3.271 \(\int \frac{x^m}{1+2 x^4+x^8} \, dx\)

Optimal. Leaf size=32 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right )}{m+1} \]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, (5 + m)/4, -x^4])/(1 + m)

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Rubi [A]  time = 0.0220683, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(1 + 2*x^4 + x^8),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, (5 + m)/4, -x^4])/(1 + m)

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Rubi in Sympy [A]  time = 4.52388, size = 24, normalized size = 0.75 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- x^{4}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(x**8+2*x**4+1),x)

[Out]

x**(m + 1)*hyper((2, m/4 + 1/4), (m/4 + 5/4,), -x**4)/(m + 1)

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Mathematica [A]  time = 0.0201887, size = 34, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+1}{4}+1;-x^4\right )}{m+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(1 + 2*x^4 + x^8),x]

[Out]

(x^(1 + m)*Hypergeometric2F1[2, (1 + m)/4, 1 + (1 + m)/4, -x^4])/(1 + m)

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}+2\,{x}^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(x^8+2*x^4+1),x)

[Out]

int(x^m/(x^8+2*x^4+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^m/(x^8 + 2*x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^m/(x^8 + 2*x^4 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\left (x^{4} + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(x**8+2*x**4+1),x)

[Out]

Integral(x**m/(x**4 + 1)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 + 2*x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^m/(x^8 + 2*x^4 + 1), x)