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

Optimal. Leaf size=16 \[ -\frac {3 \left (x^3+x\right )^{2/3}}{4 x^2} \]

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Rubi [A]  time = 0.02, antiderivative size = 16, 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 = {2014} \begin {gather*} -\frac {3 \left (x^3+x\right )^{2/3}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(x + x^3)^(1/3)),x]

[Out]

(-3*(x + x^3)^(2/3))/(4*x^2)

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^3+x\right )^{2/3}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(x + x^3)^(1/3)),x]

[Out]

(-3*(x + x^3)^(2/3))/(4*x^2)

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IntegrateAlgebraic [A]  time = 0.14, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x+x^3\right )^{2/3}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(x + x^3)^(2/3))/(4*x^2)

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fricas [A]  time = 0.44, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(x^3 + x)^(2/3)/x^2

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giac [A]  time = 0.51, size = 9, normalized size = 0.56 \begin {gather*} -\frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(1/x^2 + 1)^(2/3)

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maple [A]  time = 0.05, size = 13, normalized size = 0.81

method	result	size
trager	\(-\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}}}{4 x^{2}}\)	\(13\)
meijerg	\(-\frac {3 \left (x^{2}+1\right )^{\frac {2}{3}}}{4 x^{\frac {4}{3}}}\)	\(13\)
gosper	\(-\frac {3 \left (x^{2}+1\right )}{4 x \left (x^{3}+x \right )^{\frac {1}{3}}}\)	\(18\)
risch	\(-\frac {3 \left (x^{2}+1\right )}{4 x \left (\left (x^{2}+1\right ) x \right )^{\frac {1}{3}}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 17, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{3} + x\right )}}{4 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{\frac {7}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3\,{\left (x^3+x\right )}^{2/3}}{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(x + x^3)^(1/3)),x)

[Out]

-(3*(x + x^3)^(2/3))/(4*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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