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 = 0.0077789, 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, Rules used = {28, 364} \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{4};\frac{m+5}{4};-x^4\right )}{m+1} \]

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)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{x^m}{1+2 x^4+x^8} \, dx &=\int \frac{x^m}{\left (1+x^4\right )^2} \, dx\\ &=\frac{x^{1+m} \, _2F_1\left (2,\frac{1+m}{4};\frac{5+m}{4};-x^4\right )}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.006883, 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 = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{{x}^{8}+2\,{x}^{4}+1}}\, dx \end{align*}

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. \begin{align*} \int \frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left (\frac{x^{m}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \end{align*}

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. \begin{align*} \int \frac{x^{m}}{\left (x^{4} + 1\right )^{2}}\, dx \end{align*}

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

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)