\(\int \sqrt {\frac {b}{x}} \, dx\) [80]

Optimal result

Integrand size = 9, antiderivative size = 12 \[ \int \sqrt {\frac {b}{x}} \, dx=2 \sqrt {\frac {b}{x}} x \]

[Out]

2*x*(b/x)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt {\frac {b}{x}} \, dx=2 x \sqrt {\frac {b}{x}} \]

[In]

Int[Sqrt[b/x],x]

[Out]

2*Sqrt[b/x]*x

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

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

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\frac {b}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x}} \, dx \\ & = 2 \sqrt {\frac {b}{x}} x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \sqrt {\frac {b}{x}} \, dx=2 \sqrt {\frac {b}{x}} x \]

[In]

Integrate[Sqrt[b/x],x]

[Out]

2*Sqrt[b/x]*x

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
default	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
trager	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)
risch	\(2 x \sqrt {\frac {b}{x}}\)	\(11\)

[In]

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

[Out]

2*x*(b/x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x}} \, dx=2 \, x \sqrt {\frac {b}{x}} \]

[In]

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

[Out]

2*x*sqrt(b/x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \sqrt {\frac {b}{x}} \, dx=2 x \sqrt {\frac {b}{x}} \]

[In]

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

[Out]

2*x*sqrt(b/x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x}} \, dx=2 \, x \sqrt {\frac {b}{x}} \]

[In]

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

[Out]

2*x*sqrt(b/x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.75 \[ \int \sqrt {\frac {b}{x}} \, dx=2 \, \sqrt {b x} \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

2*sqrt(b*x)*sgn(x)

Mupad [B] (verification not implemented)

Time = 5.31 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \sqrt {\frac {b}{x}} \, dx=2\,x\,\sqrt {\frac {b}{x}} \]

[In]

int((b/x)^(1/2),x)

[Out]

2*x*(b/x)^(1/2)