3.47 \(\int \frac{d+e x^n}{a-c x^{2 n}} \, dx\)

Optimal. Leaf size=81 \[ \frac{d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{c x^{2 n}}{a}\right )}{a}+\frac{e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );\frac{c x^{2 n}}{a}\right )}{a (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0703563, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{c x^{2 n}}{a}\right )}{a}+\frac{e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );\frac{c x^{2 n}}{a}\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.3714, size = 56, normalized size = 0.69 \[ \frac{d x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{\frac{c x^{2 n}}{a}} \right )}}{a} + \frac{e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{\frac{c x^{2 n}}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0687256, size = 80, normalized size = 0.99 \[ \frac{x \left (d (n+1) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};\frac{c x^{2 n}}{a}\right )+e x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );\frac{c x^{2 n}}{a}\right )\right )}{a (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{d+e{x}^{n}}{a-c{x}^{2\,n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e*x^n)/(a-c*x^(2*n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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