3.1939 \(\int \frac{1}{(a+\frac{b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0058369, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac{2 x \sqrt{a+\frac{b}{x^2}}}{a^2}-\frac{x}{a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(-3/2),x]

[Out]

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

Rule 192

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

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

Mathematica [A]  time = 0.0120404, size = 27, normalized size = 0.77 \[ \frac{a x^2+2 b}{a^2 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(-3/2),x]

[Out]

(2*b + a*x^2)/(a^2*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.003, size = 37, normalized size = 1.1 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( a{x}^{2}+2\,b \right ) }{{x}^{3}{a}^{2}} \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)

[Out]

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

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Maxima [A]  time = 1.03776, size = 43, normalized size = 1.23 \begin{align*} \frac{\sqrt{a + \frac{b}{x^{2}}} x}{a^{2}} + \frac{b}{\sqrt{a + \frac{b}{x^{2}}} a^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4787, size = 77, normalized size = 2.2 \begin{align*} \frac{{\left (a x^{3} + 2 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a^{3} x^{2} + a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.824519, size = 42, normalized size = 1.2 \begin{align*} \frac{x^{2}}{a \sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} + \frac{2 \sqrt{b}}{a^{2} \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)

[Out]

x**2/(a*sqrt(b)*sqrt(a*x**2/b + 1)) + 2*sqrt(b)/(a**2*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}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a + b/x^2)^(-3/2), x)