\(\int x (a+b \log (c (d+\frac {e}{\sqrt {x}})^2))^p \, dx\) [550]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\text {Int}\left (x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, 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 = {} \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx \]

[In]

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

[Out]

2*Defer[Subst][Defer[Int][x^3*(a + b*Log[c*(d + e/x)^2])^p, x], x, Sqrt[x]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^2\right )\right )^p \, dx,x,\sqrt {x}\right ) \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{2}\right )\right )}^{p}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p} x \,d x } \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 2.53 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{2}\right ) + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^2\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^2\right )\right )}^p \,d x \]

[In]

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

[Out]

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