3.822 \(\int \frac{a+b x^2+c x^4}{x^6} \, dx\)

Optimal. Leaf size=23 \[ -\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x} \]

[Out]

-a/(5*x^5) - b/(3*x^3) - c/x

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Rubi [A]  time = 0.0068175, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {14} \[ -\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

-a/(5*x^5) - b/(3*x^3) - c/x

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+b x^2+c x^4}{x^6} \, dx &=\int \left (\frac{a}{x^6}+\frac{b}{x^4}+\frac{c}{x^2}\right ) \, dx\\ &=-\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x}\\ \end{align*}

Mathematica [A]  time = 0.0023745, size = 23, normalized size = 1. \[ -\frac{a}{5 x^5}-\frac{b}{3 x^3}-\frac{c}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

-a/(5*x^5) - b/(3*x^3) - c/x

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Maple [A]  time = 0.048, size = 20, normalized size = 0.9 \begin{align*} -{\frac{a}{5\,{x}^{5}}}-{\frac{b}{3\,{x}^{3}}}-{\frac{c}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6,x)

[Out]

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

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Maxima [A]  time = 0.950352, size = 28, normalized size = 1.22 \begin{align*} -\frac{15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="maxima")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

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Fricas [A]  time = 1.39925, size = 51, normalized size = 2.22 \begin{align*} -\frac{15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

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Sympy [A]  time = 0.434125, size = 22, normalized size = 0.96 \begin{align*} - \frac{3 a + 5 b x^{2} + 15 c x^{4}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**6,x)

[Out]

-(3*a + 5*b*x**2 + 15*c*x**4)/(15*x**5)

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Giac [A]  time = 1.24413, size = 28, normalized size = 1.22 \begin{align*} -\frac{15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6,x, algorithm="giac")

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5