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

Optimal. Leaf size=410 \[ \frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac{c (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c e^2 x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac{c x \left (-a e^2+c d^2-2 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{4 c e^4 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^3}+\frac{e^4 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^2} \]

[Out]

(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) +
(c*e^2*(3*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)^3) - (c*(c*d^2 - a*e^2)*(1 - 2*n)*x*Hypergeometri
c2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n)
+ (4*c*e^4*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*
e^2)^3 - (4*c^2*d*e^3*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^2*d*e*(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)^2*n*(1 + n)) + (e^4*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((
e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^2)

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Rubi [A]  time = 0.786815, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{c^2 d e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1) \left (a e^2+c d^2\right )^2}-\frac{c (1-2 n) x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n \left (a e^2+c d^2\right )^2}-\frac{4 c^2 d e^3 x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^3}+\frac{c e^2 x \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^3}+\frac{c x \left (-a e^2+c d^2-2 c d e x^n\right )}{2 a n \left (a e^2+c d^2\right )^2 \left (a+c x^{2 n}\right )}+\frac{4 c e^4 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^3}+\frac{e^4 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(c*d^2 - a*e^2 - 2*c*d*e*x^n))/(2*a*(c*d^2 + a*e^2)^2*n*(a + c*x^(2*n))) +
(c*e^2*(3*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)^3) - (c*(c*d^2 - a*e^2)*(1 - 2*n)*x*Hypergeometri
c2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*(c*d^2 + a*e^2)^2*n)
+ (4*c*e^4*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(c*d^2 + a*
e^2)^3 - (4*c^2*d*e^3*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^2*d*e*(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)^2*n*(1 + n)) + (e^4*x*Hypergeometric2F1[2, n^(-1), 1 + n^(-1), -((
e*x^n)/d)])/(d^2*(c*d^2 + a*e^2)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.32803, size = 495, normalized size = 1.21 \[ \frac{x \left (\frac{c^3 d^4}{a^2 n+a c n x^{2 n}}-\frac{2 c^3 d^3 e x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 (n+1)}+\frac{2 c^3 d^3 e x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a^2 n (n+1)}-\frac{2 c^3 d^3 e x^n}{a^2 n+a c n x^{2 n}}+\frac{c \left (a^2 e^4 (1-4 n)+6 a c d^2 e^2 n+c^2 d^4 (2 n-1)\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a^2 n}-\frac{10 c^2 d e^3 x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{2 c^2 d e^3 x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a n (n+1)}-\frac{2 c^2 d e^3 x^n}{a n+c n x^{2 n}}+\frac{2 e^4 \left (a e^2 (n-1)+c d^2 (5 n-1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 n}-\frac{a c e^4}{a n+c n x^{2 n}}+\frac{2 a e^6}{d^2 n+d e n x^n}+\frac{2 c d e^4}{d n+e n x^n}\right )}{2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*((2*c*d*e^4)/(d*n + e*n*x^n) + (2*a*e^6)/(d^2*n + d*e*n*x^n) - (a*c*e^4)/(a*n
+ c*n*x^(2*n)) - (2*c^2*d*e^3*x^n)/(a*n + c*n*x^(2*n)) + (c^3*d^4)/(a^2*n + a*c
*n*x^(2*n)) - (2*c^3*d^3*e*x^n)/(a^2*n + a*c*n*x^(2*n)) + (c*(a^2*e^4*(1 - 4*n)
+ 6*a*c*d^2*e^2*n + c^2*d^4*(-1 + 2*n))*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n
), -((c*x^(2*n))/a)])/(a^2*n) + (2*e^4*(a*e^2*(-1 + n) + c*d^2*(-1 + 5*n))*Hyper
geometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d^2*n) - (2*c^3*d^3*e*x^n*Hy
pergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*(1 + n
)) - (10*c^2*d*e^3*x^n*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*
x^(2*n))/a)])/(a*(1 + n)) + (2*c^3*d^3*e*x^n*Hypergeometric2F1[1, (1 + n)/(2*n),
(3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a^2*n*(1 + n)) + (2*c^2*d*e^3*x^n*Hypergeom
etric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*n*(1 + n))))/(2
*(c*d^2 + a*e^2)^3)

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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