3.1558 \(\int \frac {a+\frac {b}{x}}{x^6} \, dx\)

Optimal. Leaf size=17 \[ -\frac {a}{5 x^5}-\frac {b}{6 x^6} \]

[Out]

-1/6*b/x^6-1/5*a/x^5

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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} \[ -\frac {a}{5 x^5}-\frac {b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)/x^6,x]

[Out]

-b/(6*x^6) - a/(5*x^5)

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^6} \, dx &=\int \left (\frac {b}{x^7}+\frac {a}{x^6}\right ) \, dx\\ &=-\frac {b}{6 x^6}-\frac {a}{5 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{5 x^5}-\frac {b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^6,x]

[Out]

-1/6*b/x^6 - a/(5*x^5)

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fricas [A]  time = 0.88, size = 13, normalized size = 0.76 \[ -\frac {6 \, a x + 5 \, b}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^6,x, algorithm="fricas")

[Out]

-1/30*(6*a*x + 5*b)/x^6

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giac [A]  time = 0.16, size = 13, normalized size = 0.76 \[ -\frac {6 \, a x + 5 \, b}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^6,x, algorithm="giac")

[Out]

-1/30*(6*a*x + 5*b)/x^6

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ -\frac {a}{5 x^{5}}-\frac {b}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)/x^6,x)

[Out]

-1/6*b/x^6-1/5*a/x^5

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maxima [A]  time = 0.97, size = 13, normalized size = 0.76 \[ -\frac {6 \, a x + 5 \, b}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x^6,x, algorithm="maxima")

[Out]

-1/30*(6*a*x + 5*b)/x^6

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ -\frac {5\,b+6\,a\,x}{30\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)/x^6,x)

[Out]

-(5*b + 6*a*x)/(30*x^6)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.82 \[ \frac {- 6 a x - 5 b}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)/x**6,x)

[Out]

(-6*a*x - 5*b)/(30*x**6)

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