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

Optimal. Leaf size=205 \[ -\frac{2 c^2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^2}+\frac{c x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^2}+\frac{2 c e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.354646, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{2 c^2 d e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )^2}+\frac{c x \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )^2}+\frac{2 c e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{\left (a e^2+c d^2\right )^2}+\frac{e^2 x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.13004, size = 188, normalized size = 0.92 \[ \frac{x \left (e \left (-\frac{2 c^2 d x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)}+\frac{\left (a e^3 (n-1)+c d^2 e (3 n-1)\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d^2 n}+\frac{a e^3+c d^2 e}{d^2 n+d e n x^n}\right )+\frac{c \left (c d^2-a e^2\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )}{a}\right )}{\left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.231, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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