\(\int x (a+b \log (c (d+\frac {e}{x^{2/3}})))^p \, dx\) [596]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\text {Int}\left (x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p,x\right ) \]

[Out]

Unintegrable(x*(a+b*ln(c*(d+e/x^(2/3))))^p,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx \]

[In]

Int[x*(a + b*Log[c*(d + e/x^(2/3))])^p,x]

[Out]

3*Defer[Subst][Defer[Int][x^5*(a + b*Log[c*(d + e/x^2)])^p, x], x, x^(1/3)]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )\right )\right )^p \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx \]

[In]

Integrate[x*(a + b*Log[c*(d + e/x^(2/3))])^p,x]

[Out]

Integrate[x*(a + b*Log[c*(d + e/x^(2/3))])^p, x]

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )\right )\right )}^{p}d x\]

[In]

int(x*(a+b*ln(c*(d+e/x^(2/3))))^p,x)

[Out]

int(x*(a+b*ln(c*(d+e/x^(2/3))))^p,x)

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}\right ) + a\right )}^{p} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))))^p,x, algorithm="fricas")

[Out]

integral((b*log((c*d*x + c*e*x^(1/3))/x) + a)^p*x, x)

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\text {Timed out} \]

[In]

integrate(x*(a+b*ln(c*(d+e/x**(2/3))))**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}\right ) + a\right )}^{p} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))))^p,x, algorithm="maxima")

[Out]

integrate((b*log(c*(d + e/x^(2/3))) + a)^p*x, x)

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}\right ) + a\right )}^{p} x \,d x } \]

[In]

integrate(x*(a+b*log(c*(d+e/x^(2/3))))^p,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(2/3))) + a)^p*x, x)

Mupad [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,\left (d+\frac {e}{x^{2/3}}\right )\right )\right )}^p \,d x \]

[In]

int(x*(a + b*log(c*(d + e/x^(2/3))))^p,x)

[Out]

int(x*(a + b*log(c*(d + e/x^(2/3))))^p, x)