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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 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])

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]

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 = 0.01, 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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 39, normalized size = 0.95 \[ \frac {1}{135} x \left (9 x^4-10 x^2+20\right )-\frac {4}{27} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6/(2 + 3*x^2),x]

[Out]

(x*(20 - 10*x^2 + 9*x^4))/135 - (4*Sqrt[2/3]*ArcTan[Sqrt[3/2]*x])/27

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 32, normalized size = 0.78 \[ \frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {3} \sqrt {2} x\right ) + \frac {4}{27} \, x \]

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

________________________________________________________________________________________

giac [A]  time = 0.85, size = 26, normalized size = 0.63 \[ \frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {4}{27} \, x \]

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

________________________________________________________________________________________

maple [A]  time = 0.28, size = 27, normalized size = 0.66

method	result	size
default	\(\frac {4 x}{27}-\frac {2 x^{3}}{27}+\frac {x^{5}}{15}-\frac {4 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{81}\)	\(27\)
risch	\(\frac {4 x}{27}-\frac {2 x^{3}}{27}+\frac {x^{5}}{15}-\frac {4 \arctan \left (\frac {x \sqrt {6}}{2}\right ) \sqrt {6}}{81}\)	\(27\)
meijerg	\(\frac {2 \sqrt {2}\, \sqrt {3}\, \left (\frac {x \sqrt {2}\, \sqrt {3}\, \left (\frac {189}{4} x^{4}-\frac {105}{2} x^{2}+105\right )}{105}-2 \arctan \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{81}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 26, normalized size = 0.63 \[ \frac {1}{15} \, x^{5} - \frac {2}{27} \, x^{3} - \frac {4}{81} \, \sqrt {6} \arctan \left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {4}{27} \, x \]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 32, normalized size = 0.78 \[ \frac {4\,x}{27}-\frac {2\,x^3}{27}+\frac {x^5}{15}-\frac {4\,\sqrt {2}\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(3*x^2 + 2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 34, normalized size = 0.83 \[ \frac {x^{5}}{15} - \frac {2 x^{3}}{27} + \frac {4 x}{27} - \frac {4 \sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} \right )}}{81} \]

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

________________________________________________________________________________________