3.1918 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

[Out]

-(Sqrt[a + b/x^2]/b)

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Rubi [A]  time = 0.0056214, antiderivative size = 16, 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} \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x^2]*x^3),x]

[Out]

-(Sqrt[a + b/x^2]/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{1}{\sqrt{a+\frac{b}{x^2}} x^3} \, dx &=-\frac{\sqrt{a+\frac{b}{x^2}}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0085518, size = 16, normalized size = 1. \[ -\frac{\sqrt{a+\frac{b}{x^2}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x^2]*x^3),x]

[Out]

-(Sqrt[a + b/x^2]/b)

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Maple [A]  time = 0.003, size = 29, normalized size = 1.8 \begin{align*} -{\frac{a{x}^{2}+b}{b{x}^{2}}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965009, size = 19, normalized size = 1.19 \begin{align*} -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a + b/x^2)/b

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Fricas [A]  time = 1.45811, size = 35, normalized size = 2.19 \begin{align*} -\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt((a*x^2 + b)/x^2)/b

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Sympy [A]  time = 1.7236, size = 26, normalized size = 1.62 \begin{align*} \begin{cases} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{b} & \text{for}\: b \neq 0 \\- \frac{1}{2 \sqrt{a} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12839, size = 19, normalized size = 1.19 \begin{align*} -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a + b/x^2)/b