3.1557 \(\int \frac{a+\frac{b}{x}}{x^5} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{4 x^4}-\frac{b}{5 x^5} \]

[Out]

-b/(5*x^5) - a/(4*x^4)

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Rubi [A]  time = 0.0047732, 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} \[ -\frac{a}{4 x^4}-\frac{b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^5,x]

[Out]

-b/(5*x^5) - a/(4*x^4)

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^5} \, dx &=\int \left (\frac{b}{x^6}+\frac{a}{x^5}\right ) \, dx\\ &=-\frac{b}{5 x^5}-\frac{a}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0017936, size = 17, normalized size = 1. \[ -\frac{a}{4 x^4}-\frac{b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^5,x]

[Out]

-b/(5*x^5) - a/(4*x^4)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{5\,{x}^{5}}}-{\frac{a}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)/x^5,x)

[Out]

-1/5*b/x^5-1/4*a/x^4

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Maxima [A]  time = 0.949941, size = 18, normalized size = 1.06 \begin{align*} -\frac{5 \, a x + 4 \, b}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^5,x, algorithm="maxima")

[Out]

-1/20*(5*a*x + 4*b)/x^5

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Fricas [A]  time = 1.41887, size = 34, normalized size = 2. \begin{align*} -\frac{5 \, a x + 4 \, b}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^5,x, algorithm="fricas")

[Out]

-1/20*(5*a*x + 4*b)/x^5

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Sympy [A]  time = 0.265212, size = 14, normalized size = 0.82 \begin{align*} - \frac{5 a x + 4 b}{20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x**5,x)

[Out]

-(5*a*x + 4*b)/(20*x**5)

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Giac [A]  time = 1.20837, size = 18, normalized size = 1.06 \begin{align*} -\frac{5 \, a x + 4 \, b}{20 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^5,x, algorithm="giac")

[Out]

-1/20*(5*a*x + 4*b)/x^5