3.7.90 \(\int \frac {x^3}{2+3 x^4} \, dx\) [690]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {266} \begin {gather*} \frac {1}{12} \log \left (3 x^4+2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + 3*x^4),x]

[Out]

Log[2 + 3*x^4]/12

Rule 266

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*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{12} \log \left (2+3 x^4\right ) \end {gather*}

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 = 0.14, size = 11, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\ln \left (3 x^{4}+2\right )}{12}\)	\(11\)
default	\(\frac {\ln \left (3 x^{4}+2\right )}{12}\)	\(11\)
norman	\(\frac {\ln \left (3 x^{4}+2\right )}{12}\)	\(11\)
meijerg	\(\frac {\ln \left (1+\frac {3 x^{4}}{2}\right )}{12}\)	\(11\)
risch	\(\frac {\ln \left (3 x^{4}+2\right )}{12}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, \log \left (3 \, x^{4} + 2\right ) \end {gather*}

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 = 0.36, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, \log \left (3 \, x^{4} + 2\right ) \end {gather*}

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 = 0.02, size = 8, normalized size = 0.67 \begin {gather*} \frac {\log {\left (3 x^{4} + 2 \right )}}{12} \end {gather*}

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 = 0.51, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{12} \, \log \left (3 \, x^{4} + 2\right ) \end {gather*}

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)

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Mupad [B]
time = 0.98, size = 8, normalized size = 0.67 \begin {gather*} \frac {\ln \left (x^4+\frac {2}{3}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(3*x^4 + 2),x)

[Out]

log(x^4 + 2/3)/12

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