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]

-b/(6*x^6) - a/(5*x^5)

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Rubi [A]  time = 0.004701, 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} \[ -\frac{a}{5 x^5}-\frac{b}{6 x^6} \]

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*}

Mathematica [A]  time = 0.0015256, size = 17, normalized size = 1. \[ -\frac{a}{5 x^5}-\frac{b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)/x^6,x]

[Out]

-b/(6*x^6) - a/(5*x^5)

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Maple [A]  time = 0.004, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{6\,{x}^{6}}}-{\frac{a}{5\,{x}^{5}}} \end{align*}

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 = 0.952965, size = 18, normalized size = 1.06 \begin{align*} -\frac{6 \, a x + 5 \, b}{30 \, x^{6}} \end{align*}

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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Fricas [A]  time = 1.35814, size = 34, normalized size = 2. \begin{align*} -\frac{6 \, a x + 5 \, b}{30 \, x^{6}} \end{align*}

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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Sympy [A]  time = 0.271038, size = 14, normalized size = 0.82 \begin{align*} - \frac{6 a x + 5 b}{30 x^{6}} \end{align*}

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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Giac [A]  time = 1.17312, size = 18, normalized size = 1.06 \begin{align*} -\frac{6 \, a x + 5 \, b}{30 \, x^{6}} \end{align*}

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