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

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

[Out]

(2*(1 + Log[x])^(3/2))/3

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Rubi [A]  time = 0.0256897, antiderivative size = 12, normalized size of antiderivative = 1., 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 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]

Rule 30

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

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 = 0.0037376, size = 12, normalized size = 1. \[ \frac{2}{3} (\log (x)+1)^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + Log[x]]/x,x]

[Out]

(2*(1 + Log[x])^(3/2))/3

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Maple [A]  time = 0.006, size = 9, normalized size = 0.8 \begin{align*}{\frac{2}{3} \left ( 1+\ln \left ( x \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

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)

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Maxima [A]  time = 1.11695, size = 11, normalized size = 0.92 \begin{align*} \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} \end{align*}

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)

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Fricas [A]  time = 2.04066, size = 32, normalized size = 2.67 \begin{align*} \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} \end{align*}

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)

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Sympy [A]  time = 0.738572, size = 10, normalized size = 0.83 \begin{align*} \frac{2 \left (\log{\left (x \right )} + 1\right )^{\frac{3}{2}}}{3} \end{align*}

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

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Giac [A]  time = 1.29265, size = 11, normalized size = 0.92 \begin{align*} \frac{2}{3} \,{\left (\log \left (x\right ) + 1\right )}^{\frac{3}{2}} \end{align*}

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)