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

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

[Out]

Log[2 + 3*x^4]/12

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Rubi [A]  time = 0.0025986, antiderivative size = 12, 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 = {260} \[ \frac{1}{12} \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + 3*x^4),x]

[Out]

Log[2 + 3*x^4]/12

Rule 260

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

Rubi steps

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

Mathematica [A]  time = 0.0026309, size = 12, normalized size = 1. \[ \frac{1}{12} \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2 + 3*x^4),x]

[Out]

Log[2 + 3*x^4]/12

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Maple [A]  time = 0., size = 11, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.956816, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{12} \, \log \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),x, algorithm="maxima")

[Out]

1/12*log(3*x^4 + 2)

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Fricas [A]  time = 1.60579, size = 28, normalized size = 2.33 \begin{align*} \frac{1}{12} \, \log \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),x, algorithm="fricas")

[Out]

1/12*log(3*x^4 + 2)

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Sympy [A]  time = 0.088471, size = 8, normalized size = 0.67 \begin{align*} \frac{\log{\left (3 x^{4} + 2 \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x**4 + 2)/12

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Giac [A]  time = 1.17055, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{12} \, \log \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),x, algorithm="giac")

[Out]

1/12*log(3*x^4 + 2)