3.1940 \(\int \frac{1}{(a+\frac{b}{x^2})^{3/2} x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

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Rubi [A]  time = 0.0065681, antiderivative size = 19, 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 = {264} \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(3/2)*x^2),x]

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

Rule 264

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^2} \, dx &=-\frac{1}{a \sqrt{a+\frac{b}{x^2}} x}\\ \end{align*}

Mathematica [A]  time = 0.00497, size = 19, normalized size = 1. \[ -\frac{1}{a x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(a*Sqrt[a + b/x^2]*x))

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Maple [A]  time = 0.003, size = 29, normalized size = 1.5 \begin{align*} -{\frac{a{x}^{2}+b}{a{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.989726, size = 23, normalized size = 1.21 \begin{align*} -\frac{1}{\sqrt{a + \frac{b}{x^{2}}} a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(a + b/x^2)*a*x)

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Fricas [A]  time = 1.4721, size = 57, normalized size = 3. \begin{align*} -\frac{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{2} x^{2} + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.987411, size = 20, normalized size = 1.05 \begin{align*} - \frac{1}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*sqrt(b)*sqrt(a*x**2/b + 1))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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