3.385 \(\int \frac{x^m}{1-3 x^4+x^8} \, dx\)

Optimal. Leaf size=117 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3-\sqrt{5}}\right )}{\sqrt{5} \left (3-\sqrt{5}\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3+\sqrt{5}}\right )}{\sqrt{5} \left (3+\sqrt{5}\right ) (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.128404, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3-\sqrt{5}}\right )}{\sqrt{5} \left (3-\sqrt{5}\right ) (m+1)}-\frac{2 x^{m+1} \, _2F_1\left (1,\frac{m+1}{4};\frac{m+5}{4};\frac{2 x^4}{3+\sqrt{5}}\right )}{\sqrt{5} \left (3+\sqrt{5}\right ) (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(1 - 3*x^4 + x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.9538, size = 105, normalized size = 0.9 \[ - \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{- \frac{3}{2} - \frac{\sqrt{5}}{2}}} \right )}}{5 \left (\frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} + \frac{\sqrt{5} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{4} + \frac{1}{4} \\ \frac{m}{4} + \frac{5}{4} \end{matrix}\middle |{- \frac{x^{4}}{- \frac{3}{2} + \frac{\sqrt{5}}{2}}} \right )}}{5 \left (- \frac{\sqrt{5}}{2} + \frac{3}{2}\right ) \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(x**8-3*x**4+1),x)

[Out]

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

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Mathematica [C]  time = 1.0223, size = 575, normalized size = 4.91 \[ \frac{x^m \left (\frac{\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]-\text{RootSum}\left [\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\text{$\#$1}^2 m^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+3 \text{$\#$1}^2 m \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+2 \text{$\#$1}^2 \left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )+\text{$\#$1}^2 m \left (\frac{x}{\text{$\#$1}}\right )^{-m}+\text{$\#$1} m^2 x+2 \text{$\#$1} m x+m^2 x^2+m x^2}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right ]-\left (m^2+3 m+2\right ) \text{RootSum}\left [\text{$\#$1}^4+\text{$\#$1}^2-1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3+\text{$\#$1}}\&\right ]}{m^2+3 m+2}-\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2-1\&,\frac{\left (\frac{x}{x-\text{$\#$1}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{\text{$\#$1}}{x-\text{$\#$1}}\right )}{2 \text{$\#$1}^3-\text{$\#$1}}\&\right ]\right )}{4 m} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^m/(1 - 3*x^4 + x^8),x]

[Out]

(x^m*(-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*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1)
)]*#1^2)/(x/(x - #1))^m + (m^2*Hypergeometric2F1[-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)*R
ootSum[-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*Hypergeometric2F1[-m, -m, 1 - m, -(#1/(x - #1))]*#1^2)/(
x/(x - #1))^m + (m^2*Hypergeometric2F1[-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 = 0.027, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m}}{{x}^{8}-3\,{x}^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(x^8-3*x^4+1),x)

[Out]

int(x^m/(x^8-3*x^4+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^m/(x^8 - 3*x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^m/(x^8 - 3*x^4 + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(x**8-3*x**4+1),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{x^{8} - 3 \, x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(x^8 - 3*x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^m/(x^8 - 3*x^4 + 1), x)