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]

-b/(5*x^5) - a/(3*x^3)

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Rubi [A]  time = 0.0044337, 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} \[ -\frac{a}{3 x^3}-\frac{b}{5 x^5} \]

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*}

Mathematica [A]  time = 0.0020938, size = 17, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{b}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/x^4,x]

[Out]

-b/(5*x^5) - a/(3*x^3)

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Maple [A]  time = 0.003, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{5\,{x}^{5}}}-{\frac{a}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.957994, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

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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Fricas [A]  time = 1.36548, size = 36, normalized size = 2.12 \begin{align*} -\frac{5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

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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Sympy [A]  time = 0.273124, size = 15, normalized size = 0.88 \begin{align*} - \frac{5 a x^{2} + 3 b}{15 x^{5}} \end{align*}

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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Giac [A]  time = 1.12044, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, a x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

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