3.1695 \(\int \frac{\sqrt{a+\frac{b}{x}}}{x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0578148, antiderivative size = 38, 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 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.78929, size = 31, normalized size = 0.82 \[ \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0249001, size = 40, normalized size = 1.05 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (2 a^2 x^2-a b x-3 b^2\right )}{15 b^2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x]/x^3,x]

[Out]

(2*Sqrt[a + b/x]*(-3*b^2 - a*b*x + 2*a^2*x^2))/(15*b^2*x^2)

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Maple [A]  time = 0.008, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-3\,b \right ) }{15\,{b}^{2}{x}^{2}}\sqrt{{\frac{ax+b}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x)^(1/2)/x^3,x)

[Out]

2/15*(a*x+b)*(2*a*x-3*b)*((a*x+b)/x)^(1/2)/b^2/x^2

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Maxima [A]  time = 1.43872, size = 41, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227165, size = 51, normalized size = 1.34 \[ \frac{2 \,{\left (2 \, a^{2} x^{2} - a b x - 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.30176, size = 304, normalized size = 8. \[ \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{6} b x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**(11/2)*b**(3/2)*x**3*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/
2)*b**4*x**(5/2)) + 2*a**(9/2)*b**(5/2)*x**2*sqrt(a*x/b + 1)/(15*a**(7/2)*b**3*x
**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 8*a**(7/2)*b**(7/2)*x*sqrt(a*x/b + 1)/(15
*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 6*a**(5/2)*b**(9/2)*sqrt(
a*x/b + 1)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**6*b*x*
*(7/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2)) - 4*a**5*b**2*x**
(5/2)/(15*a**(7/2)*b**3*x**(7/2) + 15*a**(5/2)*b**4*x**(5/2))

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GIAC/XCAS [A]  time = 0.262161, size = 155, normalized size = 4.08 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}}{\rm sign}\left (x\right ) + 25 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b{\rm sign}\left (x\right ) + 15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{2}{\rm sign}\left (x\right ) + 3 \, b^{3}{\rm sign}\left (x\right )\right )}}{15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(15*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*sign(x) + 25*(sqrt(a)*x - sqr
t(a*x^2 + b*x))^2*a*b*sign(x) + 15*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^2*s
ign(x) + 3*b^3*sign(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^5