3.689 \(\int \frac{x^5}{2+3 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

[Out]

x^2/6 - ArcTan[Sqrt[3/2]*x^2]/(3*Sqrt[6])

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Rubi [A]  time = 0.0134381, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 321, 203} \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(2 + 3*x^4),x]

[Out]

x^2/6 - ArcTan[Sqrt[3/2]*x^2]/(3*Sqrt[6])

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{x^5}{2+3 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{6}-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.0081398, size = 29, normalized size = 1. \[ \frac{x^2}{6}-\frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(2 + 3*x^4),x]

[Out]

x^2/6 - ArcTan[Sqrt[3/2]*x^2]/(3*Sqrt[6])

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Maple [A]  time = 0.003, size = 21, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{6}}-{\frac{\sqrt{6}}{18}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(3*x^4+2),x)

[Out]

1/6*x^2-1/18*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.53849, size = 27, normalized size = 0.93 \begin{align*} \frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+2),x, algorithm="maxima")

[Out]

1/6*x^2 - 1/18*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 1.60711, size = 65, normalized size = 2.24 \begin{align*} \frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+2),x, algorithm="fricas")

[Out]

1/6*x^2 - 1/18*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Sympy [A]  time = 0.129129, size = 22, normalized size = 0.76 \begin{align*} \frac{x^{2}}{6} - \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(3*x**4+2),x)

[Out]

x**2/6 - sqrt(6)*atan(sqrt(6)*x**2/2)/18

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Giac [A]  time = 1.11551, size = 27, normalized size = 0.93 \begin{align*} \frac{1}{6} \, x^{2} - \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3*x^4+2),x, algorithm="giac")

[Out]

1/6*x^2 - 1/18*sqrt(6)*arctan(1/2*sqrt(6)*x^2)