3.1812 \(\int \frac{a+\frac{b}{x^2}}{x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{a}{x}-\frac{b}{3 x^3} \]

[Out]

-b/(3*x^3) - a/x

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Rubi [A]  time = 0.0048434, antiderivative size = 15, 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} \[ -\frac{a}{x}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^2,x]

[Out]

-b/(3*x^3) - a/x

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

Mathematica [A]  time = 0.0020442, size = 15, normalized size = 1. \[ -\frac{a}{x}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^2,x]

[Out]

-b/(3*x^3) - a/x

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Maple [A]  time = 0.005, size = 14, normalized size = 0.9 \begin{align*} -{\frac{b}{3\,{x}^{3}}}-{\frac{a}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.953534, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, a x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3

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Fricas [A]  time = 1.35321, size = 32, normalized size = 2.13 \begin{align*} -\frac{3 \, a x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3

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Sympy [A]  time = 0.258221, size = 14, normalized size = 0.93 \begin{align*} - \frac{3 a x^{2} + b}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a*x**2 + b)/(3*x**3)

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Giac [A]  time = 1.12573, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, a x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a*x^2 + b)/x^3