3.1814 \(\int \frac {a+\frac {b}{x^2}}{x^4} \, dx\)

Optimal. Leaf size=17 \[ -\frac {a}{3 x^3}-\frac {b}{5 x^5} \]

[Out]

-1/5*b/x^5-1/3*a/x^3

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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} \[ -\frac {a}{3 x^3}-\frac {b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/x^4,x]

[Out]

-b/(5*x^5) - a/(3*x^3)

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^4} \, dx &=\int \left (\frac {b}{x^6}+\frac {a}{x^4}\right ) \, dx\\ &=-\frac {b}{5 x^5}-\frac {a}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {a}{3 x^3}-\frac {b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^4,x]

[Out]

-1/5*b/x^5 - a/(3*x^3)

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fricas [A]  time = 0.79, size = 15, normalized size = 0.88 \[ -\frac {5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^4,x, algorithm="fricas")

[Out]

-1/15*(5*a*x^2 + 3*b)/x^5

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giac [A]  time = 0.15, size = 15, normalized size = 0.88 \[ -\frac {5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^4,x, algorithm="giac")

[Out]

-1/15*(5*a*x^2 + 3*b)/x^5

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \[ -\frac {a}{3 x^{3}}-\frac {b}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)/x^4,x)

[Out]

-1/5*b/x^5-1/3*a/x^3

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maxima [A]  time = 1.03, size = 15, normalized size = 0.88 \[ -\frac {5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)/x^4,x, algorithm="maxima")

[Out]

-1/15*(5*a*x^2 + 3*b)/x^5

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mupad [B]  time = 0.03, size = 15, normalized size = 0.88 \[ -\frac {5\,a\,x^2+3\,b}{15\,x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/x^4,x)

[Out]

-(3*b + 5*a*x^2)/(15*x^5)

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sympy [A]  time = 0.14, size = 15, normalized size = 0.88 \[ \frac {- 5 a x^{2} - 3 b}{15 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)/x**4,x)

[Out]

(-5*a*x**2 - 3*b)/(15*x**5)

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