3.2.55 \(\int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx\)

Optimal. Leaf size=18 \[ -\frac {3 \left (x^6+x^2\right )^{2/3}}{8 x^4} \]

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} -\frac {3 \left (x^6+x^2\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(x^2 + x^6)^(1/3)),x]

[Out]

(-3*(x^2 + x^6)^(2/3))/(8*x^4)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx &=-\frac {3 \left (x^2+x^6\right )^{2/3}}{8 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^6+x^2\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(x^2 + x^6)^(1/3)),x]

[Out]

(-3*(x^2 + x^6)^(2/3))/(8*x^4)

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IntegrateAlgebraic [A]  time = 0.31, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^2+x^6\right )^{2/3}}{8 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^3*(x^2 + x^6)^(1/3)),x]

[Out]

(-3*(x^2 + x^6)^(2/3))/(8*x^4)

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fricas [A]  time = 0.47, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6+x^2)^(1/3),x, algorithm="fricas")

[Out]

-3/8*(x^6 + x^2)^(2/3)/x^4

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giac [A]  time = 0.61, size = 9, normalized size = 0.50 \begin {gather*} -\frac {3}{8} \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6+x^2)^(1/3),x, algorithm="giac")

[Out]

-3/8*(1/x^4 + 1)^(2/3)

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maple [A]  time = 0.06, size = 13, normalized size = 0.72

method	result	size
meijerg	\(-\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{8 x^{\frac {8}{3}}}\)	\(13\)
trager	\(-\frac {3 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}}{8 x^{4}}\)	\(15\)
gosper	\(-\frac {3 \left (x^{4}+1\right )}{8 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}}\)	\(20\)
risch	\(-\frac {3 \left (x^{4}+1\right )}{8 x^{2} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(x^6+x^2)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-3/8*(x^4+1)^(2/3)/x^(8/3)

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maxima [A]  time = 0.69, size = 21, normalized size = 1.17 \begin {gather*} -\frac {3 \, {\left (x^{6} + x^{2}\right )}}{8 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {7}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(x^6+x^2)^(1/3),x, algorithm="maxima")

[Out]

-3/8*(x^6 + x^2)/((x^4 + 1)^(1/3)*(x^2)^(7/3))

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mupad [B]  time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3\,{\left (x^6+x^2\right )}^{2/3}}{8\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(x^2 + x^6)^(1/3)),x)

[Out]

-(3*(x^2 + x^6)^(2/3))/(8*x^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x^{4} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(x**6+x**2)**(1/3),x)

[Out]

Integral(1/(x**3*(x**2*(x**4 + 1))**(1/3)), x)

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