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

Optimal. Leaf size=261 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,2;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2}+\frac{e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right );-p,2;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (2 n+1)}-\frac{2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,2;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (n+1)} \]

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 2, (4 + n^(-1))/
2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) +
(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 2, (2 + n^(-1))/2, -((c*x^(2*n))/a),
(e^2*x^(2*n))/d^2])/(d^2*(1 + (c*x^(2*n))/a)^p) - (2*e*x^(1 + n)*(a + c*x^(2*n)
)^p*AppellF1[(1 + n)/(2*n), -p, 2, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n
))/d^2])/(d^3*(1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi [A]  time = 0.55777, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,2;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2}+\frac{e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2} \left (2+\frac{1}{n}\right );-p,2;\frac{1}{2} \left (4+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^4 (2 n+1)}-\frac{2 e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,2;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^3 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*AppellF1[(2 + n^(-1))/2, -p, 2, (4 + n^(-1))/
2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p) +
(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 2, (2 + n^(-1))/2, -((c*x^(2*n))/a),
(e^2*x^(2*n))/d^2])/(d^2*(1 + (c*x^(2*n))/a)^p) - (2*e*x^(1 + n)*(a + c*x^(2*n)
)^p*AppellF1[(1 + n)/(2*n), -p, 2, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n
))/d^2])/(d^3*(1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi in Sympy [A]  time = 137.215, size = 206, normalized size = 0.79 \[ \frac{x \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2 n},2,- p,\frac{n + \frac{1}{2}}{n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{2}} - \frac{2 e x^{n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{n + 1}{2 n},2,- p,\frac{3 n + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{3} \left (n + 1\right )} + \frac{e^{2} x^{2 n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{n + \frac{1}{2}}{n},2,- p,2 + \frac{1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{4} \left (2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(1 + c*x**(2*n)/a)**(-p)*(a + c*x**(2*n))**p*appellf1(1/(2*n), 2, -p, (n + 1/2
)/n, e**2*x**(2*n)/d**2, -c*x**(2*n)/a)/d**2 - 2*e*x**(n + 1)*(1 + c*x**(2*n)/a)
**(-p)*(a + c*x**(2*n))**p*appellf1((n + 1)/(2*n), 2, -p, (3*n + 1)/(2*n), e**2*
x**(2*n)/d**2, -c*x**(2*n)/a)/(d**3*(n + 1)) + e**2*x**(2*n + 1)*(1 + c*x**(2*n)
/a)**(-p)*(a + c*x**(2*n))**p*appellf1((n + 1/2)/n, 2, -p, 2 + 1/(2*n), e**2*x**
(2*n)/d**2, -c*x**(2*n)/a)/(d**4*(2*n + 1))

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Mathematica [A]  time = 0.117383, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+c{x}^{2\,n} \right ) ^{p}}{ \left ( d+e{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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