3.1726 \(\int \frac{1}{\sqrt{a+\frac{b}{x}} x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0575781, 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 \[ \frac{2 a \sqrt{a+\frac{b}{x}}}{b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x]*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.83025, size = 29, normalized size = 0.81 \[ \frac{2 a \sqrt{a + \frac{b}{x}}}{b^{2}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*sqrt(a + b/x)/b**2 - 2*(a + b/x)**(3/2)/(3*b**2)

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Mathematica [A]  time = 0.0278904, size = 29, normalized size = 0.81 \[ \frac{2 \sqrt{a+\frac{b}{x}} (2 a x-b)}{3 b^2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x]*x^3),x]

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-b \right ) }{3\,{b}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{ax+b}{x}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b/x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.4473, size = 41, normalized size = 1.14 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}}{3 \, b^{2}} + \frac{2 \, \sqrt{a + \frac{b}{x}} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.226688, size = 36, normalized size = 1. \[ \frac{2 \,{\left (2 \, a x - b\right )} \sqrt{\frac{a x + b}{x}}}{3 \, b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.43846, size = 248, normalized size = 6.89 \[ \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x}{b} + 1}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{4} b x^{\frac{5}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} - \frac{4 a^{3} b^{2} x^{\frac{3}{2}}}{3 a^{\frac{5}{2}} b^{3} x^{\frac{5}{2}} + 3 a^{\frac{3}{2}} b^{4} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(7/2)*b**(3/2)*x**2*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2) + 3*a**(3/2)*
b**4*x**(3/2)) + 2*a**(5/2)*b**(5/2)*x*sqrt(a*x/b + 1)/(3*a**(5/2)*b**3*x**(5/2)
+ 3*a**(3/2)*b**4*x**(3/2)) - 2*a**(3/2)*b**(7/2)*sqrt(a*x/b + 1)/(3*a**(5/2)*b
**3*x**(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 4*a**4*b*x**(5/2)/(3*a**(5/2)*b**3*x*
*(5/2) + 3*a**(3/2)*b**4*x**(3/2)) - 4*a**3*b**2*x**(3/2)/(3*a**(5/2)*b**3*x**(5
/2) + 3*a**(3/2)*b**4*x**(3/2))

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GIAC/XCAS [A]  time = 0.248594, size = 63, normalized size = 1.75 \[ \frac{2 \,{\left (3 \, a b^{6} \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} b^{6} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{3 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x)*x^3),x, algorithm="giac")

[Out]

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