3.1555 \(\int \frac{a+\frac{b}{x}}{x^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{2 x^2}-\frac{b}{3 x^3} \]

[Out]

-b/(3*x^3) - a/(2*x^2)

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Rubi [A]  time = 0.0048012, 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} \[ -\frac{a}{2 x^2}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0017447, size = 17, normalized size = 1. \[ -\frac{a}{2 x^2}-\frac{b}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^3,x]

[Out]

-b/(3*x^3) - a/(2*x^2)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{3\,{x}^{3}}}-{\frac{a}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)/x^3,x)

[Out]

-1/3*b/x^3-1/2/x^2*a

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Maxima [A]  time = 0.988405, size = 18, normalized size = 1.06 \begin{align*} -\frac{3 \, a x + 2 \, b}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3

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Fricas [A]  time = 1.39413, size = 32, normalized size = 1.88 \begin{align*} -\frac{3 \, a x + 2 \, b}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3

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Sympy [A]  time = 0.258369, size = 14, normalized size = 0.82 \begin{align*} - \frac{3 a x + 2 b}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*a*x + 2*b)/(6*x**3)

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Giac [A]  time = 1.1935, size = 18, normalized size = 1.06 \begin{align*} -\frac{3 \, a x + 2 \, b}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*a*x + 2*b)/x^3