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

Optimal. Leaf size=83 \[ \frac{d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a}+\frac{e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)} \]

[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 = 0.0658563, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a}+\frac{e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1)} \]

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 = 9.96631, size = 60, normalized size = 0.72 \[ \frac{d x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a} + \frac{e x^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n + 1}{2 n} \\ \frac{3 n + 1}{2 n} \end{matrix}\middle |{- \frac{c x^{2 n}}{a}} \right )}}{a \left (n + 1\right )} \]

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 = 0.0442213, size = 82, 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 = 0.059, 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)