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

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^2\right )\right )^p}{x^2} \, dx \]

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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fricas [A]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c e^{2} x^{\frac {2}{3}} + 2 \, c d e x^{\frac {1}{3}} + c d^{2}\right ) + a\right )}^{p}}{x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p}}{x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (e \,x^{\frac {1}{3}}+d \right )^{2} c \right )+a \right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{2} c\right ) + a\right )}^{p}}{x^{2}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^2\right )\right )}^p}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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