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

Optimal. Leaf size=13 \[ -\frac{1}{12 \left (3 x^4+2\right )} \]

[Out]

-1/(12*(2 + 3*x^4))

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Rubi [A]  time = 0.0025883, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {261} \[ -\frac{1}{12 \left (3 x^4+2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + 3*x^4)^2,x]

[Out]

-1/(12*(2 + 3*x^4))

Rule 261

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

Rubi steps

\begin{align*} \int \frac{x^3}{\left (2+3 x^4\right )^2} \, dx &=-\frac{1}{12 \left (2+3 x^4\right )}\\ \end{align*}

Mathematica [A]  time = 0.0030706, size = 13, normalized size = 1. \[ -\frac{1}{12 \left (3 x^4+2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2 + 3*x^4)^2,x]

[Out]

-1/(12*(2 + 3*x^4))

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Maple [A]  time = 0.001, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{36\,{x}^{4}+24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/(3*x^4+2)

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Maxima [A]  time = 1.25765, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{12 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/(3*x^4 + 2)

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Fricas [A]  time = 1.62151, size = 26, normalized size = 2. \begin{align*} -\frac{1}{12 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/(3*x^4 + 2)

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Sympy [A]  time = 0.135273, size = 8, normalized size = 0.62 \begin{align*} - \frac{1}{36 x^{4} + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(36*x**4 + 24)

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Giac [A]  time = 1.13151, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{12 \,{\left (3 \, x^{4} + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/(3*x^4 + 2)