\(\int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx\) [2383]

Optimal result

Integrand size = 15, antiderivative size = 39 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-x+\frac {3 x^{4/3}}{4}+3 \log \left (1+\sqrt [3]{x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {3 x^{4/3}}{4}+\frac {3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]

[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 45

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])

Rule 272

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]]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^4}{1+x} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {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] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {1}{4} \sqrt [3]{x} \left (-12+6 \sqrt [3]{x}-4 x^{2/3}+3 x\right )+3 \log \left (1+\sqrt [3]{x}\right ) \]

[In]

Integrate[x^(2/3)/(1 + x^(1/3)),x]

[Out]

(x^(1/3)*(-12 + 6*x^(1/3) - 4*x^(2/3) + 3*x))/4 + 3*Log[1 + x^(1/3)]

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(-3 x^{\frac {1}{3}}+\frac {3 x^{\frac {2}{3}}}{2}-x +\frac {3 x^{\frac {4}{3}}}{4}+3 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(28\)
default	\(-3 x^{\frac {1}{3}}+\frac {3 x^{\frac {2}{3}}}{2}-x +\frac {3 x^{\frac {4}{3}}}{4}+3 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(28\)
meijerg	\(-\frac {x^{\frac {1}{3}} \left (-15 x +20 x^{\frac {2}{3}}-30 x^{\frac {1}{3}}+60\right )}{20}+3 \ln \left (1+x^{\frac {1}{3}}\right )\)	\(30\)
trager	\(1-x +\left (\frac {3 x}{4}-3\right ) x^{\frac {1}{3}}+\frac {3 x^{\frac {2}{3}}}{2}+\ln \left (-3 x^{\frac {2}{3}}-3 x^{\frac {1}{3}}-x -1\right )\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {3}{4} \, {\left (x - 4\right )} x^{\frac {1}{3}} - x + \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {3 x^{\frac {4}{3}}}{4} + \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} - x + 3 \log {\left (\sqrt [3]{x} + 1 \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {3}{4} \, {\left (x^{\frac {1}{3}} + 1\right )}^{4} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{3} + 9 \, {\left (x^{\frac {1}{3}} + 1\right )}^{2} - 12 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) - 12 \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=\frac {3}{4} \, x^{\frac {4}{3}} - x + \frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx=3\,\ln \left (x^{1/3}+1\right )-x-3\,x^{1/3}+\frac {3\,x^{2/3}}{2}+\frac {3\,x^{4/3}}{4} \]

[In]

int(x^(2/3)/(x^(1/3) + 1),x)

[Out]

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