3.1822 \(\int (a+\frac{b}{x^2})^2 x \, dx\)

Optimal. Leaf size=27 \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

[Out]

-b^2/(2*x^2) + (a^2*x^2)/2 + 2*a*b*Log[x]

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Rubi [A]  time = 0.0144858, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {263, 266, 43} \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2*x,x]

[Out]

-b^2/(2*x^2) + (a^2*x^2)/2 + 2*a*b*Log[x]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x^2}\right )^2 x \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{2 x^2}+\frac{a^2 x^2}{2}+2 a b \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0044463, size = 27, normalized size = 1. \[ \frac{a^2 x^2}{2}+2 a b \log (x)-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2*x,x]

[Out]

-b^2/(2*x^2) + (a^2*x^2)/2 + 2*a*b*Log[x]

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Maple [A]  time = 0.004, size = 24, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{2\,{x}^{2}}}+{\frac{{a}^{2}{x}^{2}}{2}}+2\,ab\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.998854, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + a b \log \left (x^{2}\right ) - \frac{b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4484, size = 59, normalized size = 2.19 \begin{align*} \frac{a^{2} x^{4} + 4 \, a b x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.27389, size = 24, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{2}}{2} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + 2*a*b*log(x) - b**2/(2*x**2)

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Giac [A]  time = 1.12994, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac{b^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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