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

Optimal. Leaf size=134 \[ -\frac{d (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n}-\frac{e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1)}+\frac{x \left (d+e x^n\right )}{2 a n \left (a+c x^{2 n}\right )} \]

[Out]

(x*(d + e*x^n))/(2*a*n*(a + c*x^(2*n))) - (d*(1 - 2*n)*x*Hypergeometric2F1[1, 1/
(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n) - (e*(1 - n)*x^(1 + n)*Hyper
geometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n*(1 +
n))

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Rubi [A]  time = 0.126051, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{d (1-2 n) x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n}-\frac{e (1-n) x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{2 a^2 n (n+1)}+\frac{x \left (d+e x^n\right )}{2 a n \left (a+c x^{2 n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*(d + e*x^n))/(2*a*n*(a + c*x^(2*n))) - (d*(1 - 2*n)*x*Hypergeometric2F1[1, 1/
(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n) - (e*(1 - n)*x^(1 + n)*Hyper
geometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])/(2*a^2*n*(1 +
n))

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Rubi in Sympy [A]  time = 19.1854, size = 100, normalized size = 0.75 \[ \frac{x \left (d + e x^{n}\right )}{2 a n \left (a + c x^{2 n}\right )} - \frac{d x \left (- 2 n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{2 a^{2} n} - \frac{e x^{n + 1} \left (- n + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{2 a^{2} n \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.152691, size = 137, normalized size = 1.02 \[ \frac{x \left (d \left (2 n^2+n-1\right ) \left (a+c x^{2 n}\right ) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+e (n-1) x^n \left (a+c x^{2 n}\right ) \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )+a (n+1) \left (d+e x^n\right )\right )}{2 a^2 n (n+1) \left (a+c x^{2 n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{d+e{x}^{n}}{ \left ( a+c{x}^{2\,n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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