3.824 \(\int \frac{a+b x^2+c x^4}{x^8} \, dx\)

Optimal. Leaf size=25 \[ -\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3} \]

[Out]

-a/(7*x^7) - b/(5*x^5) - c/(3*x^3)

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Rubi [A]  time = 0.0066146, antiderivative size = 25, 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} \[ -\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-a/(7*x^7) - b/(5*x^5) - c/(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+b x^2+c x^4}{x^8} \, dx &=\int \left (\frac{a}{x^8}+\frac{b}{x^6}+\frac{c}{x^4}\right ) \, dx\\ &=-\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0024899, size = 25, normalized size = 1. \[ -\frac{a}{7 x^7}-\frac{b}{5 x^5}-\frac{c}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-a/(7*x^7) - b/(5*x^5) - c/(3*x^3)

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Maple [A]  time = 0.049, size = 20, normalized size = 0.8 \begin{align*} -{\frac{a}{7\,{x}^{7}}}-{\frac{b}{5\,{x}^{5}}}-{\frac{c}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^8,x)

[Out]

-1/7*a/x^7-1/5*b/x^5-1/3*c/x^3

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Maxima [A]  time = 0.935626, size = 28, normalized size = 1.12 \begin{align*} -\frac{35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="maxima")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

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Fricas [A]  time = 1.4354, size = 55, normalized size = 2.2 \begin{align*} -\frac{35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="fricas")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

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Sympy [A]  time = 0.487416, size = 22, normalized size = 0.88 \begin{align*} - \frac{15 a + 21 b x^{2} + 35 c x^{4}}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**8,x)

[Out]

-(15*a + 21*b*x**2 + 35*c*x**4)/(105*x**7)

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Giac [A]  time = 1.20766, size = 28, normalized size = 1.12 \begin{align*} -\frac{35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="giac")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7