\(\int (a+b \log (c (d+\frac {e}{\sqrt {x}})))^p \, dx\) [545]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\text {Int}\left (\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )\right )\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}\right ) + a\right )}^{p} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )\right )\right )^p \, dx=\int {\left (a+b\,\ln \left (c\,\left (d+\frac {e}{\sqrt {x}}\right )\right )\right )}^p \,d x \]

[In]

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

[Out]

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