3.2385 \(\int \frac{1}{(1+x^{2/3}) \sqrt [3]{x}} \, dx\)

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

[Out]

(3*Log[1 + x^(2/3)])/2

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Rubi [A]  time = 0.0027598, antiderivative size = 12, normalized size of antiderivative = 1., 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 = {260} \[ \frac{3}{2} \log \left (x^{2/3}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^(2/3))*x^(1/3)),x]

[Out]

(3*Log[1 + x^(2/3)])/2

Rule 260

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{1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx &=\frac{3}{2} \log \left (1+x^{2/3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0013949, size = 12, normalized size = 1. \[ \frac{3}{2} \log \left (x^{2/3}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Log[1 + x^(2/3)])/2

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Maple [A]  time = 0.003, size = 9, normalized size = 0.8 \begin{align*}{\frac{3}{2}\ln \left ( 1+{x}^{{\frac{2}{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+x^(2/3))/x^(1/3),x)

[Out]

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

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Maxima [A]  time = 0.97963, size = 11, normalized size = 0.92 \begin{align*} \frac{3}{2} \, \log \left (x^{\frac{2}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*log(x^(2/3) + 1)

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Fricas [A]  time = 1.45701, size = 30, normalized size = 2.5 \begin{align*} \frac{3}{2} \, \log \left (x^{\frac{2}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*log(x^(2/3) + 1)

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Sympy [A]  time = 0.163284, size = 10, normalized size = 0.83 \begin{align*} \frac{3 \log{\left (x^{\frac{2}{3}} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x**(2/3) + 1)/2

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Giac [A]  time = 1.11974, size = 11, normalized size = 0.92 \begin{align*} \frac{3}{2} \, \log \left (x^{\frac{2}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*log(x^(2/3) + 1)