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 = 0.0342108, 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, Rules used = {1357, 634, 618, 204, 628} \[ \frac{1}{10} \log \left (x^{10}+x^5+2\right )-\frac{\tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

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

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^9}{2+x^5+x^{10}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x}{2+x+x^2} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{10} \operatorname{Subst}\left (\int \frac{1}{2+x+x^2} \, dx,x,x^5\right )\right )+\frac{1}{10} \operatorname{Subst}\left (\int \frac{1+2 x}{2+x+x^2} \, dx,x,x^5\right )\\ &=\frac{1}{10} \log \left (2+x^5+x^{10}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,1+2 x^5\right )\\ &=-\frac{\tan ^{-1}\left (\frac{1+2 x^5}{\sqrt{7}}\right )}{5 \sqrt{7}}+\frac{1}{10} \log \left (2+x^5+x^{10}\right )\\ \end{align*}

Mathematica [A]  time = 0.0120461, 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 = 0.002, size = 31, normalized size = 0.8 \begin{align*}{\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 ) } \end{align*}

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 = 1.47833, size = 41, normalized size = 1.11 \begin{align*} -\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 ) \end{align*}

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 = 1.66923, size = 100, normalized size = 2.7 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Sympy [A]  time = 0.140561, size = 37, normalized size = 1. \begin{align*} \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} \end{align*}

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 [A]  time = 2.65546, size = 41, normalized size = 1.11 \begin{align*} -\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 ) \end{align*}

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*log(x^10 + x^5 + 2)