\(\int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx\) [1781]

Optimal result

Integrand size = 17, antiderivative size = 21 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x}}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {270} \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}}}{a} \]

[In]

Int[1/(Sqrt[a + b/x]*Sqrt[x]),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x}}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x}}{a} \]

[In]

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{a}\)	\(20\)
gosper	\(\frac {2 a x +2 b}{a \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}\)	\(25\)
risch	\(\frac {2 a x +2 b}{a \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}\)	\(25\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \, \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a} \]

[In]

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

[Out]

2*sqrt(x)*sqrt((a*x + b)/x)/a

Sympy [A] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \sqrt {b} \sqrt {\frac {a x}{b} + 1}}{a} \]

[In]

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

[Out]

2*sqrt(b)*sqrt(a*x/b + 1)/a

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \, \sqrt {a + \frac {b}{x}} \sqrt {x}}{a} \]

[In]

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

[Out]

2*sqrt(a + b/x)*sqrt(x)/a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2 \, {\left (\frac {\sqrt {a x + b}}{a} - \frac {\sqrt {b}}{a}\right )}}{\mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

2*(sqrt(a*x + b)/a - sqrt(b)/a)/sgn(x)

Mupad [B] (verification not implemented)

Time = 6.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}} \, dx=\frac {2\,\sqrt {x}\,\sqrt {a+\frac {b}{x}}}{a} \]

[In]

int(1/(x^(1/2)*(a + b/x)^(1/2)),x)

[Out]

(2*x^(1/2)*(a + b/x)^(1/2))/a