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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = 0.00, size = 13, normalized size = 1.00 \[ \frac {1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x^3)/(2 + 3*x^4),x]

[Out]

(d*Log[2 + 3*x^4])/12

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 11, normalized size = 0.85 \[ \frac {1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 11, normalized size = 0.85 \[ \frac {1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 12, normalized size = 0.92 \[ \frac {d \ln \left (3 x^{4}+2\right )}{12} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 11, normalized size = 0.85 \[ \frac {1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 9, normalized size = 0.69 \[ \frac {d\,\ln \left (x^4+\frac {2}{3}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^3)/(3*x^4 + 2),x)

[Out]

(d*log(x^4 + 2/3))/12

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 10, normalized size = 0.77 \[ \frac {d \log {\left (3 x^{4} + 2 \right )}}{12} \]

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

________________________________________________________________________________________