3.1828 \(\int \frac{(a+\frac{b}{x^2})^2}{x^5} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]

[Out]

-b^2/(8*x^8) - (a*b)/(3*x^6) - a^2/(4*x^4)

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Rubi [A]  time = 0.0149898, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^2/x^5,x]

[Out]

-b^2/(8*x^8) - (a*b)/(3*x^6) - a^2/(4*x^4)

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 \frac{\left (a+\frac{b}{x^2}\right )^2}{x^5} \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^9} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{x^5}+\frac{2 a b}{x^4}+\frac{a^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{8 x^8}-\frac{a b}{3 x^6}-\frac{a^2}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0008541, size = 30, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^2/x^5,x]

[Out]

-b^2/(8*x^8) - (a*b)/(3*x^6) - a^2/(4*x^4)

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{b}^{2}}{8\,{x}^{8}}}-{\frac{ab}{3\,{x}^{6}}}-{\frac{{a}^{2}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^2/x^5,x)

[Out]

-1/8*b^2/x^8-1/3*a*b/x^6-1/4*a^2/x^4

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Maxima [A]  time = 0.971239, size = 35, normalized size = 1.17 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="maxima")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8

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Fricas [A]  time = 1.41854, size = 58, normalized size = 1.93 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="fricas")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8

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Sympy [A]  time = 0.345534, size = 27, normalized size = 0.9 \begin{align*} - \frac{6 a^{2} x^{4} + 8 a b x^{2} + 3 b^{2}}{24 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**2/x**5,x)

[Out]

-(6*a**2*x**4 + 8*a*b*x**2 + 3*b**2)/(24*x**8)

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Giac [A]  time = 1.1697, size = 35, normalized size = 1.17 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^2/x^5,x, algorithm="giac")

[Out]

-1/24*(6*a^2*x^4 + 8*a*b*x^2 + 3*b^2)/x^8