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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^2\right )\right )^p}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0553495, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx &=3 \operatorname{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 [A]  time = 0.144324, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^2\right )\right )^p}{x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.312, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{2} \right ) \right ) ^{p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (\frac{c d^{2} x^{2} + 2 \, c d e x^{\frac{4}{3}} + c e^{2} x^{\frac{2}{3}}}{x^{2}}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)