3.1655 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{2 a^2}{\sqrt{x}}-\frac{4 a b}{3 x^{3/2}}-\frac{2 b^2}{5 x^{5/2}} \]

[Out]

(-2*b^2)/(5*x^(5/2)) - (4*a*b)/(3*x^(3/2)) - (2*a^2)/Sqrt[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0348254, antiderivative size = 34, 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 \[ -\frac{2 a^2}{\sqrt{x}}-\frac{4 a b}{3 x^{3/2}}-\frac{2 b^2}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x^(3/2),x]

[Out]

(-2*b^2)/(5*x^(5/2)) - (4*a*b)/(3*x^(3/2)) - (2*a^2)/Sqrt[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.62512, size = 34, normalized size = 1. \[ - \frac{2 a^{2}}{\sqrt{x}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{2 b^{2}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/sqrt(x) - 4*a*b/(3*x**(3/2)) - 2*b**2/(5*x**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0121773, size = 28, normalized size = 0.82 \[ -\frac{2 \left (15 a^2 x^2+10 a b x+3 b^2\right )}{15 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3*b^2 + 10*a*b*x + 15*a^2*x^2))/(15*x^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 25, normalized size = 0.7 \[ -{\frac{30\,{a}^{2}{x}^{2}+20\,abx+6\,{b}^{2}}{15}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x^(3/2),x)

[Out]

-2/15*(15*a^2*x^2+10*a*b*x+3*b^2)/x^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.43988, size = 32, normalized size = 0.94 \[ -\frac{2 \, a^{2}}{\sqrt{x}} - \frac{4 \, a b}{3 \, x^{\frac{3}{2}}} - \frac{2 \, b^{2}}{5 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a^2/sqrt(x) - 4/3*a*b/x^(3/2) - 2/5*b^2/x^(5/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.227801, size = 32, normalized size = 0.94 \[ -\frac{2 \,{\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*a^2*x^2 + 10*a*b*x + 3*b^2)/x^(5/2)

_______________________________________________________________________________________

Sympy [A]  time = 2.59578, size = 34, normalized size = 1. \[ - \frac{2 a^{2}}{\sqrt{x}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{2 b^{2}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**2/sqrt(x) - 4*a*b/(3*x**(3/2)) - 2*b**2/(5*x**(5/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216333, size = 32, normalized size = 0.94 \[ -\frac{2 \,{\left (15 \, a^{2} x^{2} + 10 \, a b x + 3 \, b^{2}\right )}}{15 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(15*a^2*x^2 + 10*a*b*x + 3*b^2)/x^(5/2)