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]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, 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} \[ -\frac {a}{4 x^4}-\frac {b}{5 x^5} \]

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{4 x^4}-\frac {b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^5,x]

[Out]

-1/5*b/x^5 - a/(4*x^4)

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fricas [A]  time = 0.97, size = 13, normalized size = 0.76 \[ -\frac {5 \, a x + 4 \, b}{20 \, x^{5}} \]

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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giac [A]  time = 0.17, size = 13, normalized size = 0.76 \[ -\frac {5 \, a x + 4 \, b}{20 \, x^{5}} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ -\frac {a}{4 x^{4}}-\frac {b}{5 x^{5}} \]

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 = 1.10, size = 13, normalized size = 0.76 \[ -\frac {5 \, a x + 4 \, b}{20 \, x^{5}} \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ -\frac {4\,b+5\,a\,x}{20\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)/x^5,x)

[Out]

-(4*b + 5*a*x)/(20*x^5)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.82 \[ \frac {- 5 a x - 4 b}{20 x^{5}} \]

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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