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

Optimal. Leaf size=701 \[ -\frac{c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{4 a^3 n^2 (n+1) \left (a e^2+c d^2\right )^2}+\frac{c (1-4 n) (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )^2}+\frac{2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^3}-\frac{c e^2 (1-2 n) x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^3}-\frac{c x \left ((1-4 n) \left (c d^2-a e^2\right )-2 c d e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}-\frac{6 c^2 d e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^4}+\frac{c e^2 x \left (-a e^2+3 c d^2-4 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^3 \left (a+c x^{2 n}\right )}+\frac{c x \left (-a e^2+c d^2-2 c d e x^n\right )}{4 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )^2}+\frac{6 c e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^3}+\frac{c e^4 x \left (5 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^4} \]

[Out]

(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(4*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))^2)
+ (c*e^2*x*(3*c*d^2 - a*e^2 - 4*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^3*n*(a + c*x^(2
*n))) - (c*x*((c*d^2 - a*e^2)*(1 - 4*n) - 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*(c*d^2
+ a*e^2)^2*n^2*(a + c*x^(2*n))) + (c*e^4*(5*c*d^2 - a*e^2)*x*Hypergeometric2F1[1
, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4) + (c*(c*d^2
- a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((
c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)^2*n^2) - (c*e^2*(3*c*d^2 - a*e^2)*(1 - 2*
n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*
d^2 + a*e^2)^3*n) + (6*c*e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n
)/d)])/(c*d^2 + a*e^2)^4 - (6*c^2*d*e^5*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(
2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4*(1 + n)) - (c^2*d*
e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2
, -((c*x^(2*n))/a)])/(4*a^3*(c*d^2 + a*e^2)^2*n^2*(1 + n)) + (2*c^2*d*e^3*(1 - n
)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)
])/(a^2*(c*d^2 + a*e^2)^3*n*(1 + n)) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n
^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^3)

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Rubi [A]  time = 1.49197, antiderivative size = 701, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{c^2 d e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{4 a^3 n^2 (n+1) \left (a e^2+c d^2\right )^2}+\frac{c (1-4 n) (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )^2}+\frac{2 c^2 d e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^3}-\frac{c e^2 (1-2 n) x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^3}-\frac{c x \left ((1-4 n) \left (c d^2-a e^2\right )-2 c d e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}-\frac{6 c^2 d e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^4}+\frac{c e^2 x \left (-a e^2+3 c d^2-4 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^3 \left (a+c x^{2 n}\right )}+\frac{c x \left (-a e^2+c d^2-2 c d e x^n\right )}{4 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )^2}+\frac{6 c e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^4}+\frac{e^6 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^3}+\frac{c e^4 x \left (5 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(4*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))^2)
+ (c*e^2*x*(3*c*d^2 - a*e^2 - 4*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^3*n*(a + c*x^(2
*n))) - (c*x*((c*d^2 - a*e^2)*(1 - 4*n) - 2*c*d*e*(1 - 3*n)*x^n))/(8*a^2*(c*d^2
+ a*e^2)^2*n^2*(a + c*x^(2*n))) + (c*e^4*(5*c*d^2 - a*e^2)*x*Hypergeometric2F1[1
, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4) + (c*(c*d^2
- a*e^2)*(1 - 4*n)*(1 - 2*n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((
c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e^2)^2*n^2) - (c*e^2*(3*c*d^2 - a*e^2)*(1 - 2*
n)*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*
d^2 + a*e^2)^3*n) + (6*c*e^6*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n
)/d)])/(c*d^2 + a*e^2)^4 - (6*c^2*d*e^5*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(
2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^4*(1 + n)) - (c^2*d*
e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2
, -((c*x^(2*n))/a)])/(4*a^3*(c*d^2 + a*e^2)^2*n^2*(1 + n)) + (2*c^2*d*e^3*(1 - n
)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)
])/(a^2*(c*d^2 + a*e^2)^3*n*(1 + n)) + (e^6*x*Hypergeometric2F1[2, n^(-1), 1 + n
^(-1), -((e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.60612, size = 1241, normalized size = 1.77 \[ \frac{x \left (-\frac{70 c^2 d e^5 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a (n+1)}-\frac{28 c^3 d^3 e^3 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^2 (n+1)}-\frac{6 c^4 d^5 e \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^3 (n+1)}+\frac{24 c^2 d e^5 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a n (n+1)}+\frac{32 c^3 d^3 e^3 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^2 n (n+1)}+\frac{8 c^4 d^5 e \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^3 n (n+1)}-\frac{2 c^2 d e^5 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a n^2 (n+1)}-\frac{4 c^3 d^3 e^3 \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^2 n^2 (n+1)}-\frac{2 c^4 d^5 e \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right ) x^n}{a^3 n^2 (n+1)}+\frac{2 c \left (c d^2+a e^2\right )^2 \left (c d \left (d-2 e x^n\right )-a e^2\right )}{a n \left (c x^{2 n}+a\right )^2}+\frac{c \left (c d^2+a e^2\right ) \left (a^2 (1-8 n) e^4+2 a c d \left (6 d n-e (11 n-1) x^n\right ) e^2+c^2 d^3 \left (d (4 n-1)-2 e (3 n-1) x^n\right )\right )}{a^2 n^2 \left (c x^{2 n}+a\right )}+\frac{8 c^4 d^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^3}-24 c e^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+\frac{48 c^2 d^2 e^4 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a}+\frac{32 c^3 d^4 e^2 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^2}-\frac{6 c^4 d^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^3 n}+\frac{10 c e^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{n}-\frac{2 c^2 d^2 e^4 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a n}-\frac{18 c^3 d^4 e^2 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^2 n}+\frac{c^4 d^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^3 n^2}-\frac{c e^6 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{n^2}-\frac{c^2 d^2 e^4 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a n^2}+\frac{c^3 d^4 e^2 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^2 n^2}+\frac{8 e^6 \left (c (7 n-1) d^2+a e^2 (n-1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 n}+\frac{8 a e^8+8 c d^2 e^6}{d e n x^n+d^2 n}\right )}{8 \left (c d^2+a e^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

(x*((8*c*d^2*e^6 + 8*a*e^8)/(d^2*n + d*e*n*x^n) + (2*c*(c*d^2 + a*e^2)^2*(-(a*e^
2) + c*d*(d - 2*e*x^n)))/(a*n*(a + c*x^(2*n))^2) + (c*(c*d^2 + a*e^2)*(a^2*e^4*(
1 - 8*n) + c^2*d^3*(d*(-1 + 4*n) - 2*e*(-1 + 3*n)*x^n) + 2*a*c*d*e^2*(6*d*n - e*
(-1 + 11*n)*x^n)))/(a^2*n^2*(a + c*x^(2*n))) + (8*c^4*d^6*Hypergeometric2F1[1, 1
/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a^3 + (32*c^3*d^4*e^2*Hypergeometric2F1[
1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a^2 + (48*c^2*d^2*e^4*Hypergeometric
2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a - 24*c*e^6*Hypergeometric2F1[1
, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)] + (c^4*d^6*Hypergeometric2F1[1, 1/(2*n
), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^3*n^2) + (c^3*d^4*e^2*Hypergeometric2F1[1,
1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*n^2) - (c^2*d^2*e^4*Hypergeometri
c2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n^2) - (c*e^6*Hypergeometric
2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/n^2 - (6*c^4*d^6*Hypergeometric2
F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^3*n) - (18*c^3*d^4*e^2*Hyperge
ometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*n) - (2*c^2*d^2*e^4*
Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n) + (10*c*e^6*
Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/n + (8*e^6*(a*e^2*
(-1 + n) + c*d^2*(-1 + 7*n))*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/
d)])/(d^2*n) - (6*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))
/2, -((c*x^(2*n))/a)])/(a^3*(1 + n)) - (28*c^3*d^3*e^3*x^n*Hypergeometric2F1[1,
(1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n)) - (70*c^2*d*e^5*
x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(1
+ n)) - (2*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -(
(c*x^(2*n))/a)])/(a^3*n^2*(1 + n)) - (4*c^3*d^3*e^3*x^n*Hypergeometric2F1[1, (1
+ n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*n^2*(1 + n)) - (2*c^2*d*e^5*
x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*n^
2*(1 + n)) + (8*c^4*d^5*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2
, -((c*x^(2*n))/a)])/(a^3*n*(1 + n)) + (32*c^3*d^3*e^3*x^n*Hypergeometric2F1[1,
(1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*n*(1 + n)) + (24*c^2*d*e^
5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*
n*(1 + n))))/(8*(c*d^2 + a*e^2)^4)

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Maple [F]  time = 0.274, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

(c*d^2*e^6*(7*n - 1) + a*e^8*(n - 1))*integrate(1/(c^4*d^10*n + 4*a*c^3*d^8*e^2*
n + 6*a^2*c^2*d^6*e^4*n + 4*a^3*c*d^4*e^6*n + a^4*d^2*e^8*n + (c^4*d^9*e*n + 4*a
*c^3*d^7*e^3*n + 6*a^2*c^2*d^5*e^5*n + 4*a^3*c*d^3*e^7*n + a^4*d*e^9*n)*x^n), x)
- 1/8*(2*(a*c^3*d^2*e^4*(11*n - 1) + c^4*d^4*e^2*(3*n - 1) - 4*a^2*c^2*e^6*n)*x
*x^(4*n) + (a^2*c^2*d*e^5*(8*n - 1) + 2*a*c^3*d^3*e^3*(5*n - 1) + c^4*d^5*e*(2*n
- 1))*x*x^(3*n) + (a^2*c^2*d^2*e^4*(34*n - 3) - c^4*d^6*(4*n - 1) - 2*a*c^3*d^4
*e^2*(n + 1) - 16*a^3*c*e^6*n)*x*x^(2*n) + (a^3*c*d*e^5*(10*n - 1) + 2*a^2*c^2*d
^3*e^3*(7*n - 1) + a*c^3*d^5*e*(4*n - 1))*x*x^n + (a^3*c*d^2*e^4*(10*n - 1) - a*
c^3*d^6*(6*n - 1) - 12*a^2*c^2*d^4*e^2*n - 8*a^4*e^6*n)*x)/(a^4*c^3*d^8*n^2 + 3*
a^5*c^2*d^6*e^2*n^2 + 3*a^6*c*d^4*e^4*n^2 + a^7*d^2*e^6*n^2 + (a^2*c^5*d^7*e*n^2
+ 3*a^3*c^4*d^5*e^3*n^2 + 3*a^4*c^3*d^3*e^5*n^2 + a^5*c^2*d*e^7*n^2)*x^(5*n) +
(a^2*c^5*d^8*n^2 + 3*a^3*c^4*d^6*e^2*n^2 + 3*a^4*c^3*d^4*e^4*n^2 + a^5*c^2*d^2*e
^6*n^2)*x^(4*n) + 2*(a^3*c^4*d^7*e*n^2 + 3*a^4*c^3*d^5*e^3*n^2 + 3*a^5*c^2*d^3*e
^5*n^2 + a^6*c*d*e^7*n^2)*x^(3*n) + 2*(a^3*c^4*d^8*n^2 + 3*a^4*c^3*d^6*e^2*n^2 +
3*a^5*c^2*d^4*e^4*n^2 + a^6*c*d^2*e^6*n^2)*x^(2*n) + (a^4*c^3*d^7*e*n^2 + 3*a^5
*c^2*d^5*e^3*n^2 + 3*a^6*c*d^3*e^5*n^2 + a^7*d*e^7*n^2)*x^n) - integrate(-1/8*((
8*n^2 - 6*n + 1)*c^4*d^6 + (32*n^2 - 18*n + 1)*a*c^3*d^4*e^2 + (48*n^2 - 2*n - 1
)*a^2*c^2*d^2*e^4 - (24*n^2 - 10*n + 1)*a^3*c*e^6 - 2*((3*n^2 - 4*n + 1)*c^4*d^5
*e + 2*(7*n^2 - 8*n + 1)*a*c^3*d^3*e^3 + (35*n^2 - 12*n + 1)*a^2*c^2*d*e^5)*x^n)
/(a^3*c^4*d^8*n^2 + 4*a^4*c^3*d^6*e^2*n^2 + 6*a^5*c^2*d^4*e^4*n^2 + 4*a^6*c*d^2*
e^6*n^2 + a^7*e^8*n^2 + (a^2*c^5*d^8*n^2 + 4*a^3*c^4*d^6*e^2*n^2 + 6*a^4*c^3*d^4
*e^4*n^2 + 4*a^5*c^2*d^2*e^6*n^2 + a^6*c*e^8*n^2)*x^(2*n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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