3.408 \(\int \frac{x^4}{2+x^5+x^{10}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 \tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

[Out]

(2*ArcTan[(1 + 2*x^5)/Sqrt[7]])/(5*Sqrt[7])

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Rubi [A]  time = 0.0448043, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 \tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(2 + x^5 + x^10),x]

[Out]

(2*ArcTan[(1 + 2*x^5)/Sqrt[7]])/(5*Sqrt[7])

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Rubi in Sympy [A]  time = 5.20041, size = 24, normalized size = 1.04 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{2 x^{5}}{7} + \frac{1}{7}\right ) \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(x**10+x**5+2),x)

[Out]

2*sqrt(7)*atan(sqrt(7)*(2*x**5/7 + 1/7))/35

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Mathematica [A]  time = 0.00923215, size = 23, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(2 + x^5 + x^10),x]

[Out]

(2*ArcTan[(1 + 2*x^5)/Sqrt[7]])/(5*Sqrt[7])

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Maple [A]  time = 0.003, size = 19, normalized size = 0.8 \[{\frac{2\,\sqrt{7}}{35}\arctan \left ({\frac{ \left ( 2\,{x}^{5}+1 \right ) \sqrt{7}}{7}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(x^10+x^5+2),x)

[Out]

2/35*arctan(1/7*(2*x^5+1)*7^(1/2))*7^(1/2)

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Maxima [A]  time = 0.820614, size = 24, normalized size = 1.04 \[ \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + x^5 + 2),x, algorithm="maxima")

[Out]

2/35*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^5 + 1))

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Fricas [A]  time = 0.260471, size = 24, normalized size = 1.04 \[ \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + x^5 + 2),x, algorithm="fricas")

[Out]

2/35*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^5 + 1))

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Sympy [A]  time = 0.323585, size = 27, normalized size = 1.17 \[ \frac{2 \sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x^{5}}{7} + \frac{\sqrt{7}}{7} \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(x**10+x**5+2),x)

[Out]

2*sqrt(7)*atan(2*sqrt(7)*x**5/7 + sqrt(7)/7)/35

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GIAC/XCAS [A]  time = 0.295419, size = 24, normalized size = 1.04 \[ \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(x^10 + x^5 + 2),x, algorithm="giac")

[Out]

2/35*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^5 + 1))