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

Optimal. Leaf size=582 \[ -\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1) \left (a e^2+c d^2\right )}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )}-\frac{c d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{c x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1) \left (a e^2+c d^2\right )^2}+\frac{c e^2 x \left (d-e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{c x \left (d-e x^n\right )}{4 a n \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )^2}+\frac{e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )^3}-\frac{c e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c d e^4 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3} \]

[Out]

(c*x*(d - e*x^n))/(4*a*(c*d^2 + a*e^2)*n*(a + c*x^(2*n))^2) + (c*e^2*x*(d - e*x^
n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) - (c*x*(d*(1 - 4*n) - e*(1 - 3*n)*
x^n))/(8*a^2*(c*d^2 + a*e^2)*n^2*(a + c*x^(2*n))) + (c*d*e^4*x*Hypergeometric2F1
[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3) + (c*d*(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)*n^2) - (c*d*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)^2*n) + (e^6*x*Hyp
ergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^3) - (c*
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)^3*(1 + n)) - (c*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeome
tric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e
^2)*n^2*(1 + n)) + (c*e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n),
(3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n*(1 + n))

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Rubi [A]  time = 0.941329, antiderivative size = 582, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{c e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1) \left (a e^2+c d^2\right )}+\frac{c d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{8 a^3 n^2 \left (a e^2+c d^2\right )}-\frac{c d e^2 (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{c x \left (d (1-4 n)-e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )}+\frac{c e^3 (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1) \left (a e^2+c d^2\right )^2}+\frac{c e^2 x \left (d-e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{c x \left (d-e x^n\right )}{4 a n \left (a e^2+c d^2\right ) \left (a+c x^{2 n}\right )^2}+\frac{e^6 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )^3}-\frac{c e^5 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c d e^4 x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(d - e*x^n))/(4*a*(c*d^2 + a*e^2)*n*(a + c*x^(2*n))^2) + (c*e^2*x*(d - e*x^
n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) - (c*x*(d*(1 - 4*n) - e*(1 - 3*n)*
x^n))/(8*a^2*(c*d^2 + a*e^2)*n^2*(a + c*x^(2*n))) + (c*d*e^4*x*Hypergeometric2F1
[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)^3) + (c*d*(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)*n^2) - (c*d*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)^2*n) + (e^6*x*Hyp
ergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*(c*d^2 + a*e^2)^3) - (c*
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)^3*(1 + n)) - (c*e*(1 - 3*n)*(1 - n)*x^(1 + n)*Hypergeome
tric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(8*a^3*(c*d^2 + a*e
^2)*n^2*(1 + n)) + (c*e^3*(1 - n)*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n),
(3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n*(1 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 2.25215, size = 1031, normalized size = 1.77 \[ \frac{x \left (-\frac{15 c 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{10 c^2 d^2 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{3 c^3 d^4 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{8 c 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{12 c^2 d^2 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{4 c^3 d^4 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{c 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{2 c^2 d^2 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{c^3 d^4 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{c \left (c d^2+a e^2\right ) \left (c \left (d (4 n-1)-e (3 n-1) x^n\right ) d^2+a e^2 \left (d (8 n-1)-e (7 n-1) x^n\right )\right )}{a^2 n^2 \left (c x^{2 n}+a\right )}+\frac{8 c^3 d^5 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^3}+\frac{24 c d e^4 \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a}+\frac{24 c^2 d^3 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^3 d^5 \, _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 d 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{16 c^2 d^3 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^3 d^5 \, _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 d 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{2 c^2 d^3 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 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d}+\frac{2 c \left (c d^2+a e^2\right )^2 \left (d-e x^n\right )}{a n \left (c x^{2 n}+a\right )^2}\right )}{8 \left (c d^2+a e^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*((2*c*(c*d^2 + a*e^2)^2*(d - e*x^n))/(a*n*(a + c*x^(2*n))^2) + (c*(c*d^2 + a*
e^2)*(c*d^2*(d*(-1 + 4*n) - e*(-1 + 3*n)*x^n) + a*e^2*(d*(-1 + 8*n) - e*(-1 + 7*
n)*x^n)))/(a^2*n^2*(a + c*x^(2*n))) + (8*c^3*d^5*Hypergeometric2F1[1, 1/(2*n), 1
+ 1/(2*n), -((c*x^(2*n))/a)])/a^3 + (24*c^2*d^3*e^2*Hypergeometric2F1[1, 1/(2*n
), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a^2 + (24*c*d*e^4*Hypergeometric2F1[1, 1/(2*n
), 1 + 1/(2*n), -((c*x^(2*n))/a)])/a + (c^3*d^5*Hypergeometric2F1[1, 1/(2*n), 1
+ 1/(2*n), -((c*x^(2*n))/a)])/(a^3*n^2) + (2*c^2*d^3*e^2*Hypergeometric2F1[1, 1/
(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*n^2) + (c*d*e^4*Hypergeometric2F1[1,
1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n^2) - (6*c^3*d^5*Hypergeometric2F1
[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^3*n) - (16*c^2*d^3*e^2*Hypergeom
etric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a^2*n) - (10*c*d*e^4*Hyper
geometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)])/(a*n) + (8*e^6*Hypergeo
metric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/d - (3*c^3*d^4*e*x^n*Hypergeomet
ric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*(1 + n)) - (10*
c^2*d^2*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n)
)/a)])/(a^2*(1 + n)) - (15*c*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^
(-1))/2, -((c*x^(2*n))/a)])/(a*(1 + n)) - (c^3*d^4*e*x^n*Hypergeometric2F1[1, (1
+ n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*n^2*(1 + n)) - (2*c^2*d^2*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)) - (c*e^5*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2,
-((c*x^(2*n))/a)])/(a*n^2*(1 + n)) + (4*c^3*d^4*e*x^n*Hypergeometric2F1[1, (1 +
n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^3*n*(1 + n)) + (12*c^2*d^2*e^3*
x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*
n*(1 + n)) + (8*c*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)^3)

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Maple [F]  time = 0.242, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) \left ( a+c{x}^{2\,n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ e^{6} \int \frac{1}{c^{3} d^{7} + 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} + a^{3} d e^{6} +{\left (c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}\right )} x^{n}}\,{d x} - \frac{{\left (a c^{2} e^{3}{\left (7 \, n - 1\right )} + c^{3} d^{2} e{\left (3 \, n - 1\right )}\right )} x x^{3 \, n} -{\left (a c^{2} d e^{2}{\left (8 \, n - 1\right )} + c^{3} d^{3}{\left (4 \, n - 1\right )}\right )} x x^{2 \, n} +{\left (a^{2} c e^{3}{\left (9 \, n - 1\right )} + a c^{2} d^{2} e{\left (5 \, n - 1\right )}\right )} x x^{n} -{\left (a^{2} c d e^{2}{\left (10 \, n - 1\right )} + a c^{2} d^{3}{\left (6 \, n - 1\right )}\right )} x}{8 \,{\left (a^{4} c^{2} d^{4} n^{2} + 2 \, a^{5} c d^{2} e^{2} n^{2} + a^{6} e^{4} n^{2} +{\left (a^{2} c^{4} d^{4} n^{2} + 2 \, a^{3} c^{3} d^{2} e^{2} n^{2} + a^{4} c^{2} e^{4} n^{2}\right )} x^{4 \, n} + 2 \,{\left (a^{3} c^{3} d^{4} n^{2} + 2 \, a^{4} c^{2} d^{2} e^{2} n^{2} + a^{5} c e^{4} n^{2}\right )} x^{2 \, n}\right )}} - \int -\frac{{\left (8 \, n^{2} - 6 \, n + 1\right )} c^{3} d^{5} + 2 \,{\left (12 \, n^{2} - 8 \, n + 1\right )} a c^{2} d^{3} e^{2} +{\left (24 \, n^{2} - 10 \, n + 1\right )} a^{2} c d e^{4} -{\left ({\left (3 \, n^{2} - 4 \, n + 1\right )} c^{3} d^{4} e + 2 \,{\left (5 \, n^{2} - 6 \, n + 1\right )} a c^{2} d^{2} e^{3} +{\left (15 \, n^{2} - 8 \, n + 1\right )} a^{2} c e^{5}\right )} x^{n}}{8 \,{\left (a^{3} c^{3} d^{6} n^{2} + 3 \, a^{4} c^{2} d^{4} e^{2} n^{2} + 3 \, a^{5} c d^{2} e^{4} n^{2} + a^{6} e^{6} n^{2} +{\left (a^{2} c^{4} d^{6} n^{2} + 3 \, a^{3} c^{3} d^{4} e^{2} n^{2} + 3 \, a^{4} c^{2} d^{2} e^{4} n^{2} + a^{5} c e^{6} n^{2}\right )} x^{2 \, n}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^6*integrate(1/(c^3*d^7 + 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^4 + a^3*d*e^6 + (c^3*
d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7)*x^n), x) - 1/8*((a*c^2*e^3*
(7*n - 1) + c^3*d^2*e*(3*n - 1))*x*x^(3*n) - (a*c^2*d*e^2*(8*n - 1) + c^3*d^3*(4
*n - 1))*x*x^(2*n) + (a^2*c*e^3*(9*n - 1) + a*c^2*d^2*e*(5*n - 1))*x*x^n - (a^2*
c*d*e^2*(10*n - 1) + a*c^2*d^3*(6*n - 1))*x)/(a^4*c^2*d^4*n^2 + 2*a^5*c*d^2*e^2*
n^2 + a^6*e^4*n^2 + (a^2*c^4*d^4*n^2 + 2*a^3*c^3*d^2*e^2*n^2 + a^4*c^2*e^4*n^2)*
x^(4*n) + 2*(a^3*c^3*d^4*n^2 + 2*a^4*c^2*d^2*e^2*n^2 + a^5*c*e^4*n^2)*x^(2*n)) -
integrate(-1/8*((8*n^2 - 6*n + 1)*c^3*d^5 + 2*(12*n^2 - 8*n + 1)*a*c^2*d^3*e^2
+ (24*n^2 - 10*n + 1)*a^2*c*d*e^4 - ((3*n^2 - 4*n + 1)*c^3*d^4*e + 2*(5*n^2 - 6*
n + 1)*a*c^2*d^2*e^3 + (15*n^2 - 8*n + 1)*a^2*c*e^5)*x^n)/(a^3*c^3*d^6*n^2 + 3*a
^4*c^2*d^4*e^2*n^2 + 3*a^5*c*d^2*e^4*n^2 + a^6*e^6*n^2 + (a^2*c^4*d^6*n^2 + 3*a^
3*c^3*d^4*e^2*n^2 + 3*a^4*c^2*d^2*e^4*n^2 + a^5*c*e^6*n^2)*x^(2*n)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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