3.2125 \(\int \frac{\left (a+b \sqrt{x}\right )^2}{x^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0433599, antiderivative size = 30, 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 \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2/x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.33381, size = 26, normalized size = 0.87 \[ - \frac{a^{2}}{2 x^{2}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**2/x**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0108397, size = 30, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^2/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \[ -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{4\,ab}{3}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^2/x^3,x)

[Out]

-1/2*a^2/x^2-4/3*a*b/x^(3/2)-b^2/x

_______________________________________________________________________________________

Maxima [A]  time = 1.41526, size = 32, normalized size = 1.07 \[ -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2/x^3,x, algorithm="maxima")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.232276, size = 32, normalized size = 1.07 \[ -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2/x^3,x, algorithm="fricas")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2

_______________________________________________________________________________________

Sympy [A]  time = 2.16354, size = 26, normalized size = 0.87 \[ - \frac{a^{2}}{2 x^{2}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{b^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**2/x**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220636, size = 32, normalized size = 1.07 \[ -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2/x^3,x, algorithm="giac")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2