3.2116 \(\int \frac{a+b \sqrt{x}}{x^3} \, dx\)

Optimal. Leaf size=19 \[ -\frac{a}{2 x^2}-\frac{2 b}{3 x^{3/2}} \]

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0163598, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a}{2 x^2}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])/x^3,x]

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.83381, size = 17, normalized size = 0.89 \[ - \frac{a}{2 x^{2}} - \frac{2 b}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*x**2) - 2*b/(3*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00728313, size = 19, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{2 b}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-a/(2*x^2) - (2*b)/(3*x^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \[ -{\frac{a}{2\,{x}^{2}}}-{\frac{2\,b}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))/x^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44051, size = 20, normalized size = 1.05 \[ -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.23698, size = 20, normalized size = 1.05 \[ -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2

_______________________________________________________________________________________

Sympy [A]  time = 2.10536, size = 17, normalized size = 0.89 \[ - \frac{a}{2 x^{2}} - \frac{2 b}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*x**2) - 2*b/(3*x**(3/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.213635, size = 20, normalized size = 1.05 \[ -\frac{4 \, b \sqrt{x} + 3 \, a}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(4*b*sqrt(x) + 3*a)/x^2