3.1823 \(\int (a+\frac{b}{x^2})^2 \, dx\)

Optimal. Leaf size=23 \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

________________________________________________________________________________________

Rubi [A]  time = 0.01049, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {193, 270} \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2,x]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 270

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

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right )^2 \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^4} \, dx\\ &=\int \left (a^2+\frac{b^2}{x^4}+\frac{2 a b}{x^2}\right ) \, dx\\ &=-\frac{b^2}{3 x^3}-\frac{2 a b}{x}+a^2 x\\ \end{align*}

Mathematica [A]  time = 0.0046057, size = 23, normalized size = 1. \[ a^2 x-\frac{2 a b}{x}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2,x]

[Out]

-b^2/(3*x^3) - (2*a*b)/x + a^2*x

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 22, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}}+{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^2/x^3-2*a*b/x+a^2*x

________________________________________________________________________________________

Maxima [A]  time = 0.989974, size = 28, normalized size = 1.22 \begin{align*} a^{2} x - \frac{2 \, a b}{x} - \frac{b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x - 2*a*b/x - 1/3*b^2/x^3

________________________________________________________________________________________

Fricas [A]  time = 1.4083, size = 53, normalized size = 2.3 \begin{align*} \frac{3 \, a^{2} x^{4} - 6 \, a b x^{2} - b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*a^2*x^4 - 6*a*b*x^2 - b^2)/x^3

________________________________________________________________________________________

Sympy [A]  time = 0.276088, size = 20, normalized size = 0.87 \begin{align*} a^{2} x - \frac{6 a b x^{2} + b^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x - (6*a*b*x**2 + b**2)/(3*x**3)

________________________________________________________________________________________

Giac [A]  time = 1.15834, size = 30, normalized size = 1.3 \begin{align*} a^{2} x - \frac{6 \, a b x^{2} + b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*x - 1/3*(6*a*b*x^2 + b^2)/x^3