3.1933 \(\int \frac{1}{(a+\frac{b}{x^2})^{3/2} x^3} \, dx\)

Optimal. Leaf size=15 \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

[Out]

1/(b*Sqrt[a + b/x^2])

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Rubi [A]  time = 0.0061414, antiderivative size = 15, 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} \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(3/2)*x^3),x]

[Out]

1/(b*Sqrt[a + b/x^2])

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

Mathematica [A]  time = 0.0063079, size = 15, normalized size = 1. \[ \frac{1}{b \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(b*Sqrt[a + b/x^2])

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Maple [B]  time = 0.002, size = 28, normalized size = 1.9 \begin{align*}{\frac{a{x}^{2}+b}{b{x}^{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^3,x)

[Out]

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

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Maxima [A]  time = 0.97695, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{\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)^(3/2)/x^3,x, algorithm="maxima")

[Out]

1/(sqrt(a + b/x^2)*b)

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Fricas [B]  time = 1.44293, size = 58, normalized size = 3.87 \begin{align*} \frac{x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.85188, size = 26, normalized size = 1.73 \begin{align*} \begin{cases} \frac{1}{b \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{2 a^{\frac{3}{2}} 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)**(3/2)/x**3,x)

[Out]

Piecewise((1/(b*sqrt(a + b/x**2)), Ne(b, 0)), (-1/(2*a**(3/2)*x**2), True))

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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