\(\int \frac {1}{(1+x^{2/3}) \sqrt [3]{x}} \, dx\) [2385]

Optimal result

Integrand size = 15, antiderivative size = 12 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \log \left (1+x^{2/3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266} \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \log \left (x^{2/3}+1\right ) \]

[In]

Int[1/((1 + x^(2/3))*x^(1/3)),x]

[Out]

(3*Log[1 + x^(2/3)])/2

Rule 266

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*} \text {integral}& = \frac {3}{2} \log \left (1+x^{2/3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \log \left (1+x^{2/3}\right ) \]

[In]

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

[Out]

(3*Log[1 + x^(2/3)])/2

Maple [A] (verified)

Time = 5.98 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {3 \ln \left (1+x^{\frac {2}{3}}\right )}{2}\)	\(9\)
default	\(\frac {3 \ln \left (1+x^{\frac {2}{3}}\right )}{2}\)	\(9\)
meijerg	\(\frac {3 \ln \left (1+x^{\frac {2}{3}}\right )}{2}\)	\(9\)
trager	\(\frac {\ln \left (3 x^{\frac {2}{3}}+3 x^{\frac {4}{3}}+x^{2}+1\right )}{2}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \, \log \left (x^{\frac {2}{3}} + 1\right ) \]

[In]

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

[Out]

3/2*log(x^(2/3) + 1)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3 \log {\left (x^{\frac {2}{3}} + 1 \right )}}{2} \]

[In]

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

[Out]

3*log(x**(2/3) + 1)/2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \, \log \left (x^{\frac {2}{3}} + 1\right ) \]

[In]

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

[Out]

3/2*log(x^(2/3) + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3}{2} \, \log \left (x^{\frac {2}{3}} + 1\right ) \]

[In]

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

[Out]

3/2*log(x^(2/3) + 1)

Mupad [B] (verification not implemented)

Time = 5.84 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (1+x^{2/3}\right ) \sqrt [3]{x}} \, dx=\frac {3\,\ln \left (x^{2/3}+1\right )}{2} \]

[In]

int(1/(x^(1/3)*(x^(2/3) + 1)),x)

[Out]

(3*log(x^(2/3) + 1))/2