2)*a^3 + (b
^4*c^4*d^4*n^2 - 2*b^5*c^3*d^3*e*n^2 + b^6*c^2*d^2*e^2*n^2)*a^2)*x^(4*n) + 2*(16*a^6*b*c^3*e^4*n^2 + 8*(4*b*c^
4*d^2*e^2*n^2 - 4*b^2*c^3*d*e^3*n^2 - b^3*c^2*e^4*n^2)*a^5 + (16*b*c^5*d^4*n^2 - 32*b^2*c^4*d^3*e*n^2 + 16*b^4
*c^2*d*e^3*n^2 + b^5*c*e^4*n^2)*a^4 - 2*(4*b^3*c^4*d^4*n^2 - 8*b^4*c^3*d^3*e*n^2 + 3*b^5*c^2*d^2*e^2*n^2 + b^6
*c*d*e^3*n^2)*a^3 + (b^5*c^3*d^4*n^2 - 2*b^6*c^2*d^3*e*n^2 + b^7*c*d^2*e^2*n^2)*a^2)*x^(3*n) + (32*a^7*c^3*e^4
*n^2 + 64*(c^4*d^2*e^2*n^2 - b*c^3*d*e^3*n^2)*a^6 + 2*(16*c^5*d^4*n^2 - 32*b*c^4*d^3*e*n^2 + 16*b^2*c^3*d^2*e^
2*n^2 - 3*b^4*c*e^4*n^2)*a^5 - (12*b^4*c^2*d^2*e^2*n^2 - 12*b^5*c*d*e^3*n^2 - b^6*e^4*n^2)*a^4 - 2*(3*b^4*c^3*
d^4*n^2 - 6*b^5*c^2*d^3*e*n^2 + 2*b^6*c*d^2*e^2*n^2 + b^7*d*e^3*n^2)*a^3 + (b^6*c^2*d^4*n^2 - 2*b^7*c*d^3*e*n^
2 + b^8*d^2*e^2*n^2)*a^2)*x^(2*n) + 2*(16*a^7*b*c^2*e^4*n^2 + 8*(4*b*c^3*d^2*e^2*n^2 - 4*b^2*c^2*d*e^3*n^2 - b
^3*c*e^4*n^2)*a^6 + (16*b*c^4*d^4*n^2 - 32*b^2*c^3*d^3*e*n^2 + 16*b^4*c*d*e^3*n^2 + b^5*e^4*n^2)*a^5 - 2*(4*b^
3*c^3*d^4*n^2 - 8*b^4*c^2*d^3*e*n^2 + 3*b^5*c*d^2*e^2*n^2 + b^6*d*e^3*n^2)*a^4 + (b^5*c^2*d^4*n^2 - 2*b^6*c*d^
3*e*n^2 + b^7*d^2*e^2*n^2)*a^3)*x^n) - integrate(-1/2*((2*n^2 - 3*n + 1)*b^4*c^3*d^5 - 3*(2*n^2 - 3*n + 1)*b^5
*c^2*d^4*e + 3*(2*n^2 - 3*n + 1)*b^6*c*d^3*e^2 - (2*n^2 - 3*n + 1)*b^7*d^2*e^3 + 2*(2*(24*n^2 - 10*n + 1)*c^3*
d*e^4 - (48*n^2 - 29*n + 4)*b*c^2*e^5)*a^4 + (8*(12*n^2 - 8*n + 1)*c^4*d^3*e^2 - 12*(16*n^2 - 13*n + 2)*b*c^3*
d^2*e^3 + (48*n^2 - 59*n + 11)*b^2*c^2*d*e^4 + 6*(8*n^2 - 6*n + 1)*b^3*c*e^5)*a^3 + (4*(8*n^2 - 6*n + 1)*c^5*d
^5 - 2*(48*n^2 - 41*n + 8)*b*c^4*d^4*e + 2*(24*n^2 - 19*n + 5)*b^2*c^3*d^3*e^2 + 2*(32*n^2 - 39*n + 7)*b^3*c^2
*d^2*e^3 - (42*n^2 - 53*n + 11)*b^4*c*d*e^4 - (6*n^2 - 5*n + 1)*b^5*e^5)*a^2 - ((16*n^2 - 21*n + 5)*b^2*c^4*d^
5 - 16*(3*n^2 - 4*n + 1)*b^3*c^3*d^4*e + 3*(14*n^2 - 19*n + 5)*b^4*c^2*d^3*e^2 - 2*(2*n^2 - 3*n + 1)*b^5*c*d^2
*e^3 - 2*(3*n^2 - 4*n + 1)*b^6*d*e^4)*a + ((2*n^2 - 3*n + 1)*b^3*c^4*d^5 - 3*(2*n^2 - 3*n + 1)*b^4*c^3*d^4*e +
3*(2*n^2 - 3*n + 1)*b^5*c^2*d^3*e^2 - (2*n^2 - 3*n + 1)*b^6*c*d^2*e^3 - 4*(15*n^2 - 8*n + 1)*a^4*c^3*e^5 - (8
*(5*n^2 - 6*n + 1)*c^4*d^2*e^3 - 2*(9*n^2 - 11*n + 2)*b*c^3*d*e^4 - (42*n^2 - 31*n + 5)*b^2*c^2*e^5)*a^3 - (4*
(3*n^2 - 4*n + 1)*c^5*d^4*e + 12*(n^2 - n)*b*c^4*d^3*e^2 - 2*(32*n^2 - 39*n + 7)*b^2*c^3*d^2*e^3 + 9*(4*n^2 -
5*n + 1)*b^3*c^2*d*e^4 + (6*n^2 - 5*n + 1)*b^4*c*e^5)*a^2 - (2*(7*n^2 - 9*n + 2)*b*c^5*d^5 - (42*n^2 - 55*n +
13)*b^2*c^4*d^4*e + 12*(3*n^2 - 4*n + 1)*b^3*c^3*d^3*e^2 - (2*n^2 - 3*n + 1)*b^4*c^2*d^2*e^3 - 2*(3*n^2 - 4*n
+ 1)*b^5*c*d*e^4)*a)*x^n)/(16*a^8*c^2*e^6*n^2 + 8*(6*c^3*d^2*e^4*n^2 - 6*b*c^2*d*e^5*n^2 - b^2*c*e^6*n^2)*a^7
+ (48*c^4*d^4*e^2*n^2 - 96*b*c^3*d^3*e^3*n^2 + 24*b^2*c^2*d^2*e^4*n^2 + 24*b^3*c*d*e^5*n^2 + b^4*e^6*n^2)*a^6
+ (16*c^5*d^6*n^2 - 48*b*c^4*d^5*e*n^2 + 24*b^2*c^3*d^4*e^2*n^2 + 32*b^3*c^2*d^3*e^3*n^2 - 21*b^4*c*d^2*e^4*n^
2 - 3*b^5*d*e^5*n^2)*a^5 - (8*b^2*c^4*d^6*n^2 - 24*b^3*c^3*d^5*e*n^2 + 21*b^4*c^2*d^4*e^2*n^2 - 2*b^5*c*d^3*e^
3*n^2 - 3*b^6*d^2*e^4*n^2)*a^4 + (b^4*c^3*d^6*n^2 - 3*b^5*c^2*d^5*e*n^2 + 3*b^6*c*d^4*e^2*n^2 - b^7*d^3*e^3*n^
2)*a^3 + (16*a^7*c^3*e^6*n^2 + 8*(6*c^4*d^2*e^4*n^2 - 6*b*c^3*d*e^5*n^2 - b^2*c^2*e^6*n^2)*a^6 + (48*c^5*d^4*e
^2*n^2 - 96*b*c^4*d^3*e^3*n^2 + 24*b^2*c^3*d^2*e^4*n^2 + 24*b^3*c^2*d*e^5*n^2 + b^4*c*e^6*n^2)*a^5 + (16*c^6*d
^6*n^2 - 48*b*c^5*d^5*e*n^2 + 24*b^2*c^4*d^4*e^2*n^2 + 32*b^3*c^3*d^3*e^3*n^2 - 21*b^4*c^2*d^2*e^4*n^2 - 3*b^5
*c*d*e^5*n^2)*a^4 - (8*b^2*c^5*d^6*n^2 - 24*b^3*c^4*d^5*e*n^2 + 21*b^4*c^3*d^4*e^2*n^2 - 2*b^5*c^2*d^3*e^3*n^2
- 3*b^6*c*d^2*e^4*n^2)*a^3 + (b^4*c^4*d^6*n^2 - 3*b^5*c^3*d^5*e*n^2 + 3*b^6*c^2*d^4*e^2*n^2 - b^7*c*d^3*e^3*n
^2)*a^2)*x^(2*n) + (16*a^7*b*c^2*e^6*n^2 + 8*(6*b*c^3*d^2*e^4*n^2 - 6*b^2*c^2*d*e^5*n^2 - b^3*c*e^6*n^2)*a^6 +
(48*b*c^4*d^4*e^2*n^2 - 96*b^2*c^3*d^3*e^3*n^2 + 24*b^3*c^2*d^2*e^4*n^2 + 24*b^4*c*d*e^5*n^2 + b^5*e^6*n^2)*a
^5 + (16*b*c^5*d^6*n^2 - 48*b^2*c^4*d^5*e*n^2 + 24*b^3*c^3*d^4*e^2*n^2 + 32*b^4*c^2*d^3*e^3*n^2 - 21*b^5*c*d^2
*e^4*n^2 - 3*b^6*d*e^5*n^2)*a^4 - (8*b^3*c^4*d^6*n^2 - 24*b^4*c^3*d^5*e*n^2 + 21*b^5*c^2*d^4*e^2*n^2 - 2*b^6*c
*d^3*e^3*n^2 - 3*b^7*d^2*e^4*n^2)*a^3 + (b^5*c^3*d^6*n^2 - 3*b^6*c^2*d^5*e*n^2 + 3*b^7*c*d^4*e^2*n^2 - b^8*d^3
*e^3*n^2)*a^2)*x^n), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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