\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^2))^p}{x} \, dx\) [590]

Optimal result

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

[Out]

Unintegrable((a+b*ln(c*(d+e/x^(1/3))^2))^p/x,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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx \]

[In]

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

[Out]

3*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e/x)^2])^p/x, 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}\right )^2\right )\right )^p}{x} \, dx,x,\sqrt [3]{x}\right ) \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{2}\right )\right )}^{p}}{x}d x\]

[In]

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^2))^p/x,x)

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x,x, algorithm="fricas")

[Out]

integral((b*log((c*d^2*x + 2*c*d*e*x^(2/3) + c*e^2*x^(1/3))/x) + a)^p/x, x)

Sympy [F(-1)]

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

[In]

integrate((a+b*ln(c*(d+e/x**(1/3))**2))**p/x,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x,x, algorithm="maxima")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x, x)

Giac [N/A]

Not integrable

Time = 1.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int { \frac {{\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{2}\right ) + a\right )}^{p}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(d+e/x^(1/3))^2))^p/x,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^2) + a)^p/x, x)

Mupad [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^2\right )\right )^p}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^2\right )\right )}^p}{x} \,d x \]

[In]

int((a + b*log(c*(d + e/x^(1/3))^2))^p/x,x)

[Out]

int((a + b*log(c*(d + e/x^(1/3))^2))^p/x, x)