3.2383 \(\int \frac{x^{2/3}}{1+\sqrt [3]{x}} \, dx\)

Optimal. Leaf size=39 \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

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Rubi [A]  time = 0.0155377, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^(2/3)/(1 + x^(1/3)),x]

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0132745, size = 39, normalized size = 1. \[ \frac{3 x^{4/3}}{4}+\frac{3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(2/3)/(1 + x^(1/3)),x]

[Out]

-3*x^(1/3) + (3*x^(2/3))/2 - x + (3*x^(4/3))/4 + 3*Log[1 + x^(1/3)]

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Maple [A]  time = 0.003, size = 28, normalized size = 0.7 \begin{align*} -3\,\sqrt [3]{x}+{\frac{3}{2}{x}^{{\frac{2}{3}}}}-x+{\frac{3}{4}{x}^{{\frac{4}{3}}}}+3\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01202, size = 57, normalized size = 1.46 \begin{align*} \frac{3}{4} \,{\left (x^{\frac{1}{3}} + 1\right )}^{4} - 4 \,{\left (x^{\frac{1}{3}} + 1\right )}^{3} + 9 \,{\left (x^{\frac{1}{3}} + 1\right )}^{2} - 12 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) - 12 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^(1/3) + 1)^4 - 4*(x^(1/3) + 1)^3 + 9*(x^(1/3) + 1)^2 - 12*x^(1/3) + 3*log(x^(1/3) + 1) - 12

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Fricas [A]  time = 1.5236, size = 81, normalized size = 2.08 \begin{align*} \frac{3}{4} \,{\left (x - 4\right )} x^{\frac{1}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.206077, size = 34, normalized size = 0.87 \begin{align*} \frac{3 x^{\frac{4}{3}}}{4} + \frac{3 x^{\frac{2}{3}}}{2} - 3 \sqrt [3]{x} - x + 3 \log{\left (\sqrt [3]{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x**(4/3)/4 + 3*x**(2/3)/2 - 3*x**(1/3) - x + 3*log(x**(1/3) + 1)

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Giac [A]  time = 1.19845, size = 36, normalized size = 0.92 \begin{align*} \frac{3}{4} \, x^{\frac{4}{3}} - x + \frac{3}{2} \, x^{\frac{2}{3}} - 3 \, x^{\frac{1}{3}} + 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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