3.6.46 \(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})))^p}{x} \, dx\) [546]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

2*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e/x)])^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 )\right )\right )^p}{x} \, dx &=2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x}\right )\right )\right )^p}{x} \, dx,x,\sqrt {x}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )\right )\right )^{p}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________