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

Optimal. Leaf size=15 \[ -\frac{b}{3 x^3}-\frac{c}{x} \]

[Out]

-b/(3*x^3) - c/x

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Rubi [A]  time = 0.0056849, antiderivative size = 15, 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} \[ -\frac{b}{3 x^3}-\frac{c}{x} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0020695, size = 15, normalized size = 1. \[ -\frac{b}{3 x^3}-\frac{c}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^6,x]

[Out]

-b/(3*x^3) - c/x

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Maple [A]  time = 0.046, size = 14, normalized size = 0.9 \begin{align*} -{\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)/x^6,x)

[Out]

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

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Maxima [A]  time = 0.974695, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, c x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3

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Fricas [A]  time = 1.21469, size = 32, normalized size = 2.13 \begin{align*} -\frac{3 \, c x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3

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Sympy [A]  time = 0.298958, size = 14, normalized size = 0.93 \begin{align*} - \frac{b + 3 c x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b + 3*c*x**2)/(3*x**3)

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Giac [A]  time = 1.30789, size = 18, normalized size = 1.2 \begin{align*} -\frac{3 \, c x^{2} + b}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3