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]

-1/6*b/x^6-1/4*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} \[ -\frac {a}{4 x^4}-\frac {b}{6 x^6} \]

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{4 x^4}-\frac {b}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^5,x]

[Out]

-1/6*b/x^6 - a/(4*x^4)

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fricas [A]  time = 0.61, size = 15, normalized size = 0.88 \[ -\frac {3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \]

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

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giac [A]  time = 0.20, size = 15, normalized size = 0.88 \[ -\frac {3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ -\frac {a}{4 x^{4}}-\frac {b}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.08, size = 15, normalized size = 0.88 \[ -\frac {3 \, a x^{2} + 2 \, b}{12 \, x^{6}} \]

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

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mupad [B]  time = 0.03, size = 15, normalized size = 0.88 \[ -\frac {3\,a\,x^2+2\,b}{12\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/x^5,x)

[Out]

-(2*b + 3*a*x^2)/(12*x^6)

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sympy [A]  time = 0.14, size = 15, normalized size = 0.88 \[ \frac {- 3 a x^{2} - 2 b}{12 x^{6}} \]

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)

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