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

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ a \log (x)-\frac {b}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x,x]

[Out]

-1/2*b/x^2 + a*Log[x]

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fricas [A]  time = 0.86, size = 17, normalized size = 1.31 \[ \frac {2 \, a x^{2} \log \relax (x) - b}{2 \, x^{2}} \]

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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giac [A]  time = 0.15, size = 20, normalized size = 1.54 \[ \frac {1}{2} \, a \log \left (x^{2}\right ) - \frac {a x^{2} + b}{2 \, x^{2}} \]

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

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ a \ln \relax (x )-\frac {b}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.10, size = 14, normalized size = 1.08 \[ \frac {1}{2} \, a \log \left (x^{2}\right ) - \frac {b}{2 \, x^{2}} \]

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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mupad [B]  time = 0.04, size = 11, normalized size = 0.85 \[ a\,\ln \relax (x)-\frac {b}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/x,x)

[Out]

a*log(x) - b/(2*x^2)

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sympy [A]  time = 0.11, size = 10, normalized size = 0.77 \[ a \log {\relax (x )} - \frac {b}{2 x^{2}} \]

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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