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 = 0.0218787, 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, Rules used = {1352, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{2 x^5+1}{\sqrt{7}}\right )}{5 \sqrt{7}} \]

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])

Rule 1352

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

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])

Rubi steps

\begin{align*} \int \frac{x^4}{2+x^5+x^{10}} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{1}{2+x+x^2} \, dx,x,x^5\right )\\ &=-\left (\frac{2}{5} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,1+2 x^5\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{1+2 x^5}{\sqrt{7}}\right )}{5 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0060291, 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 = 0.001, size = 19, normalized size = 0.8 \begin{align*}{\frac{2\,\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^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 = 1.51551, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \end{align*}

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 = 1.68778, size = 62, normalized size = 2.7 \begin{align*} \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \end{align*}

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 = 0.12709, size = 27, normalized size = 1.17 \begin{align*} \frac{2 \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**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 [A]  time = 2.6525, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{35} \, \sqrt{7} \arctan \left (\frac{1}{7} \, \sqrt{7}{\left (2 \, x^{5} + 1\right )}\right ) \end{align*}

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))