3.143 \(\int \frac{b x^2+c x^4}{x^7} \, dx\)

Optimal. Leaf size=17 \[ -\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

[Out]

-b/(4*x^4) - c/(2*x^2)

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Rubi [A]  time = 0.0060832, 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} \[ -\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^7,x]

[Out]

-b/(4*x^4) - c/(2*x^2)

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

Mathematica [A]  time = 0.002192, size = 17, normalized size = 1. \[ -\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^7,x]

[Out]

-b/(4*x^4) - c/(2*x^2)

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Maple [A]  time = 0.046, size = 14, normalized size = 0.8 \begin{align*} -{\frac{b}{4\,{x}^{4}}}-{\frac{c}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^7,x)

[Out]

-1/4*b/x^4-1/2*c/x^2

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Maxima [A]  time = 0.980625, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^7,x, algorithm="maxima")

[Out]

-1/4*(2*c*x^2 + b)/x^4

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Fricas [A]  time = 1.26226, size = 32, normalized size = 1.88 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^7,x, algorithm="fricas")

[Out]

-1/4*(2*c*x^2 + b)/x^4

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Sympy [A]  time = 0.331721, size = 14, normalized size = 0.82 \begin{align*} - \frac{b + 2 c x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**7,x)

[Out]

-(b + 2*c*x**2)/(4*x**4)

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Giac [A]  time = 1.30271, size = 18, normalized size = 1.06 \begin{align*} -\frac{2 \, c x^{2} + b}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^7,x, algorithm="giac")

[Out]

-1/4*(2*c*x^2 + b)/x^4