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

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

[Out]

-b/(5*x^5) - c/(3*x^3)

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Rubi [A]  time = 0.0056135, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ -\frac{b}{5 x^5}-\frac{c}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^8,x]

[Out]

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

Mathematica [A]  time = 0.0022654, size = 17, normalized size = 1. \[ -\frac{b}{5 x^5}-\frac{c}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^8,x]

[Out]

-b/(5*x^5) - c/(3*x^3)

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \begin{align*} -{\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)/x^8,x)

[Out]

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

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Maxima [A]  time = 0.993127, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5

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Fricas [A]  time = 1.22487, size = 36, normalized size = 2.12 \begin{align*} -\frac{5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5

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Sympy [A]  time = 0.349806, size = 15, normalized size = 0.88 \begin{align*} - \frac{3 b + 5 c x^{2}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*b + 5*c*x**2)/(15*x**5)

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Giac [A]  time = 1.26271, size = 20, normalized size = 1.18 \begin{align*} -\frac{5 \, c x^{2} + 3 \, b}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*c*x^2 + 3*b)/x^5