3.1753 \(\int \sqrt{a+\frac{b}{x}} \sqrt{x} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{3 a} \]

[Out]

(2*(a + b/x)^(3/2)*x^(3/2))/(3*a)

_______________________________________________________________________________________

Rubi [A]  time = 0.0247132, antiderivative size = 23, 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 \[ \frac{2 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}{3 a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]*Sqrt[x],x]

[Out]

(2*(a + b/x)^(3/2)*x^(3/2))/(3*a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.69183, size = 17, normalized size = 0.74 \[ \frac{2 x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)*(a + b/x)**(3/2)/(3*a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0292823, size = 28, normalized size = 1.22 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)}{3 a} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x))/(3*a)

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 1.1 \[{\frac{2\,ax+2\,b}{3\,a}\sqrt{{\frac{ax+b}{x}}}\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)*x^(1/2),x)

[Out]

2/3*(a*x+b)*((a*x+b)/x)^(1/2)*x^(1/2)/a

_______________________________________________________________________________________

Maxima [A]  time = 1.46116, size = 23, normalized size = 1. \[ \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a + b/x)^(3/2)*x^(3/2)/a

_______________________________________________________________________________________

Fricas [A]  time = 0.234832, size = 32, normalized size = 1.39 \[ \frac{2 \,{\left (a x + b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{3 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(a*x + b)*sqrt(x)*sqrt((a*x + b)/x)/a

_______________________________________________________________________________________

Sympy [A]  time = 9.01955, size = 39, normalized size = 1.7 \[ \frac{2 \sqrt{b} x \sqrt{\frac{a x}{b} + 1}}{3} + \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b)*x*sqrt(a*x/b + 1)/3 + 2*b**(3/2)*sqrt(a*x/b + 1)/(3*a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225942, size = 32, normalized size = 1.39 \[ \frac{2}{3} \,{\left (\frac{{\left (a x + b\right )}^{\frac{3}{2}}}{a} - \frac{b^{\frac{3}{2}}}{a}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*((a*x + b)^(3/2)/a - b^(3/2)/a)*sign(x)