\(\int \frac {1}{(1+\sqrt {x}) \sqrt {x}} \, dx\) [2251]

Optimal result

Integrand size = 15, antiderivative size = 10 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \log \left (1+\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266} \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \log \left (\sqrt {x}+1\right ) \]

[In]

Int[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps \begin{align*} \text {integral}& = 2 \log \left (1+\sqrt {x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \log \left (1+\sqrt {x}\right ) \]

[In]

Integrate[1/((1 + Sqrt[x])*Sqrt[x]),x]

[Out]

2*Log[1 + Sqrt[x]]

Maple [A] (verified)

Time = 3.70 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
default	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
meijerg	\(2 \ln \left (\sqrt {x}+1\right )\)	\(9\)
trager	\(\ln \left (x +2 \sqrt {x}+1\right )\)	\(10\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \, \log \left (\sqrt {x} + 1\right ) \]

[In]

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

[Out]

2*log(sqrt(x) + 1)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \log {\left (\sqrt {x} + 1 \right )} \]

[In]

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

[Out]

2*log(sqrt(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \, \log \left (\sqrt {x} + 1\right ) \]

[In]

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

[Out]

2*log(sqrt(x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2 \, \log \left (\sqrt {x} + 1\right ) \]

[In]

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

[Out]

2*log(sqrt(x) + 1)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (1+\sqrt {x}\right ) \sqrt {x}} \, dx=2\,\ln \left (\sqrt {x}+1\right ) \]

[In]

int(1/(x^(1/2)*(x^(1/2) + 1)),x)

[Out]

2*log(x^(1/2) + 1)