3.130 \(\int \frac {\sqrt {1+\log (x)}}{x} \, dx\)

Optimal. Leaf size=12 \[ \frac {2}{3} (\log (x)+1)^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2302, 30} \[ \frac {2}{3} (\log (x)+1)^{3/2} \]

Antiderivative was successfully verified.

[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 2302

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

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \[ \frac {2}{3} (\log (x)+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Log[x]]/x,x]

[Out]

(2*(1 + Log[x])^(3/2))/3

________________________________________________________________________________________

fricas [A]  time = 1.95, size = 8, normalized size = 0.67 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(log(x) + 1)^(3/2)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 8, normalized size = 0.67 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(log(x) + 1)^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 9, normalized size = 0.75 \[ \frac {2 \left (\ln \relax (x )+1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+ln(x))^(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.72, size = 8, normalized size = 0.67 \[ \frac {2}{3} \, {\left (\log \relax (x) + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(log(x) + 1)^(3/2)

________________________________________________________________________________________

mupad [B]  time = 0.39, size = 13, normalized size = 1.08 \[ \sqrt {\ln \relax (x)+1}\,\left (\frac {2\,\ln \relax (x)}{3}+\frac {2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) + 1)^(1/2)/x,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.80, size = 10, normalized size = 0.83 \[ \frac {2 \left (\log {\relax (x )} + 1\right )^{\frac {3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(log(x) + 1)**(3/2)/3

________________________________________________________________________________________