3.91 \(\int \frac{x^6}{2+3 x^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{x^5}{15}-\frac{2 x^3}{27}+\frac{4 x}{27}-\frac{4}{27} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

[Out]

(4*x)/27 - (2*x^3)/27 + x^5/15 - (4*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x])/27

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Rubi [A]  time = 0.0122719, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {302, 203} \[ \frac{x^5}{15}-\frac{2 x^3}{27}+\frac{4 x}{27}-\frac{4}{27} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6/(2 + 3*x^2),x]

[Out]

(4*x)/27 - (2*x^3)/27 + x^5/15 - (4*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x])/27

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^6}{2+3 x^2} \, dx &=\int \left (\frac{4}{27}-\frac{2 x^2}{9}+\frac{x^4}{3}-\frac{8}{27 \left (2+3 x^2\right )}\right ) \, dx\\ &=\frac{4 x}{27}-\frac{2 x^3}{27}+\frac{x^5}{15}-\frac{8}{27} \int \frac{1}{2+3 x^2} \, dx\\ &=\frac{4 x}{27}-\frac{2 x^3}{27}+\frac{x^5}{15}-\frac{4}{27} \sqrt{\frac{2}{3}} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\\ \end{align*}

Mathematica [A]  time = 0.0098126, size = 35, normalized size = 0.85 \[ \frac{1}{405} \left (27 x^5-30 x^3+60 x-20 \sqrt{6} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(2 + 3*x^2),x]

[Out]

(60*x - 30*x^3 + 27*x^5 - 20*Sqrt[6]*ArcTan[Sqrt[3/2]*x])/405

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Maple [A]  time = 0.003, size = 27, normalized size = 0.7 \begin{align*}{\frac{4\,x}{27}}-{\frac{2\,{x}^{3}}{27}}+{\frac{{x}^{5}}{15}}-{\frac{4\,\sqrt{6}}{81}\arctan \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(3*x^2+2),x)

[Out]

4/27*x-2/27*x^3+1/15*x^5-4/81*arctan(1/2*x*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.4167, size = 35, normalized size = 0.85 \begin{align*} \frac{1}{15} \, x^{5} - \frac{2}{27} \, x^{3} - \frac{4}{81} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{4}{27} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(3*x^2+2),x, algorithm="maxima")

[Out]

1/15*x^5 - 2/27*x^3 - 4/81*sqrt(6)*arctan(1/2*sqrt(6)*x) + 4/27*x

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Fricas [A]  time = 1.76267, size = 112, normalized size = 2.73 \begin{align*} \frac{1}{15} \, x^{5} - \frac{2}{27} \, x^{3} - \frac{4}{81} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{3} \sqrt{2} x\right ) + \frac{4}{27} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(3*x^2+2),x, algorithm="fricas")

[Out]

1/15*x^5 - 2/27*x^3 - 4/81*sqrt(3)*sqrt(2)*arctan(1/2*sqrt(3)*sqrt(2)*x) + 4/27*x

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Sympy [A]  time = 0.092286, size = 34, normalized size = 0.83 \begin{align*} \frac{x^{5}}{15} - \frac{2 x^{3}}{27} + \frac{4 x}{27} - \frac{4 \sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} \right )}}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(3*x**2+2),x)

[Out]

x**5/15 - 2*x**3/27 + 4*x/27 - 4*sqrt(6)*atan(sqrt(6)*x/2)/81

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Giac [A]  time = 1.05415, size = 35, normalized size = 0.85 \begin{align*} \frac{1}{15} \, x^{5} - \frac{2}{27} \, x^{3} - \frac{4}{81} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x\right ) + \frac{4}{27} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(3*x^2+2),x, algorithm="giac")

[Out]

1/15*x^5 - 2/27*x^3 - 4/81*sqrt(6)*arctan(1/2*sqrt(6)*x) + 4/27*x