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

Optimal. Leaf size=167 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d}-\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \]

[Out]

(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, (2 + n^(-1))/2, -((c*x^(2*n))/a),
(e^2*x^(2*n))/d^2])/(d*(1 + (c*x^(2*n))/a)^p) - (e*x^(1 + n)*(a + c*x^(2*n))^p*A
ppellF1[(1 + n)/(2*n), -p, 1, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^
2])/(d^2*(1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi [A]  time = 0.331645, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{1}{2 n};-p,1;\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d}-\frac{e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac{n+1}{2 n};-p,1;\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a},\frac{e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + c*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, (2 + n^(-1))/2, -((c*x^(2*n))/a),
(e^2*x^(2*n))/d^2])/(d*(1 + (c*x^(2*n))/a)^p) - (e*x^(1 + n)*(a + c*x^(2*n))^p*A
ppellF1[(1 + n)/(2*n), -p, 1, (3 + n^(-1))/2, -((c*x^(2*n))/a), (e^2*x^(2*n))/d^
2])/(d^2*(1 + n)*(1 + (c*x^(2*n))/a)^p)

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Rubi in Sympy [A]  time = 70.9471, size = 128, normalized size = 0.77 \[ \frac{x \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2 n},1,- p,\frac{n + \frac{1}{2}}{n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d} - \frac{e x^{n + 1} \left (1 + \frac{c x^{2 n}}{a}\right )^{- p} \left (a + c x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{n + 1}{2 n},1,- p,\frac{3 n + 1}{2 n},\frac{e^{2} x^{2 n}}{d^{2}},- \frac{c x^{2 n}}{a} \right )}}{d^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.069954, size = 0, normalized size = 0. \[ \int \frac{\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.13, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+c{x}^{2\,n} \right ) ^{p}}{d+e{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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