3.1656 \(\int \frac{\left (a+\frac{b}{x}\right )^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=36 \[ -\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{5 x^{5/2}}-\frac{2 b^2}{7 x^{7/2}} \]

[Out]

(-2*b^2)/(7*x^(7/2)) - (4*a*b)/(5*x^(5/2)) - (2*a^2)/(3*x^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0340564, antiderivative size = 36, 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 \[ -\frac{2 a^2}{3 x^{3/2}}-\frac{4 a b}{5 x^{5/2}}-\frac{2 b^2}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^2/x^(5/2),x]

[Out]

(-2*b^2)/(7*x^(7/2)) - (4*a*b)/(5*x^(5/2)) - (2*a^2)/(3*x^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.52632, size = 36, normalized size = 1. \[ - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 a b}{5 x^{\frac{5}{2}}} - \frac{2 b^{2}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x)**2/x**(5/2),x)

[Out]

-2*a**2/(3*x**(3/2)) - 4*a*b/(5*x**(5/2)) - 2*b**2/(7*x**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0127769, size = 28, normalized size = 0.78 \[ -\frac{2 \left (35 a^2 x^2+42 a b x+15 b^2\right )}{105 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x)^2/x^(5/2),x]

[Out]

(-2*(15*b^2 + 42*a*b*x + 35*a^2*x^2))/(105*x^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \[ -{\frac{70\,{a}^{2}{x}^{2}+84\,abx+30\,{b}^{2}}{105}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^2/x^(5/2),x)

[Out]

-2/105*(35*a^2*x^2+42*a*b*x+15*b^2)/x^(7/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.4417, size = 32, normalized size = 0.89 \[ -\frac{2 \, a^{2}}{3 \, x^{\frac{3}{2}}} - \frac{4 \, a b}{5 \, x^{\frac{5}{2}}} - \frac{2 \, b^{2}}{7 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*a^2/x^(3/2) - 4/5*a*b/x^(5/2) - 2/7*b^2/x^(7/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.226834, size = 32, normalized size = 0.89 \[ -\frac{2 \,{\left (35 \, a^{2} x^{2} + 42 \, a b x + 15 \, b^{2}\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^(5/2),x, algorithm="fricas")

[Out]

-2/105*(35*a^2*x^2 + 42*a*b*x + 15*b^2)/x^(7/2)

_______________________________________________________________________________________

Sympy [A]  time = 4.9372, size = 36, normalized size = 1. \[ - \frac{2 a^{2}}{3 x^{\frac{3}{2}}} - \frac{4 a b}{5 x^{\frac{5}{2}}} - \frac{2 b^{2}}{7 x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x)**2/x**(5/2),x)

[Out]

-2*a**2/(3*x**(3/2)) - 4*a*b/(5*x**(5/2)) - 2*b**2/(7*x**(7/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221695, size = 32, normalized size = 0.89 \[ -\frac{2 \,{\left (35 \, a^{2} x^{2} + 42 \, a b x + 15 \, b^{2}\right )}}{105 \, x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x)^2/x^(5/2),x, algorithm="giac")

[Out]

-2/105*(35*a^2*x^2 + 42*a*b*x + 15*b^2)/x^(7/2)