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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0449333, 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 = {} \[ \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.64589, size = 0, normalized size = 0. \[ \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.483, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (\frac{c d x + c e \sqrt{x}}{x}\right ) + a\right )}^{p} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)