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

Optimal. Leaf size=127 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+i\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+i\right ) (m+1)} \]

[Out]

(2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*x^4)/(1 - I*Sqrt[3])])/(Sqrt[3]*(I + Sqrt[3])*(1 +
m)) - (2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*x^4)/(1 + I*Sqrt[3])])/(Sqrt[3]*(I - Sqrt[3
])*(1 + m))

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Rubi [A]  time = 0.0779896, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1375, 364} \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+i\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+i\right ) (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^m/(1 + x^4 + x^8),x]

[Out]

(2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*x^4)/(1 - I*Sqrt[3])])/(Sqrt[3]*(I + Sqrt[3])*(1 +
m)) - (2*x^(1 + m)*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, (-2*x^4)/(1 + I*Sqrt[3])])/(Sqrt[3]*(I - Sqrt[3
])*(1 + m))

Rule 1375

Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]
}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

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+x^4+x^8} \, dx &=-\frac{i \int \frac{x^m}{\frac{1}{2}-\frac{i \sqrt{3}}{2}+x^4} \, dx}{\sqrt{3}}+\frac{i \int \frac{x^m}{\frac{1}{2}+\frac{i \sqrt{3}}{2}+x^4} \, dx}{\sqrt{3}}\\ &=\frac{2 x^{1+m} \, _2F_1\left (1,\frac{1+m}{4};\frac{5+m}{4};-\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (i+\sqrt{3}\right ) (1+m)}-\frac{2 x^{1+m} \, _2F_1\left (1,\frac{1+m}{4};\frac{5+m}{4};-\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (i-\sqrt{3}\right ) (1+m)}\\ \end{align*}

Mathematica [C]  time = 0.40294, size = 212, normalized size = 1.67 \[ \frac{1}{4} x^{m+1} \left (\frac{x^2 \text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\, _2F_1\left (1,m+3;m+4;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2-2}\& \right ]}{m+3}-\frac{\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2-2}\& \right ]}{m+1}+\frac{3 i \left (\sqrt{3}+i\right ) \, _2F_1\left (1,m+1;m+2;-\sqrt [3]{-1} x\right )+3 i \left (\sqrt{3}+i\right ) \, _2F_1\left (1,m+1;m+2;\sqrt [3]{-1} x\right )-6 \left (\, _2F_1\left (1,m+1;m+2;-(-1)^{2/3} x\right )+\, _2F_1\left (1,m+1;m+2;(-1)^{2/3} x\right )\right )}{6 \left (\sqrt [3]{-1}-2\right ) (m+1)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^m/(1 + x^4 + x^8),x]

[Out]

(x^(1 + m)*(((3*I)*(I + Sqrt[3])*Hypergeometric2F1[1, 1 + m, 2 + m, -((-1)^(1/3)*x)] + (3*I)*(I + Sqrt[3])*Hyp
ergeometric2F1[1, 1 + m, 2 + m, (-1)^(1/3)*x] - 6*(Hypergeometric2F1[1, 1 + m, 2 + m, -((-1)^(2/3)*x)] + Hyper
geometric2F1[1, 1 + m, 2 + m, (-1)^(2/3)*x]))/(6*(-2 + (-1)^(1/3))*(1 + m)) - RootSum[1 - #1^2 + #1^4 & , Hype
rgeometric2F1[1, 1 + m, 2 + m, x/#1]/(-2 + #1^2) & ]/(1 + m) + (x^2*RootSum[1 - #1^2 + #1^4 & , Hypergeometric
2F1[1, 3 + m, 4 + m, x/#1]/(-2 + #1^2) & ])/(3 + m)))/4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m/(x^8+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} + x^{4} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(x^8+x^4+1),x, algorithm="maxima")

[Out]

integrate(x^m/(x^8 + 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} + x^{4} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(x^8+x^4+1),x, algorithm="fricas")

[Out]

integral(x^m/(x^8 + 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^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(x**8+x**4+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(x^8+x^4+1),x, algorithm="giac")

[Out]

integrate(x^m/(x^8 + x^4 + 1), x)