3.1932 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^5} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0646196, antiderivative size = 34, 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 \[ -\frac{a}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.94936, size = 29, normalized size = 0.85 \[ - \frac{a}{b^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0276238, size = 28, normalized size = 0.82 \[ \frac{-2 a x^2-b}{b^2 x^2 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^5),x]

[Out]

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

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Maple [A]  time = 0.008, size = 37, normalized size = 1.1 \[ -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}+b \right ) }{{b}^{2}{x}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.42762, size = 41, normalized size = 1.21 \[ -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{2}} - \frac{a}{\sqrt{a + \frac{b}{x^{2}}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235011, size = 50, normalized size = 1.47 \[ -\frac{{\left (2 \, a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a b^{2} x^{2} + b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.1242, size = 48, normalized size = 1.41 \[ \begin{cases} - \frac{2 a}{b^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{1}{b x^{2} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{3}{2}} x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**5,x)

[Out]

Piecewise((-2*a/(b**2*sqrt(a + b/x**2)) - 1/(b*x**2*sqrt(a + b/x**2)), Ne(b, 0))
, (-1/(4*a**(3/2)*x**4), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^5),x, algorithm="giac")

[Out]

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