3.1821 \(\int (a+\frac{b}{x^2})^2 x^2 \, dx\)

Optimal. Leaf size=24 \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

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Rubi [A]  time = 0.01072, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 270} \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2*x^2,x]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

Rule 263

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

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

Mathematica [A]  time = 0.0009351, size = 24, normalized size = 1. \[ \frac{a^2 x^3}{3}+2 a b x-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*x^2,x]

[Out]

-(b^2/x) + 2*a*b*x + (a^2*x^3)/3

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Maple [A]  time = 0.003, size = 23, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{x}}+2\,xab+{\frac{{x}^{3}{a}^{2}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971741, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, a^{2} x^{3} + 2 \, a b x - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3 + 2*a*b*x - b^2/x

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Fricas [A]  time = 1.36477, size = 50, normalized size = 2.08 \begin{align*} \frac{a^{2} x^{4} + 6 \, a b x^{2} - 3 \, b^{2}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^2*x^4 + 6*a*b*x^2 - 3*b^2)/x

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Sympy [A]  time = 0.243745, size = 19, normalized size = 0.79 \begin{align*} \frac{a^{2} x^{3}}{3} + 2 a b x - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + 2*a*b*x - b**2/x

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Giac [A]  time = 1.18107, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, a^{2} x^{3} + 2 \, a b x - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3 + 2*a*b*x - b^2/x