3.1813 \(\int \frac{a+\frac{b}{x^2}}{x^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

[Out]

-b/(4*x^4) - a/(2*x^2)

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Rubi [A]  time = 0.0050026, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^3,x]

[Out]

-b/(4*x^4) - a/(2*x^2)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{a+\frac{b}{x^2}}{x^3} \, dx &=\int \left (\frac{b}{x^5}+\frac{a}{x^3}\right ) \, dx\\ &=-\frac{b}{4 x^4}-\frac{a}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0019782, size = 17, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{b}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^3,x]

[Out]

-b/(4*x^4) - a/(2*x^2)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{4\,{x}^{4}}}-{\frac{a}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b/x^4-1/2/x^2*a

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Maxima [A]  time = 0.958889, size = 19, normalized size = 1.12 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{2}}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a + b/x^2)^2/b

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Fricas [A]  time = 1.42089, size = 32, normalized size = 1.88 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*x^2 + b)/x^4

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Sympy [A]  time = 0.261185, size = 14, normalized size = 0.82 \begin{align*} - \frac{2 a x^{2} + b}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a*x**2 + b)/(4*x**4)

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Giac [A]  time = 1.14286, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \, a x^{2} + b}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*x^2 + b)/x^4