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

Optimal. Leaf size=2446 \[ \text{result too large to display} \]

[Out]

-(x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*
e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^n))/(2*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))^2) - (e^2*x*(5*b^3*c*d*e - 14*a*b*c^2*d*e - 2*b^4*e^2 - b^2*c*(3*c*d^2 - 7*a*e^2) + 2*a*c^2*
(3*c*d^2 - a*e^2) + c*(5*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(3*c*d^2 - 5*a*e^2))*x^n))/(a*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)^3*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*(a*e^2*(13 - 37*n) - 5*c*d^2*(1 - 3*n))
- b^4*c*(a*e^2*(7 - 17*n) - c*d^2*(1 - 2*n)) - 4*a^2*b*c^3*d*e*(4 - 11*n) + 6*a*b^3*c^2*d*e*(2 - 5*n) + 4*a^2*
c^3*(c*d^2 - a*e^2)*(1 - 4*n) - 2*b^5*c*d*e*(1 - 2*n) + b^6*e^2*(1 - 2*n) + c*(2*a*b*c^2*(a*e^2*(4 - 13*n) - c
*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1
- 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*n^2
*(a + b*x^n + c*x^(2*n))) - (c*e^4*(10*c^2*d^2 + 3*b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d + 3*Sqrt[b^2 -
4*a*c]*d + a*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)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) + 2*Sqrt[b^2 - 4*a*c]*
d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) + 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - n))
+ b*c*(c*d*(4*a*e*(5 - 8*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*b^4*e^2
*(1 - n) - b^3*e*(5*c*d - 2*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)^3*n)
+ (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*
c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) - (b^4*c*(4*
a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2) - 8*a^2
*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*n + 18*
n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometri
c2F1[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 - Sqrt[b^2 - 4*a*c]
)*(c*d^2 - b*d*e + a*e^2)^2*n^2) - (c*e^4*(10*c^2*d^2 + 3*b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d - 3*Sqr
t[b^2 - 4*a*c]*d + a*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)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) - 2*Sqrt[b^2 -
4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) - 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(
1 - n)) + b*c*(c*d*(4*a*e*(5 - 8*n) - 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) + 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*
b^4*e^2*(1 - n) - b^3*e*(5*c*d + 2*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
)^3*n) + (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2
*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) + (b^
4*c*(4*a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2)
- 8*a^2*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*
n + 18*n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hyperg
eometric2F1[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 + Sqrt[b^2 -
4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n^2) + (3*e^6*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e
*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^4) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2
*(c*d^2 - b*d*e + a*e^2)^3)

________________________________________________________________________________________

Rubi [A]  time = 8.93528, antiderivative size = 2446, normalized size of antiderivative = 1., number of steps used = 16, 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)^2*(a + b*x^n + c*x^(2*n))^3),x]

[Out]

-(x*(2*b^3*c*d*e - 6*a*b*c^2*d*e - b^4*e^2 - b^2*c*(c*d^2 - 4*a*e^2) + 2*a*c^2*(c*d^2 - a*e^2) + c*(2*b^2*c*d*
e - 4*a*c^2*d*e - b^3*e^2 - b*c*(c*d^2 - 3*a*e^2))*x^n))/(2*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))^2) - (e^2*x*(5*b^3*c*d*e - 14*a*b*c^2*d*e - 2*b^4*e^2 - b^2*c*(3*c*d^2 - 7*a*e^2) + 2*a*c^2*
(3*c*d^2 - a*e^2) + c*(5*b^2*c*d*e - 8*a*c^2*d*e - 2*b^3*e^2 - b*c*(3*c*d^2 - 5*a*e^2))*x^n))/(a*(b^2 - 4*a*c)
*(c*d^2 - b*d*e + a*e^2)^3*n*(a + b*x^n + c*x^(2*n))) - (x*(a*b^2*c^2*(a*e^2*(13 - 37*n) - 5*c*d^2*(1 - 3*n))
- b^4*c*(a*e^2*(7 - 17*n) - c*d^2*(1 - 2*n)) - 4*a^2*b*c^3*d*e*(4 - 11*n) + 6*a*b^3*c^2*d*e*(2 - 5*n) + 4*a^2*
c^3*(c*d^2 - a*e^2)*(1 - 4*n) - 2*b^5*c*d*e*(1 - 2*n) + b^6*e^2*(1 - 2*n) + c*(2*a*b*c^2*(a*e^2*(4 - 13*n) - c
*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1
- 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*x^n))/(2*a^2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^2*n^2
*(a + b*x^n + c*x^(2*n))) - (c*e^4*(10*c^2*d^2 + 3*b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d + 3*Sqrt[b^2 -
4*a*c]*d + a*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)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) + 2*Sqrt[b^2 - 4*a*c]*
d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) + 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - n))
+ b*c*(c*d*(4*a*e*(5 - 8*n) + 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*b^4*e^2
*(1 - n) - b^3*e*(5*c*d - 2*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)^3*n)
+ (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2*a*b^2*
c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) - (b^4*c*(4*
a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2) - 8*a^2
*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*n + 18*
n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hypergeometri
c2F1[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 - Sqrt[b^2 - 4*a*c]
)*(c*d^2 - b*d*e + a*e^2)^2*n^2) - (c*e^4*(10*c^2*d^2 + 3*b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(5*b*d - 3*Sqr
t[b^2 - 4*a*c]*d + a*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)^4) + (c*e^2*(4*a*c^2*(e*(a*e*(1 - 2*n) - 2*Sqrt[b^2 -
4*a*c]*d*(1 - n)) - 3*c*d^2*(1 - 2*n)) - b^2*c*(e*(a*e*(9 - 13*n) - 5*Sqrt[b^2 - 4*a*c]*d*(1 - n)) - 3*c*d^2*(
1 - n)) + b*c*(c*d*(4*a*e*(5 - 8*n) - 3*Sqrt[b^2 - 4*a*c]*d*(1 - n)) + 5*a*Sqrt[b^2 - 4*a*c]*e^2*(1 - n)) + 2*
b^4*e^2*(1 - n) - b^3*e*(5*c*d + 2*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
)^3*n) + (c*((2*a*b*c^2*(a*e^2*(4 - 13*n) - c*d^2*(2 - 7*n)) - b^3*c*(2*a*e^2*(3 - 8*n) - c*d^2*(1 - 2*n)) + 2
*a*b^2*c^2*d*e*(5 - 14*n) - 8*a^2*c^3*d*e*(1 - 3*n) - 2*b^4*c*d*e*(1 - 2*n) + b^5*e^2*(1 - 2*n))*(1 - n) + (b^
4*c*(4*a*e^2*(2 - 5*n) - c*d^2*(1 - 2*n))*(1 - n) + 2*b^5*c*d*e*(1 - 3*n + 2*n^2) - b^6*e^2*(1 - 3*n + 2*n^2)
- 8*a^2*c^3*(c*d^2 - a*e^2)*(1 - 6*n + 8*n^2) + 8*a^2*b*c^3*d*e*(3 - 13*n + 13*n^2) - 2*a*b^3*c^2*d*e*(7 - 25*
n + 18*n^2) + 2*a*b^2*c^2*(3*c*d^2*(1 - 4*n + 3*n^2) - a*e^2*(9 - 38*n + 35*n^2)))/Sqrt[b^2 - 4*a*c])*x*Hyperg
eometric2F1[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 + Sqrt[b^2 -
4*a*c])*(c*d^2 - b*d*e + a*e^2)^2*n^2) + (3*e^6*(2*c*d - b*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e
*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^4) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2
*(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 )^2 \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 )^2}-\frac{3 e^6 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^4 \left (d+e x^n\right )}+\frac{c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^n+c x^{2 n}\right )^3}+\frac{e^2 \left (3 c^2 d^2-5 b c d e+2 b^2 e^2-a c e^2+\left (-4 c^2 d e+2 b c e^2\right ) 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 )^2}+\frac{e^4 \left (5 c^2 d^2-8 b c d e+3 b^2 e^2-a c e^2+\left (-6 c^2 d e+3 b c e^2\right ) x^n\right )}{\left (c d^2-b d e+a e^2\right )^4 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx\\ &=\frac{e^4 \int \frac{5 c^2 d^2-8 b c d e+3 b^2 e^2-a c e^2+\left (-6 c^2 d e+3 b c e^2\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{\left (c d^2-b d e+a e^2\right )^4}+\frac{\left (3 e^6 (2 c d-b e)\right ) \int \frac{1}{d+e x^n} \, dx}{\left (c d^2-b d e+a e^2\right )^4}+\frac{e^2 \int \frac{3 c^2 d^2-5 b c d e+2 b^2 e^2-a c e^2+\left (-4 c^2 d e+2 b c e^2\right ) 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 )^3}+\frac{e^6 \int \frac{1}{\left (d+e x^n\right )^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{\left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{2 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 )^2}-\frac{e^2 x \left (5 b^3 c d e-14 a b c^2 d e-2 b^4 e^2-b^2 c \left (3 c d^2-7 a e^2\right )+2 a c^2 \left (3 c d^2-a e^2\right )+c \left (5 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (3 c d^2-5 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 n \left (a+b x^n+c x^{2 n}\right )}+\frac{3 e^6 (2 c d-b e) 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 )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^3}-\frac{\left (c e^4 \left (10 c^2 d^2+3 b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d-3 \sqrt{b^2-4 a c} d+a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^4}+\frac{\left (c e^4 \left (10 c^2 d^2+3 b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d+3 \sqrt{b^2-4 a c} d+a e\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )^4}-\frac{e^2 \int \frac{-b^2 c \left (a e^2 (7-9 n)-3 c d^2 (1-n)\right )+2 a b c^2 d e (7-10 n)-2 a c^2 \left (3 c d^2-a e^2\right ) (1-2 n)-5 b^3 c d e (1-n)+2 b^4 e^2 (1-n)-c \left (5 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (3 c d^2-5 a e^2\right )\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 )^3 n}-\frac{\int \frac{a b c e (2 c d-b e)-2 a c \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right ) (1-4 n)+b^2 \left (c^2 d^2+b^2 e^2-c e (2 b d+a e)\right ) (1-2 n)-c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\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 )^2 n}\\ &=-\frac{x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{2 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 )^2}-\frac{e^2 x \left (5 b^3 c d e-14 a b c^2 d e-2 b^4 e^2-b^2 c \left (3 c d^2-7 a e^2\right )+2 a c^2 \left (3 c d^2-a e^2\right )+c \left (5 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (3 c d^2-5 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 \left (a e^2 (13-37 n)-5 c d^2 (1-3 n)\right )-b^4 c \left (a e^2 (7-17 n)-c d^2 (1-2 n)\right )-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left (c d^2-a e^2\right ) (1-4 n)-2 b^5 c d e (1-2 n)+b^6 e^2 (1-2 n)+c \left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (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 )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c e^4 \left (10 c^2 d^2+3 b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d+3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}-\frac{c e^4 \left (10 c^2 d^2+3 b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d-3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}+\frac{3 e^6 (2 c d-b e) 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 )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\int \frac{-b^4 c \left (a e^2 (7-18 n)-c d^2 (1-2 n)\right ) (1-n)-2 b^5 c d e \left (1-3 n+2 n^2\right )+b^6 e^2 \left (1-3 n+2 n^2\right )+4 a b^3 c^2 d e \left (3-11 n+8 n^2\right )+4 a^2 c^3 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )-4 a^2 b c^3 d e \left (4-17 n+16 n^2\right )-a b^2 c^2 \left (c d^2 \left (5-21 n+16 n^2\right )-a e^2 \left (13-55 n+48 n^2\right )\right )+c \left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (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 )^2 n^2}-\frac{\left (c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right )-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d-2 \sqrt{b^2-4 a c} e\right ) (1-n)\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 )^{3/2} \left (c d^2-b d e+a e^2\right )^3 n}+\frac{\left (c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )+5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d+2 \sqrt{b^2-4 a c} e\right ) (1-n)\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 )^{3/2} \left (c d^2-b d e+a e^2\right )^3 n}\\ &=-\frac{x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{2 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 )^2}-\frac{e^2 x \left (5 b^3 c d e-14 a b c^2 d e-2 b^4 e^2-b^2 c \left (3 c d^2-7 a e^2\right )+2 a c^2 \left (3 c d^2-a e^2\right )+c \left (5 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (3 c d^2-5 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 \left (a e^2 (13-37 n)-5 c d^2 (1-3 n)\right )-b^4 c \left (a e^2 (7-17 n)-c d^2 (1-2 n)\right )-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left (c d^2-a e^2\right ) (1-4 n)-2 b^5 c d e (1-2 n)+b^6 e^2 (1-2 n)+c \left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (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 )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c e^4 \left (10 c^2 d^2+3 b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d+3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}-\frac{c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right )-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d-2 \sqrt{b^2-4 a c} e\right ) (1-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 )^3 n}-\frac{c e^4 \left (10 c^2 d^2+3 b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d-3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}+\frac{c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )+5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d+2 \sqrt{b^2-4 a c} e\right ) (1-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 )^3 n}+\frac{3 e^6 (2 c d-b e) 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 )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^3}+\frac{\left (c \left (\left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (1-2 n)\right ) (1-n)-\frac{b^4 c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n)+2 b^5 c d e \left (1-3 n+2 n^2\right )-b^6 e^2 \left (1-3 n+2 n^2\right )-8 a^2 c^3 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+8 a^2 b c^3 d e \left (3-13 n+13 n^2\right )-2 a b^3 c^2 d e \left (7-25 n+18 n^2\right )+2 a b^2 c^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (9-38 n+35 n^2\right )\right )}{\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}{4 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 n^2}+\frac{\left (c \left (\left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (1-2 n)\right ) (1-n)+\frac{b^4 c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n)+2 b^5 c d e \left (1-3 n+2 n^2\right )-b^6 e^2 \left (1-3 n+2 n^2\right )-8 a^2 c^3 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+8 a^2 b c^3 d e \left (3-13 n+13 n^2\right )-2 a b^3 c^2 d e \left (7-25 n+18 n^2\right )+2 a b^2 c^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (9-38 n+35 n^2\right )\right )}{\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}{4 a^2 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 n^2}\\ &=-\frac{x \left (2 b^3 c d e-6 a b c^2 d e-b^4 e^2-b^2 c \left (c d^2-4 a e^2\right )+2 a c^2 \left (c d^2-a e^2\right )+c \left (2 b^2 c d e-4 a c^2 d e-b^3 e^2-b c \left (c d^2-3 a e^2\right )\right ) x^n\right )}{2 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 )^2}-\frac{e^2 x \left (5 b^3 c d e-14 a b c^2 d e-2 b^4 e^2-b^2 c \left (3 c d^2-7 a e^2\right )+2 a c^2 \left (3 c d^2-a e^2\right )+c \left (5 b^2 c d e-8 a c^2 d e-2 b^3 e^2-b c \left (3 c d^2-5 a e^2\right )\right ) x^n\right )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 \left (a e^2 (13-37 n)-5 c d^2 (1-3 n)\right )-b^4 c \left (a e^2 (7-17 n)-c d^2 (1-2 n)\right )-4 a^2 b c^3 d e (4-11 n)+6 a b^3 c^2 d e (2-5 n)+4 a^2 c^3 \left (c d^2-a e^2\right ) (1-4 n)-2 b^5 c d e (1-2 n)+b^6 e^2 (1-2 n)+c \left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (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 )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c e^4 \left (10 c^2 d^2+3 b \left (b+\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d+3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}-\frac{c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)+2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)+5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)+3 \sqrt{b^2-4 a c} d (1-n)\right )-5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d-2 \sqrt{b^2-4 a c} e\right ) (1-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 )^3 n}+\frac{c \left (\left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (1-2 n)\right ) (1-n)-\frac{b^4 c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n)+2 b^5 c d e \left (1-3 n+2 n^2\right )-b^6 e^2 \left (1-3 n+2 n^2\right )-8 a^2 c^3 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+8 a^2 b c^3 d e \left (3-13 n+13 n^2\right )-2 a b^3 c^2 d e \left (7-25 n+18 n^2\right )+2 a b^2 c^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (9-38 n+35 n^2\right )\right )}{\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 )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n^2}-\frac{c e^4 \left (10 c^2 d^2+3 b \left (b-\sqrt{b^2-4 a c}\right ) e^2-2 c e \left (5 b d-3 \sqrt{b^2-4 a c} d+a e\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 )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^4}+\frac{c e^2 \left (4 a c^2 \left (e \left (a e (1-2 n)-2 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-2 n)\right )-b^2 c \left (e \left (a e (9-13 n)-5 \sqrt{b^2-4 a c} d (1-n)\right )-3 c d^2 (1-n)\right )+b c \left (c d \left (4 a e (5-8 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )+5 a \sqrt{b^2-4 a c} e^2 (1-n)\right )+2 b^4 e^2 (1-n)-b^3 e \left (5 c d+2 \sqrt{b^2-4 a c} e\right ) (1-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 )^3 n}+\frac{c \left (\left (2 a b c^2 \left (a e^2 (4-13 n)-c d^2 (2-7 n)\right )-b^3 c \left (2 a e^2 (3-8 n)-c d^2 (1-2 n)\right )+2 a b^2 c^2 d e (5-14 n)-8 a^2 c^3 d e (1-3 n)-2 b^4 c d e (1-2 n)+b^5 e^2 (1-2 n)\right ) (1-n)+\frac{b^4 c \left (4 a e^2 (2-5 n)-c d^2 (1-2 n)\right ) (1-n)+2 b^5 c d e \left (1-3 n+2 n^2\right )-b^6 e^2 \left (1-3 n+2 n^2\right )-8 a^2 c^3 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+8 a^2 b c^3 d e \left (3-13 n+13 n^2\right )-2 a b^3 c^2 d e \left (7-25 n+18 n^2\right )+2 a b^2 c^2 \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (9-38 n+35 n^2\right )\right )}{\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 )}{2 a^2 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2 n^2}+\frac{3 e^6 (2 c d-b e) 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 )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}

Mathematica [B]  time = 10.1094, size = 39502, normalized size = 16.15 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [F]  time = 0.448, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \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)^2/(a+b*x^n+c*x^(2*n))^3,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

(c*d^2*e^6*(7*n - 1) - b*d*e^7*(4*n - 1) + a*e^8*(n - 1))*integrate(1/(c^4*d^10*n - 4*b*c^3*d^9*e*n + 6*b^2*c^
2*d^8*e^2*n - 4*b^3*c*d^7*e^3*n + b^4*d^6*e^4*n + a^4*d^2*e^8*n + 4*(c*d^4*e^6*n - b*d^3*e^7*n)*a^3 + 6*(c^2*d
^6*e^4*n - 2*b*c*d^5*e^5*n + b^2*d^4*e^6*n)*a^2 + 4*(c^3*d^8*e^2*n - 3*b*c^2*d^7*e^3*n + 3*b^2*c*d^6*e^4*n - b
^3*d^5*e^5*n)*a + (c^4*d^9*e*n - 4*b*c^3*d^8*e^2*n + 6*b^2*c^2*d^7*e^3*n - 4*b^3*c*d^6*e^4*n + b^4*d^5*e^5*n +
a^4*d*e^9*n + 4*(c*d^3*e^7*n - b*d^2*e^8*n)*a^3 + 6*(c^2*d^5*e^5*n - 2*b*c*d^4*e^6*n + b^2*d^3*e^7*n)*a^2 + 4
*(c^3*d^7*e^3*n - 3*b*c^2*d^6*e^4*n + 3*b^2*c*d^5*e^5*n - b^3*d^4*e^6*n)*a)*x^n), x) + 1/2*((b^3*c^5*d^5*e*(2*
n - 1) - 3*b^4*c^4*d^4*e^2*(2*n - 1) + 3*b^5*c^3*d^3*e^3*(2*n - 1) - b^6*c^2*d^2*e^4*(2*n - 1) + 32*a^4*c^4*e^
6*n + 2*(b*c^4*d*e^5*(33*n - 4) - 4*c^5*d^2*e^4*(11*n - 1) - 8*b^2*c^3*e^6*n)*a^3 + 2*(b^2*c^4*d^2*e^4*(29*n -
1) - 3*b^3*c^3*d*e^5*(7*n - 1) - 4*c^6*d^4*e^2*(3*n - 1) + 6*b*c^5*d^3*e^3*(n - 1) + b^4*c^2*e^6*n)*a^2 - (3*
b^3*c^4*d^3*e^3*(12*n - 5) + 2*b*c^6*d^5*e*(7*n - 2) - b^5*c^2*d*e^5*(6*n - 1) - 14*b^2*c^5*d^4*e^2*(3*n - 1)
- 2*b^4*c^3*d^2*e^4*(n - 2))*a)*x*x^(4*n) + (b^3*c^5*d^6*(2*n - 1) - b^4*c^4*d^5*e*(2*n - 1) - 3*b^5*c^3*d^4*e
^2*(2*n - 1) + 5*b^6*c^2*d^3*e^3*(2*n - 1) - 2*b^7*c*d^2*e^4*(2*n - 1) - 4*(c^4*d*e^5*(8*n - 1) - 16*b*c^3*e^6
*n)*a^4 + (b^2*c^3*d*e^5*(163*n - 21) - 6*b*c^4*d^2*e^4*(27*n - 2) - 8*c^5*d^3*e^3*(5*n - 1) - 32*b^3*c^2*e^6*
n)*a^3 - (b^4*c^2*d*e^5*(89*n - 13) - b^3*c^3*d^2*e^4*(77*n + 5) - 2*b^2*c^4*d^3*e^3*(50*n - 19) + 8*b*c^5*d^4
*e^2*(9*n - 2) + 4*c^6*d^5*e*(2*n - 1) - 4*b^5*c*e^6*n)*a^2 - (b^4*c^3*d^3*e^3*(73*n - 29) - b^3*c^4*d^4*e^2*(
51*n - 16) - b^2*c^5*d^5*e*(13*n - 5) - b^5*c^2*d^2*e^4*(11*n - 10) + 2*b*c^6*d^6*(7*n - 2) - 2*b^6*c*d*e^5*(6
*n - 1))*a)*x*x^(3*n) + (2*b^4*c^4*d^6*(2*n - 1) - 5*b^5*c^3*d^5*e*(2*n - 1) + 3*b^6*c^2*d^4*e^2*(2*n - 1) + b
^7*c*d^3*e^3*(2*n - 1) - b^8*d^2*e^4*(2*n - 1) + 64*a^5*c^3*e^6*n - 2*(2*c^4*d^2*e^4*(34*n - 3) - b*c^3*d*e^5*
(23*n - 2))*a^4 + (b^2*c^3*d^2*e^4*(81*n - 11) + b^3*c^2*d*e^5*(48*n - 7) - 8*b*c^4*d^3*e^3*(18*n - 1) + 8*c^5
*d^4*e^2*(n + 1) - 12*b^4*c*e^6*n)*a^3 - (2*b*c^5*d^5*e*(43*n - 14) + b^4*c^2*d^2*e^4*(21*n - 10) + 2*b^5*c*d*
e^5*(20*n - 3) - 5*b^3*c^3*d^3*e^3*(19*n - 2) - 4*c^6*d^6*(4*n - 1) - 10*b^2*c^4*d^4*e^2*(4*n - 3) - 2*b^6*e^6
*n)*a^2 - (b^4*c^3*d^4*e^2*(39*n - 19) + b^2*c^5*d^6*(29*n - 9) + b^5*c^2*d^3*e^3*(25*n - 6) - 3*b^3*c^4*d^5*e
*(25*n - 9) - b^7*d*e^5*(6*n - 1) - 6*b^6*c*d^2*e^4*(2*n - 1))*a)*x*x^(2*n) + (b^5*c^3*d^6*(2*n - 1) - 3*b^6*c
^2*d^5*e*(2*n - 1) + 3*b^7*c*d^4*e^2*(2*n - 1) - b^8*d^3*e^3*(2*n - 1) - 4*(c^3*d*e^5*(10*n - 1) - 16*b*c^2*e^
6*n)*a^5 + (b^2*c^2*d*e^5*(115*n - 13) - 2*b*c^3*d^2*e^4*(55*n - 4) - 8*c^4*d^3*e^3*(7*n - 1) - 32*b^3*c*e^6*n
)*a^4 - (b^4*c*d*e^5*(55*n - 7) - 3*b^3*c^2*d^2*e^4*(35*n - 2) + 2*b^2*c^3*d^3*e^3*(8*n + 7) + 4*c^5*d^5*e*(4*
n - 1) + 8*b*c^4*d^4*e^2*(n - 1) - 4*b^5*e^6*n)*a^3 + (b^3*c^3*d^4*e^2*(41*n - 26) - b^5*c*d^2*e^4*(31*n - 1)
- b^2*c^4*d^5*e*(23*n - 11) + b^4*c^2*d^3*e^3*(8*n + 15) + b^6*d*e^5*(7*n - 1) - 2*b*c^5*d^6*n)*a^2 + (3*b^4*c
^3*d^5*e*(13*n - 5) - 3*b^5*c^2*d^4*e^2*(13*n - 6) + b^6*c*d^3*e^3*(9*n - 7) - 4*b^3*c^4*d^6*(3*n - 1) + 3*b^7
*d^2*e^4*n)*a)*x*x^n + (32*a^6*c^2*e^6*n - 4*(c^3*d^2*e^4*(10*n - 1) + 4*b^2*c*e^6*n)*a^5 + (b^2*c^2*d^2*e^4*(
115*n - 13) - 12*b*c^3*d^3*e^3*(13*n - 1) + 48*c^4*d^4*e^2*n + 2*b^4*e^6*n)*a^4 + (b^3*c^2*d^3*e^3*(57*n + 1)
- b^4*c*d^2*e^4*(55*n - 7) - 4*b*c^4*d^5*e*(23*n - 5) + 6*b^2*c^3*d^4*e^2*(11*n - 4) + 4*c^5*d^6*(6*n - 1))*a^
3 + (b^3*c^3*d^5*e*(65*n - 17) - b^2*c^4*d^6*(21*n - 5) - 6*b^4*c^2*d^4*e^2*(10*n - 3) + b^5*c*d^3*e^3*(9*n -
5) + b^6*d^2*e^4*(7*n - 1))*a^2 + (b^4*c^3*d^6*(3*n - 1) - 3*b^5*c^2*d^5*e*(3*n - 1) + 3*b^6*c*d^4*e^2*(3*n -
1) - b^7*d^3*e^3*(3*n - 1))*a)*x)/(16*a^9*c^2*d^2*e^6*n^2 + 8*(6*c^3*d^4*e^4*n^2 - 6*b*c^2*d^3*e^5*n^2 - b^2*c
*d^2*e^6*n^2)*a^8 + (48*c^4*d^6*e^2*n^2 - 96*b*c^3*d^5*e^3*n^2 + 24*b^2*c^2*d^4*e^4*n^2 + 24*b^3*c*d^3*e^5*n^2
+ b^4*d^2*e^6*n^2)*a^7 + (16*c^5*d^8*n^2 - 48*b*c^4*d^7*e*n^2 + 24*b^2*c^3*d^6*e^2*n^2 + 32*b^3*c^2*d^5*e^3*n
^2 - 21*b^4*c*d^4*e^4*n^2 - 3*b^5*d^3*e^5*n^2)*a^6 - (8*b^2*c^4*d^8*n^2 - 24*b^3*c^3*d^7*e*n^2 + 21*b^4*c^2*d^
6*e^2*n^2 - 2*b^5*c*d^5*e^3*n^2 - 3*b^6*d^4*e^4*n^2)*a^5 + (b^4*c^3*d^8*n^2 - 3*b^5*c^2*d^7*e*n^2 + 3*b^6*c*d^
6*e^2*n^2 - b^7*d^5*e^3*n^2)*a^4 + (16*a^7*c^4*d*e^7*n^2 + 8*(6*c^5*d^3*e^5*n^2 - 6*b*c^4*d^2*e^6*n^2 - b^2*c^
3*d*e^7*n^2)*a^6 + (48*c^6*d^5*e^3*n^2 - 96*b*c^5*d^4*e^4*n^2 + 24*b^2*c^4*d^3*e^5*n^2 + 24*b^3*c^3*d^2*e^6*n^
2 + b^4*c^2*d*e^7*n^2)*a^5 + (16*c^7*d^7*e*n^2 - 48*b*c^6*d^6*e^2*n^2 + 24*b^2*c^5*d^5*e^3*n^2 + 32*b^3*c^4*d^
4*e^4*n^2 - 21*b^4*c^3*d^3*e^5*n^2 - 3*b^5*c^2*d^2*e^6*n^2)*a^4 - (8*b^2*c^6*d^7*e*n^2 - 24*b^3*c^5*d^6*e^2*n^
2 + 21*b^4*c^4*d^5*e^3*n^2 - 2*b^5*c^3*d^4*e^4*n^2 - 3*b^6*c^2*d^3*e^5*n^2)*a^3 + (b^4*c^5*d^7*e*n^2 - 3*b^5*c
^4*d^6*e^2*n^2 + 3*b^6*c^3*d^5*e^3*n^2 - b^7*c^2*d^4*e^4*n^2)*a^2)*x^(5*n) + (16*(c^4*d^2*e^6*n^2 + 2*b*c^3*d*
e^7*n^2)*a^7 + 8*(6*c^5*d^4*e^4*n^2 + 6*b*c^4*d^3*e^5*n^2 - 13*b^2*c^3*d^2*e^6*n^2 - 2*b^3*c^2*d*e^7*n^2)*a^6
+ (48*c^6*d^6*e^2*n^2 - 168*b^2*c^4*d^4*e^4*n^2 + 72*b^3*c^3*d^3*e^5*n^2 + 49*b^4*c^2*d^2*e^6*n^2 + 2*b^5*c*d*
e^7*n^2)*a^5 + (16*c^7*d^8*n^2 - 16*b*c^6*d^7*e*n^2 - 72*b^2*c^5*d^6*e^2*n^2 + 80*b^3*c^4*d^5*e^3*n^2 + 43*b^4
*c^3*d^4*e^4*n^2 - 45*b^5*c^2*d^3*e^5*n^2 - 6*b^6*c*d^2*e^6*n^2)*a^4 - (8*b^2*c^6*d^8*n^2 - 8*b^3*c^5*d^7*e*n^
2 - 27*b^4*c^4*d^6*e^2*n^2 + 40*b^5*c^3*d^5*e^3*n^2 - 7*b^6*c^2*d^4*e^4*n^2 - 6*b^7*c*d^3*e^5*n^2)*a^3 + (b^4*
c^5*d^8*n^2 - b^5*c^4*d^7*e*n^2 - 3*b^6*c^3*d^6*e^2*n^2 + 5*b^7*c^2*d^5*e^3*n^2 - 2*b^8*c*d^4*e^4*n^2)*a^2)*x^
(4*n) + (32*a^8*c^3*d*e^7*n^2 + 32*(3*c^4*d^3*e^5*n^2 - 2*b*c^3*d^2*e^6*n^2)*a^7 + 2*(48*c^5*d^5*e^3*n^2 - 48*
b*c^4*d^4*e^4*n^2 - 8*b^3*c^2*d^2*e^6*n^2 - 3*b^4*c*d*e^7*n^2)*a^6 + (32*c^6*d^7*e*n^2 - 96*b^2*c^4*d^5*e^3*n^
2 + 16*b^3*c^3*d^4*e^4*n^2 + 30*b^4*c^2*d^3*e^5*n^2 + 20*b^5*c*d^2*e^6*n^2 + b^6*d*e^7*n^2)*a^5 + (32*b*c^6*d^
8*n^2 - 96*b^2*c^5*d^7*e*n^2 + 48*b^3*c^4*d^6*e^2*n^2 + 46*b^4*c^3*d^5*e^3*n^2 - 6*b^5*c^2*d^4*e^4*n^2 - 21*b^
6*c*d^3*e^5*n^2 - 3*b^7*d^2*e^6*n^2)*a^4 - (16*b^3*c^5*d^8*n^2 - 42*b^4*c^4*d^7*e*n^2 + 24*b^5*c^3*d^6*e^2*n^2
+ 11*b^6*c^2*d^5*e^3*n^2 - 6*b^7*c*d^4*e^4*n^2 - 3*b^8*d^3*e^5*n^2)*a^3 + (2*b^5*c^4*d^8*n^2 - 5*b^6*c^3*d^7*
e*n^2 + 3*b^7*c^2*d^6*e^2*n^2 + b^8*c*d^5*e^3*n^2 - b^9*d^4*e^4*n^2)*a^2)*x^(3*n) + (32*(c^3*d^2*e^6*n^2 + b*c
^2*d*e^7*n^2)*a^8 + 16*(6*c^4*d^4*e^4*n^2 - 6*b^2*c^2*d^2*e^6*n^2 - b^3*c*d*e^7*n^2)*a^7 + 2*(48*c^5*d^6*e^2*n
^2 - 48*b*c^4*d^5*e^3*n^2 - 48*b^2*c^3*d^4*e^4*n^2 + 24*b^3*c^2*d^3*e^5*n^2 + 21*b^4*c*d^2*e^6*n^2 + b^5*d*e^7
*n^2)*a^6 + (32*c^6*d^8*n^2 - 64*b*c^5*d^7*e*n^2 + 16*b^3*c^3*d^5*e^3*n^2 + 46*b^4*c^2*d^4*e^4*n^2 - 24*b^5*c*
d^3*e^5*n^2 - 5*b^6*d^2*e^6*n^2)*a^5 - (16*b^3*c^4*d^7*e*n^2 - 30*b^4*c^3*d^6*e^2*n^2 + 6*b^5*c^2*d^5*e^3*n^2
+ 11*b^6*c*d^4*e^4*n^2 - 3*b^7*d^3*e^5*n^2)*a^4 - (6*b^4*c^4*d^8*n^2 - 20*b^5*c^3*d^7*e*n^2 + 21*b^6*c^2*d^6*e
^2*n^2 - 6*b^7*c*d^5*e^3*n^2 - b^8*d^4*e^4*n^2)*a^3 + (b^6*c^3*d^8*n^2 - 3*b^7*c^2*d^7*e*n^2 + 3*b^8*c*d^6*e^2
*n^2 - b^9*d^5*e^3*n^2)*a^2)*x^(2*n) + (16*a^9*c^2*d*e^7*n^2 + 8*(6*c^3*d^3*e^5*n^2 - 2*b*c^2*d^2*e^6*n^2 - b^
2*c*d*e^7*n^2)*a^8 + (48*c^4*d^5*e^3*n^2 - 72*b^2*c^2*d^3*e^5*n^2 + 8*b^3*c*d^2*e^6*n^2 + b^4*d*e^7*n^2)*a^7 +
(16*c^5*d^7*e*n^2 + 48*b*c^4*d^6*e^2*n^2 - 168*b^2*c^3*d^5*e^3*n^2 + 80*b^3*c^2*d^4*e^4*n^2 + 27*b^4*c*d^3*e^
5*n^2 - b^5*d^2*e^6*n^2)*a^6 + (32*b*c^5*d^8*n^2 - 104*b^2*c^4*d^7*e*n^2 + 72*b^3*c^3*d^6*e^2*n^2 + 43*b^4*c^2
*d^5*e^3*n^2 - 40*b^5*c*d^4*e^4*n^2 - 3*b^6*d^3*e^5*n^2)*a^5 - (16*b^3*c^4*d^8*n^2 - 49*b^4*c^3*d^7*e*n^2 + 45
*b^5*c^2*d^6*e^2*n^2 - 7*b^6*c*d^5*e^3*n^2 - 5*b^7*d^4*e^4*n^2)*a^4 + 2*(b^5*c^3*d^8*n^2 - 3*b^6*c^2*d^7*e*n^2
+ 3*b^7*c*d^6*e^2*n^2 - b^8*d^5*e^3*n^2)*a^3)*x^n) + integrate(1/2*((2*n^2 - 3*n + 1)*b^4*c^4*d^6 - 4*(2*n^2
- 3*n + 1)*b^5*c^3*d^5*e + 6*(2*n^2 - 3*n + 1)*b^6*c^2*d^4*e^2 - 4*(2*n^2 - 3*n + 1)*b^7*c*d^3*e^3 + (2*n^2 -
3*n + 1)*b^8*d^2*e^4 - 4*(24*n^2 - 10*n + 1)*a^5*c^3*e^6 + (4*(48*n^2 - 2*n - 1)*c^4*d^2*e^4 - 4*(96*n^2 - 29*
n + 2)*b*c^3*d*e^5 + (240*n^2 - 115*n + 13)*b^2*c^2*e^6)*a^4 + (4*(32*n^2 - 18*n + 1)*c^5*d^4*e^2 - 8*(48*n^2
- 37*n + 4)*b*c^4*d^3*e^3 + (288*n^2 - 337*n + 49)*b^2*c^3*d^2*e^4 + 2*(32*n^2 + 29*n - 7)*b^3*c^2*d*e^5 - (10
2*n^2 - 55*n + 7)*b^4*c*e^6)*a^3 + (4*(8*n^2 - 6*n + 1)*c^6*d^6 - 4*(32*n^2 - 29*n + 6)*b*c^5*d^5*e + (128*n^2
- 137*n + 39)*b^2*c^4*d^4*e^2 + 8*(8*n^2 - 7*n - 1)*b^3*c^3*d^3*e^3 - 4*(37*n^2 - 43*n + 6)*b^4*c^2*d^2*e^4 +
4*(10*n^2 - 16*n + 3)*b^5*c*d*e^5 + (12*n^2 - 7*n + 1)*b^6*e^6)*a^2 - ((16*n^2 - 21*n + 5)*b^2*c^5*d^6 - 2*(3
2*n^2 - 43*n + 11)*b^3*c^4*d^5*e + 2*(44*n^2 - 61*n + 17)*b^4*c^3*d^4*e^2 - 20*(2*n^2 - 3*n + 1)*b^5*c^2*d^3*e
^3 - (8*n^2 - 7*n - 1)*b^6*c*d^2*e^4 + 2*(4*n^2 - 5*n + 1)*b^7*d*e^5)*a + ((2*n^2 - 3*n + 1)*b^3*c^5*d^6 - 4*(
2*n^2 - 3*n + 1)*b^4*c^4*d^5*e + 6*(2*n^2 - 3*n + 1)*b^5*c^3*d^4*e^2 - 4*(2*n^2 - 3*n + 1)*b^6*c^2*d^3*e^3 + (
2*n^2 - 3*n + 1)*b^7*c*d^2*e^4 - 2*(4*(35*n^2 - 12*n + 1)*c^4*d*e^5 - (81*n^2 - 37*n + 4)*b*c^3*e^6)*a^4 - 2*(
8*(7*n^2 - 8*n + 1)*c^5*d^3*e^3 - (83*n^2 - 97*n + 14)*b*c^4*d^2*e^4 - (44*n^2 + 7*n - 3)*b^2*c^3*d*e^5 + 3*(1
5*n^2 - 8*n + 1)*b^3*c^2*e^6)*a^3 - (8*(3*n^2 - 4*n + 1)*c^6*d^5*e - 2*(11*n^2 - 19*n + 8)*b*c^5*d^4*e^2 - 4*(
22*n^2 - 23*n + 1)*b^2*c^4*d^3*e^3 + (136*n^2 - 159*n + 23)*b^3*c^3*d^2*e^4 - 2*(16*n^2 - 27*n + 5)*b^4*c^2*d*
e^5 - (12*n^2 - 7*n + 1)*b^5*c*e^6)*a^2 - 2*((7*n^2 - 9*n + 2)*b*c^6*d^6 - (28*n^2 - 37*n + 9)*b^2*c^5*d^5*e +
2*(19*n^2 - 26*n + 7)*b^3*c^4*d^4*e^2 - 8*(2*n^2 - 3*n + 1)*b^4*c^3*d^3*e^3 - 5*(n^2 - n)*b^5*c^2*d^2*e^4 + (
4*n^2 - 5*n + 1)*b^6*c*d*e^5)*a)*x^n)/(16*a^9*c^2*e^8*n^2 + 8*(8*c^3*d^2*e^6*n^2 - 8*b*c^2*d*e^7*n^2 - b^2*c*e
^8*n^2)*a^8 + (96*c^4*d^4*e^4*n^2 - 192*b*c^3*d^3*e^5*n^2 + 64*b^2*c^2*d^2*e^6*n^2 + 32*b^3*c*d*e^7*n^2 + b^4*
e^8*n^2)*a^7 + 4*(16*c^5*d^6*e^2*n^2 - 48*b*c^4*d^5*e^3*n^2 + 36*b^2*c^3*d^4*e^4*n^2 + 8*b^3*c^2*d^3*e^5*n^2 -
11*b^4*c*d^2*e^6*n^2 - b^5*d*e^7*n^2)*a^6 + 2*(8*c^6*d^8*n^2 - 32*b*c^5*d^7*e*n^2 + 32*b^2*c^4*d^6*e^2*n^2 +
16*b^3*c^3*d^5*e^3*n^2 - 37*b^4*c^2*d^4*e^4*n^2 + 10*b^5*c*d^3*e^5*n^2 + 3*b^6*d^2*e^6*n^2)*a^5 - 4*(2*b^2*c^5
*d^8*n^2 - 8*b^3*c^4*d^7*e*n^2 + 11*b^4*c^3*d^6*e^2*n^2 - 5*b^5*c^2*d^5*e^3*n^2 - b^6*c*d^4*e^4*n^2 + b^7*d^3*
e^5*n^2)*a^4 + (b^4*c^4*d^8*n^2 - 4*b^5*c^3*d^7*e*n^2 + 6*b^6*c^2*d^6*e^2*n^2 - 4*b^7*c*d^5*e^3*n^2 + b^8*d^4*
e^4*n^2)*a^3 + (16*a^8*c^3*e^8*n^2 + 8*(8*c^4*d^2*e^6*n^2 - 8*b*c^3*d*e^7*n^2 - b^2*c^2*e^8*n^2)*a^7 + (96*c^5
*d^4*e^4*n^2 - 192*b*c^4*d^3*e^5*n^2 + 64*b^2*c^3*d^2*e^6*n^2 + 32*b^3*c^2*d*e^7*n^2 + b^4*c*e^8*n^2)*a^6 + 4*
(16*c^6*d^6*e^2*n^2 - 48*b*c^5*d^5*e^3*n^2 + 36*b^2*c^4*d^4*e^4*n^2 + 8*b^3*c^3*d^3*e^5*n^2 - 11*b^4*c^2*d^2*e
^6*n^2 - b^5*c*d*e^7*n^2)*a^5 + 2*(8*c^7*d^8*n^2 - 32*b*c^6*d^7*e*n^2 + 32*b^2*c^5*d^6*e^2*n^2 + 16*b^3*c^4*d^
5*e^3*n^2 - 37*b^4*c^3*d^4*e^4*n^2 + 10*b^5*c^2*d^3*e^5*n^2 + 3*b^6*c*d^2*e^6*n^2)*a^4 - 4*(2*b^2*c^6*d^8*n^2
- 8*b^3*c^5*d^7*e*n^2 + 11*b^4*c^4*d^6*e^2*n^2 - 5*b^5*c^3*d^5*e^3*n^2 - b^6*c^2*d^4*e^4*n^2 + b^7*c*d^3*e^5*n
^2)*a^3 + (b^4*c^5*d^8*n^2 - 4*b^5*c^4*d^7*e*n^2 + 6*b^6*c^3*d^6*e^2*n^2 - 4*b^7*c^2*d^5*e^3*n^2 + b^8*c*d^4*e
^4*n^2)*a^2)*x^(2*n) + (16*a^8*b*c^2*e^8*n^2 + 8*(8*b*c^3*d^2*e^6*n^2 - 8*b^2*c^2*d*e^7*n^2 - b^3*c*e^8*n^2)*a
^7 + (96*b*c^4*d^4*e^4*n^2 - 192*b^2*c^3*d^3*e^5*n^2 + 64*b^3*c^2*d^2*e^6*n^2 + 32*b^4*c*d*e^7*n^2 + b^5*e^8*n
^2)*a^6 + 4*(16*b*c^5*d^6*e^2*n^2 - 48*b^2*c^4*d^5*e^3*n^2 + 36*b^3*c^3*d^4*e^4*n^2 + 8*b^4*c^2*d^3*e^5*n^2 -
11*b^5*c*d^2*e^6*n^2 - b^6*d*e^7*n^2)*a^5 + 2*(8*b*c^6*d^8*n^2 - 32*b^2*c^5*d^7*e*n^2 + 32*b^3*c^4*d^6*e^2*n^2
+ 16*b^4*c^3*d^5*e^3*n^2 - 37*b^5*c^2*d^4*e^4*n^2 + 10*b^6*c*d^3*e^5*n^2 + 3*b^7*d^2*e^6*n^2)*a^4 - 4*(2*b^3*
c^5*d^8*n^2 - 8*b^4*c^4*d^7*e*n^2 + 11*b^5*c^3*d^6*e^2*n^2 - 5*b^6*c^2*d^5*e^3*n^2 - b^7*c*d^4*e^4*n^2 + b^8*d
^3*e^5*n^2)*a^3 + (b^5*c^4*d^8*n^2 - 4*b^6*c^3*d^7*e*n^2 + 6*b^7*c^2*d^6*e^2*n^2 - 4*b^8*c*d^5*e^3*n^2 + b^9*d
^4*e^4*n^2)*a^2)*x^n), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{3} e^{2} x^{5 \, n} + a^{3} d^{2} +{\left (c^{3} e^{2} x^{2 \, n} + 2 \, c^{3} d e x^{n} + c^{3} d^{2}\right )} x^{6 \, n} + 3 \,{\left (b c^{2} e^{2} x^{3 \, n} + a c^{2} d^{2} +{\left (2 \, b c^{2} d e + a c^{2} e^{2}\right )} x^{2 \, n} +{\left (b c^{2} d^{2} + 2 \, a c^{2} d e\right )} x^{n}\right )} x^{4 \, n} +{\left (2 \, b^{3} d e + 3 \, a b^{2} e^{2}\right )} x^{4 \, n} +{\left (b^{3} d^{2} + 6 \, a b^{2} d e + 3 \, a^{2} b e^{2}\right )} x^{3 \, n} + 3 \,{\left (b^{2} c e^{2} x^{4 \, n} + a^{2} c d^{2} + 2 \,{\left (b^{2} c d e + a b c e^{2}\right )} x^{3 \, n} +{\left (b^{2} c d^{2} + 4 \, a b c d e + a^{2} c e^{2}\right )} x^{2 \, n} + 2 \,{\left (a b c d^{2} + a^{2} c d e\right )} x^{n}\right )} x^{2 \, n} +{\left (3 \, a b^{2} d^{2} + 6 \, a^{2} b d e + a^{3} e^{2}\right )} x^{2 \, n} +{\left (3 \, a^{2} b d^{2} + 2 \, a^{3} d 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)^2/(a+b*x^n+c*x^(2*n))^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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 )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+e*x^n)^2/(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)^2), x)