3.1704 \(\int \frac{\left (a+\frac{b}{x}\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0248153, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x)^(3/2)/x^2,x]

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.18905, size = 14, normalized size = 0.78 \[ - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b/x)**(5/2)/(5*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0219896, size = 18, normalized size = 1. \[ -\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b/x)^(5/2))/(5*b)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 25, normalized size = 1.4 \[ -{\frac{2\,ax+2\,b}{5\,bx} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(3/2)/x^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43651, size = 19, normalized size = 1.06 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*(a + b/x)^(5/2)/b

_______________________________________________________________________________________

Fricas [A]  time = 0.221582, size = 47, normalized size = 2.61 \[ -\frac{2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{5 \, b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.1317, size = 65, normalized size = 3.61 \[ - \frac{2 a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x}}}{5 b} - \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x}}}{5 x} - \frac{2 \sqrt{a} b \sqrt{1 + \frac{b}{a x}}}{5 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/2)*sqrt(1 + b/(a*x))/(5*b) - 4*a**(3/2)*sqrt(1 + b/(a*x))/(5*x) - 2*sqr
t(a)*b*sqrt(1 + b/(a*x))/(5*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.256799, size = 196, normalized size = 10.89 \[ \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2}{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{2}{\rm sign}\left (x\right ) + 5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{3}{\rm sign}\left (x\right ) + b^{4}{\rm sign}\left (x\right )\right )}}{5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*sign(x) + 10*(sqrt(a)*x - sqrt(a*x^
2 + b*x))^3*a^(3/2)*b*sign(x) + 10*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*sign(
x) + 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*sign(x) + b^4*sign(x))/(sqrt(
a)*x - sqrt(a*x^2 + b*x))^5