3.7.99 \(\int \frac {x^3}{(2+3 x^4)^2} \, dx\) [699]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, 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 = {267} \begin {gather*} -\frac {1}{12 \left (3 x^4+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(2 + 3*x^4)^2,x]

[Out]

-1/12*1/(2 + 3*x^4)

Rule 267

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*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{12 \left (2+3 x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(2 + 3*x^4)^2,x]

[Out]

-1/12*1/(2 + 3*x^4)

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Maple [A]
time = 0.13, size = 12, normalized size = 0.92

method	result	size
risch	\(-\frac {1}{36 \left (x^{4}+\frac {2}{3}\right )}\)	\(10\)
gosper	\(-\frac {1}{12 \left (3 x^{4}+2\right )}\)	\(12\)
derivativedivides	\(-\frac {1}{12 \left (3 x^{4}+2\right )}\)	\(12\)
default	\(-\frac {1}{12 \left (3 x^{4}+2\right )}\)	\(12\)
norman	\(\frac {x^{4}}{24 x^{4}+16}\)	\(15\)
meijerg	\(\frac {x^{4}}{24 x^{4}+16}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{12 \, {\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)^2,x, algorithm="maxima")

[Out]

-1/12/(3*x^4 + 2)

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Fricas [A]
time = 0.35, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{12 \, {\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)^2,x, algorithm="fricas")

[Out]

-1/12/(3*x^4 + 2)

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Sympy [A]
time = 0.03, size = 8, normalized size = 0.62 \begin {gather*} - \frac {1}{36 x^{4} + 24} \end {gather*}

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 = 0.51, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{12 \, {\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)^2,x, algorithm="giac")

[Out]

-1/12/(3*x^4 + 2)

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Mupad [B]
time = 1.00, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{36\,x^4+24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(3*x^4 + 2)^2,x)

[Out]

-1/(36*x^4 + 24)

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