\(\int \frac {\sqrt {1+\log (x)}}{x} \, dx\) [130]

Optimal result

Integrand size = 12, antiderivative size = 12 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} (1+\log (x))^{3/2} \]

[Out]

2/3*(1+ln(x))^(3/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2339, 30} \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} (\log (x)+1)^{3/2} \]

[In]

Int[Sqrt[1 + Log[x]]/x,x]

[Out]

(2*(1 + Log[x])^(3/2))/3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {x} \, dx,x,1+\log (x)\right ) \\ & = \frac {2}{3} (1+\log (x))^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} (1+\log (x))^{3/2} \]

[In]

Integrate[Sqrt[1 + Log[x]]/x,x]

[Out]

(2*(1 + Log[x])^(3/2))/3

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75

method	result	size
derivativedivides	\(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}\)	\(9\)
default	\(\frac {2 \left (1+\ln \left (x \right )\right )^{\frac {3}{2}}}{3}\)	\(9\)

[In]

int((1+ln(x))^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

2/3*(1+ln(x))^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} \]

[In]

integrate((1+log(x))^(1/2)/x,x, algorithm="fricas")

[Out]

2/3*(log(x) + 1)^(3/2)

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2 \left (\log {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} \]

[In]

integrate((1+ln(x))**(1/2)/x,x)

[Out]

2*(log(x) + 1)**(3/2)/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} \]

[In]

integrate((1+log(x))^(1/2)/x,x, algorithm="maxima")

[Out]

2/3*(log(x) + 1)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\frac {2}{3} \, {\left (\log \left (x\right ) + 1\right )}^{\frac {3}{2}} \]

[In]

integrate((1+log(x))^(1/2)/x,x, algorithm="giac")

[Out]

2/3*(log(x) + 1)^(3/2)

Mupad [B] (verification not implemented)

Time = 1.84 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {1+\log (x)}}{x} \, dx=\sqrt {\ln \left (x\right )+1}\,\left (\frac {2\,\ln \left (x\right )}{3}+\frac {2}{3}\right ) \]

[In]

int((log(x) + 1)^(1/2)/x,x)

[Out]

(log(x) + 1)^(1/2)*((2*log(x))/3 + 2/3)