3.1903 \(\int \frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{x^3} \, dx\)

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

[Out]

-(a + b/x^2)^(5/2)/(5*b)

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Rubi [A]  time = 0.0295815, 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 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(3/2)/x^3,x]

[Out]

-(a + b/x^2)^(5/2)/(5*b)

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Rubi in Sympy [A]  time = 2.12385, size = 14, normalized size = 0.78 \[ - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{5}{2}}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b/x**2)**(5/2)/(5*b)

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Mathematica [A]  time = 0.0260863, size = 30, normalized size = 1.67 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )^2}{5 b x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 29, normalized size = 1.6 \[ -{\frac{a{x}^{2}+b}{5\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(3/2)/x^3,x)

[Out]

-1/5*(a*x^2+b)/x^2/b*((a*x^2+b)/x^2)^(3/2)

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Maxima [A]  time = 1.43721, size = 19, normalized size = 1.06 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}}}{5 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*(a + b/x^2)^(5/2)/b

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Fricas [A]  time = 0.247137, size = 53, normalized size = 2.94 \[ -\frac{{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \, b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.54414, size = 68, normalized size = 3.78 \[ - \frac{a^{\frac{5}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 b} - \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{5 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.251341, size = 124, normalized size = 6.89 \[ \frac{2 \,{\left (5 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{5}{2}}{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{5}{2}} b^{2}{\rm sign}\left (x\right ) + a^{\frac{5}{2}} b^{4}{\rm sign}\left (x\right )\right )}}{5 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(5*(sqrt(a)*x - sqrt(a*x^2 + b))^8*a^(5/2)*sign(x) + 10*(sqrt(a)*x - sqrt(a*
x^2 + b))^4*a^(5/2)*b^2*sign(x) + a^(5/2)*b^4*sign(x))/((sqrt(a)*x - sqrt(a*x^2
+ b))^2 - b)^5