\(\int \frac {(a+b \log (c (d+\frac {e}{x^{2/3}})^2))^p}{x^2} \, dx\) [605]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx \]

[In]

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

[Out]

3*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e/x^2)^2])^p/x^4, x], x, x^(1/3)]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^2}\right )^2\right )\right )^p}{x^4} \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral((b*log((c*d^2*x^2 + 2*c*d*e*x^(4/3) + c*e^2*x^(2/3))/x^2) + a)^p/x^2, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^2\right )\right )}^p}{x^2} \,d x \]

[In]

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

[Out]

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