3.687 \(\int \frac{x^9}{2+3 x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^6}{18}-\frac{x^2}{9}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

[Out]

-x^2/9 + x^6/18 + (Sqrt[2/3]*ArcTan[Sqrt[3/2]*x^2])/9

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Rubi [A]  time = 0.0219432, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {275, 302, 203} \[ \frac{x^6}{18}-\frac{x^2}{9}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(2 + 3*x^4),x]

[Out]

-x^2/9 + x^6/18 + (Sqrt[2/3]*ArcTan[Sqrt[3/2]*x^2])/9

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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^9}{2+3 x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{2}{9}+\frac{x^2}{3}+\frac{4}{9 \left (2+3 x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{9}+\frac{x^6}{18}+\frac{2}{9} \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )\\ &=-\frac{x^2}{9}+\frac{x^6}{18}+\frac{1}{9} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0144178, size = 34, normalized size = 0.89 \[ \frac{1}{54} \left (3 x^6-6 x^2+2 \sqrt{6} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(2 + 3*x^4),x]

[Out]

(-6*x^2 + 3*x^6 + 2*Sqrt[6]*ArcTan[Sqrt[3/2]*x^2])/54

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Maple [A]  time = 0.003, size = 26, normalized size = 0.7 \begin{align*} -{\frac{{x}^{2}}{9}}+{\frac{{x}^{6}}{18}}+{\frac{\sqrt{6}}{27}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(3*x^4+2),x)

[Out]

-1/9*x^2+1/18*x^6+1/27*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.53237, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{18} \, x^{6} - \frac{1}{9} \, x^{2} + \frac{1}{27} \, \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^9/(3*x^4+2),x, algorithm="maxima")

[Out]

1/18*x^6 - 1/9*x^2 + 1/27*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 1.69063, size = 101, normalized size = 2.66 \begin{align*} \frac{1}{18} \, x^{6} - \frac{1}{9} \, x^{2} + \frac{1}{27} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(3*x^4+2),x, algorithm="fricas")

[Out]

1/18*x^6 - 1/9*x^2 + 1/27*sqrt(3)*sqrt(2)*arctan(1/2*sqrt(3)*sqrt(2)*x^2)

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Sympy [A]  time = 0.130031, size = 27, normalized size = 0.71 \begin{align*} \frac{x^{6}}{18} - \frac{x^{2}}{9} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(3*x**4+2),x)

[Out]

x**6/18 - x**2/9 + sqrt(6)*atan(sqrt(6)*x**2/2)/27

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Giac [A]  time = 1.12248, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{18} \, x^{6} - \frac{1}{9} \, x^{2} + \frac{1}{27} \, \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^9/(3*x^4+2),x, algorithm="giac")

[Out]

1/18*x^6 - 1/9*x^2 + 1/27*sqrt(6)*arctan(1/2*sqrt(6)*x^2)