3.1781 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx\)

Optimal. Leaf size=21 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{a} \]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

________________________________________________________________________________________

Rubi [A]  time = 0.005363, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*Sqrt[x]),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

Rule 264

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*} \int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx &=\frac{2 \sqrt{a+\frac{b}{x}} \sqrt{x}}{a}\\ \end{align*}

Mathematica [A]  time = 0.0089242, size = 21, normalized size = 1. \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 1.2 \begin{align*} 2\,{\frac{ax+b}{a\sqrt{x}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.96491, size = 23, normalized size = 1.1 \begin{align*} \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a + b/x)*sqrt(x)/a

________________________________________________________________________________________

Fricas [A]  time = 1.46166, size = 42, normalized size = 2. \begin{align*} \frac{2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 2.08187, size = 17, normalized size = 0.81 \begin{align*} \frac{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b)*sqrt(a*x/b + 1)/a

________________________________________________________________________________________

Giac [A]  time = 1.15257, size = 28, normalized size = 1.33 \begin{align*} \frac{2 \, \sqrt{a x + b}}{a} - \frac{2 \, \sqrt{b}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a*x + b)/a - 2*sqrt(b)/a