3.1924 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

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

[Out]

(Sqrt[a + b/x^2]*x)/a

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Rubi [A]  time = 0.0023042, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {191} \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^2}}} \, dx &=\frac{\sqrt{a+\frac{b}{x^2}} x}{a}\\ \end{align*}

Mathematica [A]  time = 0.0057822, size = 16, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x^2}}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x)/a

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Maple [A]  time = 0.001, size = 28, normalized size = 1.8 \begin{align*}{\frac{a{x}^{2}+b}{ax}{\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)

[Out]

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

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Maxima [A]  time = 0.99024, size = 19, normalized size = 1.19 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a + b/x^2)*x/a

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Fricas [A]  time = 1.50766, size = 36, normalized size = 2.25 \begin{align*} \frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt((a*x^2 + b)/x^2)/a

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Sympy [A]  time = 0.689915, size = 17, normalized size = 1.06 \begin{align*} \frac{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(a*x**2/b + 1)/a

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Giac [A]  time = 1.20908, size = 38, normalized size = 2.38 \begin{align*} -\frac{\sqrt{b} \mathrm{sgn}\left (x\right )}{a} + \frac{\sqrt{a x^{2} + b}}{a \mathrm{sgn}\left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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