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

Optimal. Leaf size=552 \[ \frac{e^2 x \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^3}+\frac{e^2 x (2 c d-b e) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^3 \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((c*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 -
4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(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)^3))
- (c*(2*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d - Sqrt[b^2 -
4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*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)
+ (e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*x*Hypergeometric2F1[1, n^(-1),
1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3) + (e^2*(2*c*d - b*e)*x
*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a
*e^2)^2) + (e^2*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^3*(
c*d^2 - b*d*e + a*e^2))

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Rubi [A]  time = 2.35493, antiderivative size = 552, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{e^2 x \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (-3 c^2 d e \left (d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (3 b \left (d \sqrt{b^2-4 a c}+a e\right )+a e \sqrt{b^2-4 a c}+3 b^2 d\right )-b^2 e^3 \left (\sqrt{b^2-4 a c}+b\right )+2 c^3 d^3\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (-3 c^2 d e \left (-d \sqrt{b^2-4 a c}+2 a e+b d\right )+c e^2 \left (-3 b d \sqrt{b^2-4 a c}-a e \sqrt{b^2-4 a c}+3 a b e+3 b^2 d\right )-b^2 e^3 \left (b-\sqrt{b^2-4 a c}\right )+2 c^3 d^3\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^3}+\frac{e^2 x (2 c d-b e) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^3 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((c*(2*c^3*d^3 - b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d + Sqrt[b^2 -
4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d + a*Sqrt[b^2 - 4*a*c]*e + 3*b*(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)^3))
- (c*(2*c^3*d^3 - b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 3*c^2*d*e*(b*d - Sqrt[b^2 -
4*a*c]*d + 2*a*e) + c*e^2*(3*b^2*d - 3*b*Sqrt[b^2 - 4*a*c]*d + 3*a*b*e - a*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)
+ (e^2*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*x*Hypergeometric2F1[1, n^(-1),
1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2)^3) + (e^2*(2*c*d - b*e)*x
*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 - b*d*e + a
*e^2)^2) + (e^2*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^3*(
c*d^2 - b*d*e + a*e^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 6.47417, size = 4111, normalized size = 7.45 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

((-(a*c*d^2*e^2) + a*b*d*e^3 - a^2*e^4 + 7*a*c*d^2*e^2*n - 5*a*b*d*e^3*n + 3*a^2
*e^4*n - 2*c^2*d^4*n^2 + 4*b*c*d^3*e*n^2 - 2*b^2*d^2*e^2*n^2 - 4*a*c*d^2*e^2*n^2
+ 4*a*b*d*e^3*n^2 - 2*a^2*e^4*n^2)*x)/(2*a*d^3*(c*d^2 - b*d*e + a*e^2)^2*n^2) +
((a*c*d^2*e^2 - a*b*d*e^3 + a^2*e^4 - 7*a*c*d^2*e^2*n + 5*a*b*d*e^3*n - 3*a^2*e
^4*n + 2*c^2*d^4*n^2 - 4*b*c*d^3*e*n^2 + 2*b^2*d^2*e^2*n^2 + 4*a*c*d^2*e^2*n^2 -
4*a*b*d*e^3*n^2 + 2*a^2*e^4*n^2)*x)/(2*a*d^3*(c*d^2 - b*d*e + a*e^2)^2*n^2) + (
e^2*x)/(2*d*(c*d^2 - b*d*e + a*e^2)*n*(d + e*x^n)^2) + ((-(c*d^2*e^2) + b*d*e^3
- a*e^4 + 6*c*d^2*e^2*n - 4*b*d*e^3*n + 2*a*e^4*n)*x)/(2*d^2*(c*d^2 - b*d*e + a*
e^2)^2*n^2*(d + e*x^n)) + ((c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4 + 2*a*c*d^
2*e^4 - 2*a*b*d*e^5 + a^2*e^6 - 7*c^2*d^4*e^2*n + 12*b*c*d^3*e^3*n - 5*b^2*d^2*e
^4*n - 10*a*c*d^2*e^4*n + 8*a*b*d*e^5*n - 3*a^2*e^6*n + 12*c^2*d^4*e^2*n^2 - 16*
b*c*d^3*e^3*n^2 + 6*b^2*d^2*e^4*n^2 + 6*a*c*d^2*e^4*n^2 - 6*a*b*d*e^5*n^2 + 2*a^
2*e^6*n^2)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(2*d^3*(c*d
^2 - b*d*e + a*e^2)^3*n^2) - (3*c^3*d^2*e*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(
-(Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c
*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[
b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 +
n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/
(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(c*d^
2 - b*d*e + a*e^2)^3 + (3*b*c^2*d*e^2*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hy
pergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(
-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2
- 4*a*c])/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/
n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqr
t[b^2 - 4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(c*d^2 -
b*d*e + a*e^2)^3 - (b^2*c*e^3*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeome
tric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqr
t[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4*a*c]
)/(2*c) + x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b
+ Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 -
4*a*c]*(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(c*d^2 - b*d*e +
a*e^2)^3 + (a*c^2*e^3*x^(1 + n)*(x^n)^(n^(-1) - (1 + n)/n)*(-(Hypergeometric2F1[
-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 -
4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c)
+ x^n))^n^(-1))) + Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b
^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(Sqrt[b^2 - 4*a*c]*(
x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))))/(c*d^2 - b*d*e + a*e^2)^3
- (c^3*d^3*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[
b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b
^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - S
qrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n
, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/
(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(
2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(c*d^2 - b*d*e + a*e^2)^3 + (3*b*c^2*
d^2*e*x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 -
4*a*c])/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4
*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^
2 - 4*a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b
+ Sqrt[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b
+ Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) +
(-b + Sqrt[b^2 - 4*a*c])^2/(2*c))))/(c*d^2 - b*d*e + a*e^2)^3 - (3*b^2*c*d*e^2*
x*((1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c]
)/(2*c*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])
/(2*c) + x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*
a*c])^2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqr
t[b^2 - 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt
[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b +
Sqrt[b^2 - 4*a*c])^2/(2*c))))/(c*d^2 - b*d*e + a*e^2)^3 + (3*a*c^2*d*e^2*x*((1
- Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c
*(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c)
+ x^n))^n^(-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^
2/(2*c)) + (1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2
- 4*a*c])/(2*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 -
4*a*c])/(2*c) + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[
b^2 - 4*a*c])^2/(2*c))))/(c*d^2 - b*d*e + a*e^2)^3 + (b^3*e^3*x*((1 - Hypergeome
tric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqr
t[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(
-1))/((b*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (
1 - Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2
*c*(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*
c) + x^n))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c]
)^2/(2*c))))/(c*d^2 - b*d*e + a*e^2)^3 - (2*a*b*c*e^3*x*((1 - Hypergeometric2F1[
-n^(-1), -n^(-1), (-1 + n)/n, -(-b - Sqrt[b^2 - 4*a*c])/(2*c*(-(-b - Sqrt[b^2 -
4*a*c])/(2*c) + x^n))]/(x^n/(-(-b - Sqrt[b^2 - 4*a*c])/(2*c) + x^n))^n^(-1))/((b
*(-b - Sqrt[b^2 - 4*a*c]))/(2*c) + (-b - Sqrt[b^2 - 4*a*c])^2/(2*c)) + (1 - Hype
rgeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, -(-b + Sqrt[b^2 - 4*a*c])/(2*c*(-(-b
+ Sqrt[b^2 - 4*a*c])/(2*c) + x^n))]/(x^n/(-(-b + Sqrt[b^2 - 4*a*c])/(2*c) + x^n
))^n^(-1))/((b*(-b + Sqrt[b^2 - 4*a*c]))/(2*c) + (-b + Sqrt[b^2 - 4*a*c])^2/(2*c
))))/(c*d^2 - b*d*e + a*e^2)^3

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Maple [F]  time = 0.234, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{3} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

((12*n^2 - 7*n + 1)*c^2*d^4*e^2 - 2*(8*n^2 - 6*n + 1)*b*c*d^3*e^3 + (6*n^2 - 5*n
+ 1)*b^2*d^2*e^4 + (2*n^2 - 3*n + 1)*a^2*e^6 + 2*((3*n^2 - 5*n + 1)*c*d^2*e^4 -
(3*n^2 - 4*n + 1)*b*d*e^5)*a)*integrate(1/2/(c^3*d^9*n^2 - 3*b*c^2*d^8*e*n^2 +
3*b^2*c*d^7*e^2*n^2 - b^3*d^6*e^3*n^2 + a^3*d^3*e^6*n^2 + 3*(c*d^5*e^4*n^2 - b*d
^4*e^5*n^2)*a^2 + 3*(c^2*d^7*e^2*n^2 - 2*b*c*d^6*e^3*n^2 + b^2*d^5*e^4*n^2)*a +
(c^3*d^8*e*n^2 - 3*b*c^2*d^7*e^2*n^2 + 3*b^2*c*d^6*e^3*n^2 - b^3*d^5*e^4*n^2 + a
^3*d^2*e^7*n^2 + 3*(c*d^4*e^5*n^2 - b*d^3*e^6*n^2)*a^2 + 3*(c^2*d^6*e^3*n^2 - 2*
b*c*d^5*e^4*n^2 + b^2*d^4*e^5*n^2)*a)*x^n), x) + 1/2*((c*d^2*e^3*(6*n - 1) - b*d
*e^4*(4*n - 1) + a*e^5*(2*n - 1))*x*x^n + (c*d^3*e^2*(7*n - 1) - b*d^2*e^3*(5*n
- 1) + a*d*e^4*(3*n - 1))*x)/(c^2*d^8*n^2 - 2*b*c*d^7*e*n^2 + b^2*d^6*e^2*n^2 +
a^2*d^4*e^4*n^2 + 2*(c*d^6*e^2*n^2 - b*d^5*e^3*n^2)*a + (c^2*d^6*e^2*n^2 - 2*b*c
*d^5*e^3*n^2 + b^2*d^4*e^4*n^2 + a^2*d^2*e^6*n^2 + 2*(c*d^4*e^4*n^2 - b*d^3*e^5*
n^2)*a)*x^(2*n) + 2*(c^2*d^7*e*n^2 - 2*b*c*d^6*e^2*n^2 + b^2*d^5*e^3*n^2 + a^2*d
^3*e^5*n^2 + 2*(c*d^5*e^3*n^2 - b*d^4*e^4*n^2)*a)*x^n) + integrate((c^3*d^3 - 3*
b*c^2*d^2*e + 3*b^2*c*d*e^2 - b^3*e^3 - (3*c^2*d*e^2 - 2*b*c*e^3)*a - (3*c^3*d^2
*e - 3*b*c^2*d*e^2 + b^2*c*e^3 - a*c^2*e^3)*x^n)/(a^4*e^6 + 3*(c*d^2*e^4 - b*d*e
^5)*a^3 + 3*(c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4)*a^2 + (c^3*d^6 - 3*b*c^2
*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*a + (c^4*d^6 - 3*b*c^3*d^5*e + 3*b^2*c^2
*d^4*e^2 - b^3*c*d^3*e^3 + a^3*c*e^6 + 3*(c^2*d^2*e^4 - b*c*d*e^5)*a^2 + 3*(c^3*
d^4*e^2 - 2*b*c^2*d^3*e^3 + b^2*c*d^2*e^4)*a)*x^(2*n) + (b*c^3*d^6 - 3*b^2*c^2*d
^5*e + 3*b^3*c*d^4*e^2 - b^4*d^3*e^3 + a^3*b*e^6 + 3*(b*c*d^2*e^4 - b^2*d*e^5)*a
^2 + 3*(b*c^2*d^4*e^2 - 2*b^2*c*d^3*e^3 + b^3*d^2*e^4)*a)*x^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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