\(\int \log (\sqrt {x}+x) \, dx\) [273]

Optimal result

Integrand size = 8, antiderivative size = 29 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=\sqrt {x}-x-\log \left (1+\sqrt {x}\right )+x \log \left (\sqrt {x}+x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2628, 383, 78} \[ \int \log \left (\sqrt {x}+x\right ) \, dx=-x+\sqrt {x}+x \log \left (x+\sqrt {x}\right )-\log \left (\sqrt {x}+1\right ) \]

[In]

Int[Log[Sqrt[x] + x],x]

[Out]

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

Rule 78

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]))))

Rule 383

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 2628

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

Rubi steps \begin{align*} \text {integral}& = 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 \text {Subst}\left (\int \frac {x (1+2 x)}{2+2 x} \, dx,x,\sqrt {x}\right ) \\ & = x \log \left (\sqrt {x}+x\right )-2 \text {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] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=\sqrt {x}-x-\log \left (1+\sqrt {x}\right )+x \log \left (\sqrt {x}+x\right ) \]

[In]

Integrate[Log[Sqrt[x] + x],x]

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-x -\ln \left (1+\sqrt {x}\right )+x \ln \left (x +\sqrt {x}\right )+\sqrt {x}\)	\(24\)
default	\(-x -\ln \left (1+\sqrt {x}\right )+x \ln \left (x +\sqrt {x}\right )+\sqrt {x}\)	\(24\)
parts	\(-x -\ln \left (1+\sqrt {x}\right )+x \ln \left (x +\sqrt {x}\right )+\sqrt {x}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \log \left (\sqrt {x}+x\right ) \, dx={\left (x + 1\right )} \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - 2 \, \log \left (\sqrt {x} + 1\right ) - \log \left (\sqrt {x}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=\sqrt {x} + x \log {\left (\sqrt {x} + x \right )} - x + \log {\left (- \frac {1}{\sqrt {x}} \right )} - \log {\left (-1 - \frac {1}{\sqrt {x}} \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=x \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - \log \left (\sqrt {x} + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=x \log \left (x + \sqrt {x}\right ) - x + \sqrt {x} - \log \left (\sqrt {x} + 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \log \left (\sqrt {x}+x\right ) \, dx=\sqrt {x}-\ln \left (\sqrt {x}+1\right )-x+x\,\ln \left (x+\sqrt {x}\right ) \]

[In]

int(log(x + x^(1/2)),x)

[Out]

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