3.1897 \(\int \frac{\sqrt{a+\frac{b}{x^2}}}{x^3} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

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Rubi [A]  time = 0.0059974, 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, Rules used = {261} \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]/x^3,x]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^2}}}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0110041, size = 18, normalized size = 1. \[ -\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]/x^3,x]

[Out]

-(a + b/x^2)^(3/2)/(3*b)

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Maple [A]  time = 0.003, size = 29, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}+b}{3\,b{x}^{2}}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01209, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)/x^3,x, algorithm="maxima")

[Out]

-1/3*(a + b/x^2)^(3/2)/b

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Fricas [A]  time = 1.48516, size = 65, normalized size = 3.61 \begin{align*} -\frac{{\left (a x^{2} + b\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \, b x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.941374, size = 42, normalized size = 2.33 \begin{align*} - \frac{a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{3 b} - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{3 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.21999, size = 85, normalized size = 4.72 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + a^{\frac{3}{2}} b^{2} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^(1/2)/x^3,x, algorithm="giac")

[Out]

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