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

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 + 1
3*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*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
- 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*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hyp
ergeometric2F1[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*Hyp
ergeometric2F1[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*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 +
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*Hyper
geometric2F1[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)

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Rubi [A]  time = 24.8268, 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 \[ \text{result too large to display} \]

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 + 1
3*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*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
- 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*Sqrt[b^2 - 4*a*c]*d + a*e))*x*Hyp
ergeometric2F1[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*Hyp
ergeometric2F1[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*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 +
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*Hyper
geometric2F1[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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x^{n}\right )^{2} \left (a + b x^{n} + c x^{2 n}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 9.88568, size = 56566, normalized size = 23.13 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

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Maple [F]  time = 0.608, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)^3*(e*x^n + d)^2),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) + 6
4*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 - 4
8*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) + integr
ate(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 - (102*n^2 - 55*n + 7)*b^4*c*e^6)*a^3 + (4*(8*n^2 - 6*n + 1)*c^6*d^6 - 4*(3
2*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*(32*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*(15*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 + (13
6*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + b*x^n + a)^3*(e*x^n + d)^2),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) + 3*(b^2*c + a*c^2)*d^2 + 2*(b^3 + 6*a*b*c)*d
*e + 3*(a*b^2 + a^2*c)*e^2 + 3*(2*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*x^(2*n) + 3*(
b*c^2*d^2 + 2*a*c^2*d*e)*x^n)*x^(4*n) + (b^3*d^2 + 6*a*b^2*d*e + 3*a^2*b*e^2 + 6
*(b^2*c*d*e + a*b*c*e^2)*x^(2*n))*x^(3*n) + (6*a^2*b*d*e + a^3*e^2 + 3*(a*b^2 +
a^2*c)*d^2 + 6*(a*b*c*d^2 + a^2*c*d*e)*x^n)*x^(2*n) + (3*a^2*b*d^2 + 2*a^3*d*e)*
x^n), x)

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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