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

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

[Out]

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

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Rubi [A]  time = 0.0598604, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.164973, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^2\right )\right )^p}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{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^(1/2))^2))^p/x,x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*(d + e/sqrt(x))^2) + a)^p/x, 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 \, c d e \sqrt{x} + c e^{2}}{x}\right ) + a\right )}^{p}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log((c*d^2*x + 2*c*d*e*sqrt(x) + c*e^2)/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((a+b*ln(c*(d+e/x**(1/2))**2))**p/x,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}{\sqrt{x}}\right )}^{2}\right ) + a\right )}^{p}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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