3.1811 \(\int \frac{a+\frac{b}{x^2}}{x} \, dx\)

Optimal. Leaf size=13 \[ a \log (x)-\frac{b}{2 x^2} \]

[Out]

-b/(2*x^2) + a*Log[x]

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Rubi [A]  time = 0.0047612, antiderivative size = 13, 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} \[ a \log (x)-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x,x]

[Out]

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

Mathematica [A]  time = 0.0023529, size = 13, normalized size = 1. \[ a \log (x)-\frac{b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x,x]

[Out]

-b/(2*x^2) + a*Log[x]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.9 \begin{align*} -{\frac{b}{2\,{x}^{2}}}+a\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.981223, size = 19, normalized size = 1.46 \begin{align*} \frac{1}{2} \, a \log \left (x^{2}\right ) - \frac{b}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*log(x^2) - 1/2*b/x^2

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Fricas [A]  time = 1.42833, size = 41, normalized size = 3.15 \begin{align*} \frac{2 \, a x^{2} \log \left (x\right ) - b}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*x^2*log(x) - b)/x^2

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Sympy [A]  time = 0.248663, size = 10, normalized size = 0.77 \begin{align*} a \log{\left (x \right )} - \frac{b}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - b/(2*x**2)

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Giac [A]  time = 1.25137, size = 27, normalized size = 2.08 \begin{align*} \frac{1}{2} \, a \log \left (x^{2}\right ) - \frac{a x^{2} + b}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*log(x^2) - 1/2*(a*x^2 + b)/x^2