3.1816 \(\int \frac{a+\frac{b}{x^2}}{x^6} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{5 x^5}-\frac{b}{7 x^7} \]

[Out]

-b/(7*x^7) - a/(5*x^5)

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Rubi [A]  time = 0.0047044, 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} \[ -\frac{a}{5 x^5}-\frac{b}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^6,x]

[Out]

-b/(7*x^7) - 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^2}}{x^6} \, dx &=\int \left (\frac{b}{x^8}+\frac{a}{x^6}\right ) \, dx\\ &=-\frac{b}{7 x^7}-\frac{a}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0019654, size = 17, normalized size = 1. \[ -\frac{a}{5 x^5}-\frac{b}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^6,x]

[Out]

-b/(7*x^7) - a/(5*x^5)

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Maple [A]  time = 0.005, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{7\,{x}^{7}}}-{\frac{a}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)/x^6,x)

[Out]

-1/7*b/x^7-1/5*a/x^5

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Maxima [A]  time = 0.962349, size = 20, normalized size = 1.18 \begin{align*} -\frac{7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^6,x, algorithm="maxima")

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7

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Fricas [A]  time = 1.4024, size = 36, normalized size = 2.12 \begin{align*} -\frac{7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^6,x, algorithm="fricas")

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7

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Sympy [A]  time = 0.288041, size = 15, normalized size = 0.88 \begin{align*} - \frac{7 a x^{2} + 5 b}{35 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/x**6,x)

[Out]

-(7*a*x**2 + 5*b)/(35*x**7)

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Giac [A]  time = 1.17797, size = 20, normalized size = 1.18 \begin{align*} -\frac{7 \, a x^{2} + 5 \, b}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^6,x, algorithm="giac")

[Out]

-1/35*(7*a*x^2 + 5*b)/x^7