\(\int x^2 (a+b \log (c (d+e x^{2/3})^2))^p \, dx\) [579]

Optimal result

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

[Out]

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

[In]

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

[Out]

3*Defer[Subst][Defer[Int][x^8*(a + b*Log[c*(d + e*x^2)^2])^p, x], x, x^(1/3)]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^2\right )\right )^p \, 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 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{2} c\right ) + a\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^2\right )\right )^p \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^2\right )\right )}^p \,d x \]

[In]

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

[Out]

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