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

Optimal. Leaf size=18 \[ -\frac{a}{3 x^3}-\frac{b}{x}+c x \]

[Out]

-a/(3*x^3) - b/x + c*x

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Rubi [A]  time = 0.0066929, antiderivative size = 18, 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} \[ -\frac{a}{3 x^3}-\frac{b}{x}+c x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0040091, size = 18, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{b}{x}+c x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^4,x]

[Out]

-a/(3*x^3) - b/x + c*x

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Maple [A]  time = 0.046, size = 17, normalized size = 0.9 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{x}}+cx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.959433, size = 23, normalized size = 1.28 \begin{align*} c x - \frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3

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Fricas [A]  time = 1.4283, size = 45, normalized size = 2.5 \begin{align*} \frac{3 \, c x^{4} - 3 \, b x^{2} - a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*c*x^4 - 3*b*x^2 - a)/x^3

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Sympy [A]  time = 0.303946, size = 15, normalized size = 0.83 \begin{align*} c x - \frac{a + 3 b 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+a)/x**4,x)

[Out]

c*x - (a + 3*b*x**2)/(3*x**3)

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Giac [A]  time = 1.22534, size = 23, normalized size = 1.28 \begin{align*} c x - \frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x - 1/3*(3*b*x^2 + a)/x^3