3.1815 \(\int \frac{a+\frac{b}{x^2}}{x^5} \, dx\)

Optimal. Leaf size=17 \[ -\frac{a}{4 x^4}-\frac{b}{6 x^6} \]

[Out]

-b/(6*x^6) - a/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0048508, 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} \[ -\frac{a}{4 x^4}-\frac{b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^5,x]

[Out]

-b/(6*x^6) - 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^2}}{x^5} \, dx &=\int \left (\frac{b}{x^7}+\frac{a}{x^5}\right ) \, dx\\ &=-\frac{b}{6 x^6}-\frac{a}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.0019672, size = 17, normalized size = 1. \[ -\frac{a}{4 x^4}-\frac{b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^5,x]

[Out]

-b/(6*x^6) - a/(4*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{6\,{x}^{6}}}-{\frac{a}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)/x^5,x)

[Out]

-1/6*b/x^6-1/4*a/x^4

________________________________________________________________________________________

Maxima [A]  time = 0.974659, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^5,x, algorithm="maxima")

[Out]

-1/12*(3*a*x^2 + 2*b)/x^6

________________________________________________________________________________________

Fricas [A]  time = 1.38158, size = 36, normalized size = 2.12 \begin{align*} -\frac{3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^5,x, algorithm="fricas")

[Out]

-1/12*(3*a*x^2 + 2*b)/x^6

________________________________________________________________________________________

Sympy [A]  time = 0.276931, size = 15, normalized size = 0.88 \begin{align*} - \frac{3 a x^{2} + 2 b}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/x**5,x)

[Out]

-(3*a*x**2 + 2*b)/(12*x**6)

________________________________________________________________________________________

Giac [A]  time = 1.14871, size = 20, normalized size = 1.18 \begin{align*} -\frac{3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^5,x, algorithm="giac")

[Out]

-1/12*(3*a*x^2 + 2*b)/x^6