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

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

[Out]

(d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.0037983, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 260} \[ \frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(d*x^3)/(2 + 3*x^4),x]

[Out]

(d*Log[2 + 3*x^4])/12

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0028887, size = 13, normalized size = 1. \[ \frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x^3)/(2 + 3*x^4),x]

[Out]

(d*Log[2 + 3*x^4])/12

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Maple [A]  time = 0.039, size = 12, normalized size = 0.9 \begin{align*}{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.943257, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*d*log(3*x^4 + 2)

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Fricas [A]  time = 1.39175, size = 31, normalized size = 2.38 \begin{align*} \frac{1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*d*log(3*x^4 + 2)

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Sympy [A]  time = 0.076402, size = 10, normalized size = 0.77 \begin{align*} \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(3*x**4 + 2)/12

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Giac [A]  time = 1.07502, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*d*log(3*x^4 + 2)