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

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0547608, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.719293, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c e x^{\frac{2}{3}} + c d\right ) + a\right )}^{p} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x^{\frac{2}{3}} + d\right )} c\right ) + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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