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

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

[Out]

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

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Rubi [A]  time = 0.0537709, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

3*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e*x)^2])^p/x^4, x], x, x^(1/3)]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^2\right )\right )^p}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.115935, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e\sqrt [3]{x} \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^(1/3))^2))^p/x^2,x)

[Out]

int((a+b*ln(c*(d+e*x^(1/3))^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 ({\left (e x^{\frac{1}{3}} + d\right )}^{2} c\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^(1/3))^2))^p/x^2,x, algorithm="maxima")

[Out]

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

[Out]

integral((b*log(c*e^2*x^(2/3) + 2*c*d*e*x^(1/3) + c*d^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**(1/3))**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 ({\left (e x^{\frac{1}{3}} + d\right )}^{2} c\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^(1/3))^2))^p/x^2,x, algorithm="giac")

[Out]

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