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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x} \, dx \]

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 = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p}{x} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (\frac {c d^{2} x + 2 \, c d e \sqrt {x} + c e^{2}}{x}\right ) + a\right )}^{p}}{x}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (\left (d +\frac {e}{\sqrt {x}}\right )^{2} c \right )+a \right )^{p}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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**(1/2))**2))**p/x,x)

[Out]

Timed out

________________________________________________________________________________________