3.407 \(\int \frac{x^9}{2+x^5+x^{10}} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{10} \log \left (x^{10}+x^5+2\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[7]]/(5*Sqrt[7]) + Log[2 + x^5 + x^10]/10

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Rubi [A]  time = 0.0685276, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{10} \log \left (x^{10}+x^5+2\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(2 + x^5 + x^10),x]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[7]]/(5*Sqrt[7]) + Log[2 + x^5 + x^10]/10

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Rubi in Sympy [A]  time = 8.26654, size = 34, normalized size = 0.92 \[ \frac{\log{\left (x^{10} + x^{5} + 2 \right )}}{10} - \frac{\sqrt{7} \operatorname{atan}{\left (\sqrt{7} \left (\frac{2 x^{5}}{7} + \frac{1}{7}\right ) \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(x**10+x**5+2),x)

[Out]

log(x**10 + x**5 + 2)/10 - sqrt(7)*atan(sqrt(7)*(2*x**5/7 + 1/7))/35

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Mathematica [A]  time = 0.0187539, size = 37, normalized size = 1. \[ \frac{1}{10} \log \left (x^{10}+x^5+2\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(2 + x^5 + x^10),x]

[Out]

-ArcTan[(1 + 2*x^5)/Sqrt[7]]/(5*Sqrt[7]) + Log[2 + x^5 + x^10]/10

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \[{\frac{\ln \left ({x}^{10}+{x}^{5}+2 \right ) }{10}}-{\frac{\sqrt{7}}{35}\arctan \left ({\frac{ \left ( 2\,{x}^{5}+1 \right ) \sqrt{7}}{7}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(x^10+x^5+2),x)

[Out]

1/10*ln(x^10+x^5+2)-1/35*arctan(1/7*(2*x^5+1)*7^(1/2))*7^(1/2)

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Maxima [A]  time = 0.821823, size = 41, normalized size = 1.11 \[ -\frac{1}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) + \frac{1}{10} \, \log \left (x^{10} + x^{5} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^10 + x^5 + 2),x, algorithm="maxima")

[Out]

-1/35*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^5 + 1)) + 1/10*log(x^10 + x^5 + 2)

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Fricas [A]  time = 0.298392, size = 46, normalized size = 1.24 \[ \frac{1}{70} \, \sqrt{7}{\left (\sqrt{7} \log \left (x^{10} + x^{5} + 2\right ) - 2 \, \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^10 + x^5 + 2),x, algorithm="fricas")

[Out]

1/70*sqrt(7)*(sqrt(7)*log(x^10 + x^5 + 2) - 2*arctan(1/7*sqrt(7)*(2*x^5 + 1)))

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Sympy [A]  time = 0.351268, size = 37, normalized size = 1. \[ \frac{\log{\left (x^{10} + x^{5} + 2 \right )}}{10} - \frac{\sqrt{7} \operatorname{atan}{\left (\frac{2 \sqrt{7} x^{5}}{7} + \frac{\sqrt{7}}{7} \right )}}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(x**10+x**5+2),x)

[Out]

log(x**10 + x**5 + 2)/10 - sqrt(7)*atan(2*sqrt(7)*x**5/7 + sqrt(7)/7)/35

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GIAC/XCAS [A]  time = 0.271432, size = 41, normalized size = 1.11 \[ -\frac{1}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) + \frac{1}{10} \,{\rm ln}\left (x^{10} + x^{5} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(x^10 + x^5 + 2),x, algorithm="giac")

[Out]

-1/35*sqrt(7)*arctan(1/7*sqrt(7)*(2*x^5 + 1)) + 1/10*ln(x^10 + x^5 + 2)