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

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

[Out]

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

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Rubi [A]  time = 0.0469373, 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 = {} \[ \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*(a + b*Log[c*(d + e/x^(1/3))^2])^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*} \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \, dx &=3 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c \left (d+\frac{e}{x}\right )^2\right )\right )^p \, dx,x,\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.228788, size = 0, normalized size = 0. \[ \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^2\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.574, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \right ) ^{2} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{2}\right ) + a\right )}^{p} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (\frac{c d^{2} x + 2 \, c d e x^{\frac{2}{3}} + c e^{2} x^{\frac{1}{3}}}{x}\right ) + a\right )}^{p} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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*(a+b*ln(c*(d+e/x**(1/3))**2))**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 (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{2}\right ) + a\right )}^{p} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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