3.1653 \(\int \left (a+\frac{b}{x}\right )^2 \sqrt{x} \, dx\)

Optimal. Leaf size=32 \[ \frac{2}{3} a^2 x^{3/2}+4 a b \sqrt{x}-\frac{2 b^2}{\sqrt{x}} \]

[Out]

(-2*b^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*a^2*x^(3/2))/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0349985, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2}{3} a^2 x^{3/2}+4 a b \sqrt{x}-\frac{2 b^2}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2*Sqrt[x],x]

[Out]

(-2*b^2)/Sqrt[x] + 4*a*b*Sqrt[x] + (2*a^2*x^(3/2))/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.58337, size = 31, normalized size = 0.97 \[ \frac{2 a^{2} x^{\frac{3}{2}}}{3} + 4 a b \sqrt{x} - \frac{2 b^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*b*sqrt(x) - 2*b**2/sqrt(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.013038, size = 27, normalized size = 0.84 \[ \frac{2 \left (a^2 x^2+6 a b x-3 b^2\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2*Sqrt[x],x]

[Out]

(2*(-3*b^2 + 6*a*b*x + a^2*x^2))/(3*Sqrt[x])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 24, normalized size = 0.8 \[{\frac{2\,{a}^{2}{x}^{2}+12\,abx-6\,{b}^{2}}{3}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2*x^(1/2),x)

[Out]

2/3*(a^2*x^2+6*a*b*x-3*b^2)/x^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.44129, size = 34, normalized size = 1.06 \[ \frac{2}{3} \,{\left (a^{2} + \frac{6 \, a b}{x}\right )} x^{\frac{3}{2}} - \frac{2 \, b^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2*sqrt(x),x, algorithm="maxima")

[Out]

2/3*(a^2 + 6*a*b/x)*x^(3/2) - 2*b^2/sqrt(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.22572, size = 31, normalized size = 0.97 \[ \frac{2 \,{\left (a^{2} x^{2} + 6 \, a b x - 3 \, b^{2}\right )}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2*sqrt(x),x, algorithm="fricas")

[Out]

2/3*(a^2*x^2 + 6*a*b*x - 3*b^2)/sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 1.89636, size = 31, normalized size = 0.97 \[ \frac{2 a^{2} x^{\frac{3}{2}}}{3} + 4 a b \sqrt{x} - \frac{2 b^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*x**(3/2)/3 + 4*a*b*sqrt(x) - 2*b**2/sqrt(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.225498, size = 32, normalized size = 1. \[ \frac{2}{3} \, a^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, b^{2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2*sqrt(x),x, algorithm="giac")

[Out]

2/3*a^2*x^(3/2) + 4*a*b*sqrt(x) - 2*b^2/sqrt(x)