3.7.87 \(\int \frac {x^9}{2+3 x^4} \, dx\) [687]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {281, 308, 209} \begin {gather*} \frac {1}{9} \sqrt {\frac {2}{3}} \text {ArcTan}\left (\sqrt {\frac {3}{2}} x^2\right )+\frac {x^6}{18}-\frac {x^2}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/(2 + 3*x^4),x]

[Out]

-1/9*x^2 + x^6/18 + (Sqrt[2/3]*ArcTan[Sqrt[3/2]*x^2])/9

Rule 209

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

Rule 281

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 308

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^9}{2+3 x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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} \text {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*}

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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{54} \left (-6 x^2+3 x^6+2 \sqrt {6} \tan ^{-1}\left (\sqrt {\frac {3}{2}} x^2\right )\right ) \end {gather*}

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 = 0.15, size = 26, normalized size = 0.68

method	result	size
default	\(-\frac {x^{2}}{9}+\frac {x^{6}}{18}+\frac {\arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{27}\)	\(26\)
risch	\(-\frac {x^{2}}{9}+\frac {x^{6}}{18}+\frac {\arctan \left (\frac {x^{2} \sqrt {6}}{2}\right ) \sqrt {6}}{27}\)	\(26\)
meijerg	\(\frac {\sqrt {6}\, \left (-\frac {x^{2} \sqrt {2}\, \sqrt {3}\, \left (-\frac {15 x^{4}}{2}+15\right )}{15}+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, x^{2}}{2}\right )\right )}{54}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.51, size = 25, normalized size = 0.66 \begin {gather*} \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 {gather*}

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 = 0.36, size = 31, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.03, size = 27, normalized size = 0.71 \begin {gather*} \frac {x^{6}}{18} - \frac {x^{2}}{9} + \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x^{2}}{2} \right )}}{27} \end {gather*}

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 = 0.53, size = 25, normalized size = 0.66 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.04, size = 25, normalized size = 0.66 \begin {gather*} \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x^2}{2}\right )}{27}-\frac {x^2}{9}+\frac {x^6}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(3*x^4 + 2),x)

[Out]

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

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