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

Optimal. Leaf size=1708 \[ \text{result too large to display} \]

[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)

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Rubi [A]  time = 5.07442, antiderivative size = 1708, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1436, 245, 1430, 1422} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[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 1436

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]))

Rule 245

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 1430

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] + Dist
[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 1422

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])

Rubi steps

\begin{align*} \int \frac{1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\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}\\ &=\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^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 (c d (2-4 n)-\sqrt{b^2-4 a c} e (1-n)\right )+b^2 \left (c d-\sqrt{b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\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 )}{a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n}-\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^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 ) 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 )}{a \left (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 )^2 n}+\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 \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 )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right ) n^2}+\frac{\left (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 )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} \left (c d^2-b d e+a e^2\right ) n^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{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^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 (c d (2-4 n)-\sqrt{b^2-4 a c} e (1-n)\right )+b^2 \left (c d-\sqrt{b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\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 )}{a \left (b^2-4 a c\right )^{3/2} \left (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 \, _2F_1\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 )^{5/2} \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-e (b d-a e)\right ) n^2}-\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^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 ) 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 )}{a \left (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 )^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 \, _2F_1\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 )^{5/2} \left (b+\sqrt{b^2-4 a c}\right ) \left (c d^2-e (b d-a e)\right ) n^2}+\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}\\ \end{align*}

Mathematica [B]  time = 8.65581, size = 30258, normalized size = 17.72 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.329, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+e{x}^{n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} e x^{4 \, n} + a^{3} d +{\left (c^{3} e x^{n} + c^{3} d\right )} x^{6 \, n} + 3 \,{\left (b c^{2} e x^{2 \, n} + a c^{2} d +{\left (b c^{2} d + a c^{2} e\right )} x^{n}\right )} x^{4 \, n} +{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{3 \, n} + 3 \,{\left (b^{2} c e x^{3 \, n} + a^{2} c d +{\left (b^{2} c d + 2 \, a b c e\right )} x^{2 \, n} +{\left (2 \, a b c d + a^{2} c e\right )} x^{n}\right )} x^{2 \, n} + 3 \,{\left (a b^{2} d + a^{2} b e\right )} x^{2 \, n} +{\left (3 \, a^{2} b d + a^{3} e\right )} x^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}{\left (e x^{n} + d\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)