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 = 0.151875, 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 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1-i \sqrt{3}}\right )}{\sqrt{3} \left (\sqrt{3}+i\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};-\frac{2 x^4}{1+i \sqrt{3}}\right )}{\sqrt{3} \left (-\sqrt{3}+i\right ) (m+1)} \]

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))

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Rubi in Sympy [A]  time = 17.576, size = 102, normalized size = 0.8 \[ \frac{2 \sqrt{3} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{x^{4} \left (- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right )} \right )}}{3 \left (\sqrt{3} + i\right ) \left (m + 1\right )} + \frac{2 \sqrt{3} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{x^{4} \left (- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right )} \right )}}{3 \left (\sqrt{3} - i\right ) \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(3)*x**(m + 1)*hyper((1, m/4 + 1/4), (m/4 + 5/4,), x**4*(-1/2 - sqrt(3)*I/
2))/(3*(sqrt(3) + I)*(m + 1)) + 2*sqrt(3)*x**(m + 1)*hyper((1, m/4 + 1/4), (m/4
+ 5/4,), x**4*(-1/2 + sqrt(3)*I/2))/(3*(sqrt(3) - I)*(m + 1))

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Mathematica [C]  time = 2.42724, size = 488, normalized size = 3.84 \[ \frac{x^m \left (-\frac{\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]}{m^2+3 m+2}+\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]-\frac{i \left (\left (\frac{x}{x-\sqrt [3]{-1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt [3]{-1}}{\sqrt [3]{-1}-x}\right )+\left (\frac{x}{x-(-1)^{2/3}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{(-1)^{2/3}}{(-1)^{2/3}-x}\right )-\left (\frac{x}{x+\sqrt [3]{-1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{\sqrt [3]{-1}}{x+\sqrt [3]{-1}}\right )-\left (\frac{x}{x+(-1)^{2/3}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{(-1)^{2/3}}{x+(-1)^{2/3}}\right )\right )}{\sqrt{3}}\right )}{4 m} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x^m*(((-I)*(Hypergeometric2F1[-m, -m, 1 - m, (-1)^(1/3)/((-1)^(1/3) - x)]/(x/(-
(-1)^(1/3) + x))^m + Hypergeometric2F1[-m, -m, 1 - m, (-1)^(2/3)/((-1)^(2/3) - x
)]/(x/(-(-1)^(2/3) + x))^m - Hypergeometric2F1[-m, -m, 1 - m, (-1)^(1/3)/((-1)^(
1/3) + x)]/(x/((-1)^(1/3) + x))^m - Hypergeometric2F1[-m, -m, 1 - m, (-1)^(2/3)/
((-1)^(2/3) + x)]/(x/((-1)^(2/3) + x))^m))/Sqrt[3] + RootSum[1 - #1^2 + #1^4 & ,
Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]/((x/(x - #1))^m*(-#1 + 2*#1^3)
) & ] - RootSum[1 - #1^2 + #1^4 & , (m*x^2 + m^2*x^2 + 2*m*x*#1 + m^2*x*#1 + (2*
Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (3*m*Hyp
ergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m^2*Hyperg
eometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(x/(x - #1))^m + (m*#1^2)/(x/#1
)^m)/(-#1 + 2*#1^3) & ]/(2 + 3*m + m^2)))/(4*m)

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}+{x}^{4}+1}}\, dx \]

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. \[ \int \frac{x^{m}}{x^{8} + x^{4} + 1}\,{d x} \]

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. \[{\rm integral}\left (\frac{x^{m}}{x^{8} + x^{4} + 1}, x\right ) \]

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. \[ \int \frac{x^{m}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

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

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)