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]

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

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Rubi [A]  time = 0.08, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [C]  time = 1.13, 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 + Hypergeom
etric2F1[-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*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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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{x^{8} + x^{4} + 1}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} + x^{4} + 1}\,{d x} \]

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)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8}+x^{4}+1}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{x^{8} + x^{4} + 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m}{x^8+x^4+1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(x^4 + x^8 + 1),x)

[Out]

int(x^m/(x^4 + x^8 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

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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