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

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

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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^3+x^2\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*(x^2 + x^3)^(1/3)),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 (x+1)}{2 \sqrt [3]{x^2 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(x^2 + x^3)^(1/3)),x]

[Out]

(-3*(1 + x))/(2*(x^2*(1 + x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.12, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^2+x^3\right )^{2/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(x^2 + x^3)^(2/3))/(2*x^2)

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fricas [A]  time = 0.43, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3 \, {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x^3 + x^2)^(2/3)/x^2

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giac [A]  time = 0.38, size = 9, normalized size = 0.50 \begin {gather*} -\frac {3}{2} \, {\left (\frac {1}{x} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(1/x + 1)^(2/3)

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maple [A]  time = 0.05, size = 11, normalized size = 0.61

method	result	size
meijerg	\(-\frac {3 \left (1+x \right )^{\frac {2}{3}}}{2 x^{\frac {2}{3}}}\)	\(11\)
gosper	\(-\frac {3 \left (1+x \right )}{2 \left (x^{3}+x^{2}\right )^{\frac {1}{3}}}\)	\(15\)
trager	\(-\frac {3 \left (x^{3}+x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(15\)
risch	\(-\frac {3 \left (1+x \right )}{2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((x^3 + x^2)^(1/3)*x), x)

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mupad [B]  time = 0.17, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3\,{\left (x^3+x^2\right )}^{2/3}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(x^2 + x^3)^(1/3)),x)

[Out]

-(3*(x^2 + x^3)^(2/3))/(2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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