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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = 0.01, size = 15, normalized size = 1.00 \[ \frac {1}{b \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/(b*Sqrt[a + b/x^2])

________________________________________________________________________________________

fricas [B]  time = 0.99, size = 29, normalized size = 1.93 \[ \frac {x^{2} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{a b x^{2} + b^{2}} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.19, size = 44, normalized size = 2.93 \[ \frac {1}{{\left ({\left (\sqrt {a} x^{2} - \sqrt {a x^{4} + b x^{2}}\right )} \sqrt {a} + b\right )} \sqrt {a}} - \frac {1}{\sqrt {a} b} \]

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]

1/(((sqrt(a)*x^2 - sqrt(a*x^4 + b*x^2))*sqrt(a) + b)*sqrt(a)) - 1/(sqrt(a)*b)

________________________________________________________________________________________

maple [B]  time = 0.00, size = 28, normalized size = 1.87 \[ \frac {a \,x^{2}+b}{\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {3}{2}} b \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.88, size = 13, normalized size = 0.87 \[ \frac {1}{\sqrt {a + \frac {b}{x^{2}}} b} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.25, size = 18, normalized size = 1.20 \[ \frac {\sqrt {x^2}}{b\,\sqrt {a\,x^2+b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b/x^2)^(3/2)),x)

[Out]

(x^2)^(1/2)/(b*(b + a*x^2)^(1/2))

________________________________________________________________________________________

sympy [A]  time = 2.37, size = 26, normalized size = 1.73 \[ \begin {cases} \frac {1}{b \sqrt {a + \frac {b}{x^{2}}}} & \text {for}\: b \neq 0 \\- \frac {1}{2 a^{\frac {3}{2}} x^{2}} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________