3.1946 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^5} \, dx\)

Optimal. Leaf size=35 \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0656304, antiderivative size = 35, 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 \[ \frac{1}{b^2 \sqrt{a+\frac{b}{x^2}}}-\frac{a}{3 b^2 \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.008, size = 31, normalized size = 0.89 \[ - \frac{a}{3 b^{2} \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} + \frac{1}{b^{2} \sqrt{a + \frac{b}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0260047, size = 37, normalized size = 1.06 \[ \frac{2 a x^2+3 b}{3 b^2 \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 39, normalized size = 1.1 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 2\,a{x}^{2}+3\,b \right ) }{3\,{b}^{2}{x}^{4}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(5/2)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.44786, size = 39, normalized size = 1.11 \[ \frac{1}{\sqrt{a + \frac{b}{x^{2}}} b^{2}} - \frac{a}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.245109, size = 72, normalized size = 2.06 \[ \frac{{\left (2 \, a x^{4} + 3 \, b x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.0675, size = 94, normalized size = 2.69 \[ \begin{cases} \frac{2 a x^{2}}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{3 b}{3 a b^{2} x^{2} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{3} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{4 a^{\frac{5}{2}} x^{4}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a*x**2/(3*a*b**2*x**2*sqrt(a + b/x**2) + 3*b**3*sqrt(a + b/x**2)) +
3*b/(3*a*b**2*x**2*sqrt(a + b/x**2) + 3*b**3*sqrt(a + b/x**2)), Ne(b, 0)), (-1/
(4*a**(5/2)*x**4), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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