\(\int \frac {1}{(a+\frac {b}{x^2})^2 x^3} \, dx\) [1867]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=\frac {1}{2 b \left (a+\frac {b}{x^2}\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=\frac {1}{2 b \left (a+\frac {b}{x^2}\right )} \]

[In]

Int[1/((a + b/x^2)^2*x^3),x]

[Out]

1/(2*b*(a + b/x^2))

Rule 267

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*} \text {integral}& = \frac {1}{2 b \left (a+\frac {b}{x^2}\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=-\frac {1}{2 a \left (b+a x^2\right )} \]

[In]

Integrate[1/((a + b/x^2)^2*x^3),x]

[Out]

-1/2*1/(a*(b + a*x^2))

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {1}{2 \left (a \,x^{2}+b \right ) a}\)	\(15\)
derivativedivides	\(\frac {1}{2 b \left (a +\frac {b}{x^{2}}\right )}\)	\(15\)
default	\(-\frac {1}{2 \left (a \,x^{2}+b \right ) a}\)	\(15\)
norman	\(-\frac {1}{2 \left (a \,x^{2}+b \right ) a}\)	\(15\)
risch	\(-\frac {1}{2 \left (a \,x^{2}+b \right ) a}\)	\(15\)
parallelrisch	\(-\frac {1}{2 \left (a \,x^{2}+b \right ) a}\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=-\frac {1}{2 \, {\left (a^{2} x^{2} + a b\right )}} \]

[In]

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

[Out]

-1/2/(a^2*x^2 + a*b)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=- \frac {1}{2 a^{2} x^{2} + 2 a b} \]

[In]

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

[Out]

-1/(2*a**2*x**2 + 2*a*b)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=\frac {1}{2 \, {\left (a + \frac {b}{x^{2}}\right )} b} \]

[In]

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

[Out]

1/2/((a + b/x^2)*b)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=-\frac {1}{2 \, {\left (a x^{2} + b\right )} a} \]

[In]

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

[Out]

-1/2/((a*x^2 + b)*a)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^3} \, dx=-\frac {1}{2\,a\,\left (a\,x^2+b\right )} \]

[In]

int(1/(x^3*(a + b/x^2)^2),x)

[Out]

-1/(2*a*(b + a*x^2))