3.1779 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} \sqrt{x}} \, dx\)

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 = 0.0247353, 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 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x])/a

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.08494, size = 15, normalized size = 0.71 \[ \frac{2 \sqrt{x} \sqrt{a + \frac{b}{x}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x)**(1/2)/x**(1/2),x)

[Out]

2*sqrt(x)*sqrt(a + b/x)/a

_______________________________________________________________________________________

Mathematica [A]  time = 0.0276699, size = 23, normalized size = 1.1 \[ \frac{2 \sqrt{x} \sqrt{\frac{a x+b}{x}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*Sqrt[(b + a*x)/x])/a

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 1.2 \[ 2\,{\frac{ax+b}{a\sqrt{x}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

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 = 1.44128, size = 23, normalized size = 1.1 \[ \frac{2 \, \sqrt{a + \frac{b}{x}} \sqrt{x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*sqrt(x)),x, algorithm="maxima")

[Out]

2*sqrt(a + b/x)*sqrt(x)/a

_______________________________________________________________________________________

Fricas [A]  time = 0.229055, size = 26, normalized size = 1.24 \[ \frac{2 \, \sqrt{x} \sqrt{\frac{a x + b}{x}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*sqrt(x)),x, algorithm="fricas")

[Out]

2*sqrt(x)*sqrt((a*x + b)/x)/a

_______________________________________________________________________________________

Sympy [A]  time = 8.94276, size = 17, normalized size = 0.81 \[ \frac{2 \sqrt{b} \sqrt{\frac{a x}{b} + 1}}{a} \]

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/XCAS [A]  time = 0.23001, size = 28, normalized size = 1.33 \[ \frac{2 \, \sqrt{a x + b}}{a} - \frac{2 \, \sqrt{b}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*sqrt(x)),x, algorithm="giac")

[Out]

2*sqrt(a*x + b)/a - 2*sqrt(b)/a