3.273 \(\int \log (\sqrt{x}+x) \, dx\)

Optimal. Leaf size=29 \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]

[Out]

Sqrt[x] - x - Log[1 + Sqrt[x]] + x*Log[Sqrt[x] + x]

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Rubi [A]  time = 0.0180576, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2548, 376, 77} \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[Sqrt[x] + x],x]

[Out]

Sqrt[x] - x - Log[1 + Sqrt[x]] + x*Log[Sqrt[x] + x]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0116458, size = 29, normalized size = 1. \[ -x+\sqrt{x}+x \log \left (x+\sqrt{x}\right )-\log \left (\sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[Sqrt[x] + x],x]

[Out]

Sqrt[x] - x - Log[1 + Sqrt[x]] + x*Log[Sqrt[x] + x]

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Maple [A]  time = 0.002, size = 24, normalized size = 0.8 \begin{align*} -x-\ln \left ( 1+\sqrt{x} \right ) +x\ln \left ( x+\sqrt{x} \right ) +\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03459, size = 31, normalized size = 1.07 \begin{align*} x \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x + sqrt(x)) - x + sqrt(x) - log(sqrt(x) + 1)

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Fricas [A]  time = 2.23564, size = 103, normalized size = 3.55 \begin{align*}{\left (x + 1\right )} \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - 2 \, \log \left (\sqrt{x} + 1\right ) - \log \left (\sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*log(x + sqrt(x)) - x + sqrt(x) - 2*log(sqrt(x) + 1) - log(sqrt(x))

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Sympy [A]  time = 4.50717, size = 24, normalized size = 0.83 \begin{align*} \sqrt{x} + x \log{\left (\sqrt{x} + x \right )} - x - \log{\left (\sqrt{x} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x) + x*log(sqrt(x) + x) - x - log(sqrt(x) + 1)

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Giac [A]  time = 1.34515, size = 31, normalized size = 1.07 \begin{align*} x \log \left (x + \sqrt{x}\right ) - x + \sqrt{x} - \log \left (\sqrt{x} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x + sqrt(x)) - x + sqrt(x) - log(sqrt(x) + 1)