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

Optimal. Leaf size=25 \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

[Out]

-a/(6*x^6) - b/(4*x^4) - c/(2*x^2)

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Rubi [A]  time = 0.0069301, 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} \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0023377, size = 25, normalized size = 1. \[ -\frac{a}{6 x^6}-\frac{b}{4 x^4}-\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^7,x]

[Out]

-a/(6*x^6) - b/(4*x^4) - c/(2*x^2)

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Maple [A]  time = 0.046, size = 20, normalized size = 0.8 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\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+a)/x^7,x)

[Out]

-1/6*a/x^6-1/4*b/x^4-1/2*c/x^2

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Maxima [A]  time = 0.95056, size = 28, normalized size = 1.12 \begin{align*} -\frac{6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

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Fricas [A]  time = 1.39779, size = 50, normalized size = 2. \begin{align*} -\frac{6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6

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Sympy [A]  time = 0.486558, size = 22, normalized size = 0.88 \begin{align*} - \frac{2 a + 3 b x^{2} + 6 c x^{4}}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*a + 3*b*x**2 + 6*c*x**4)/(12*x**6)

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Giac [A]  time = 1.24525, size = 28, normalized size = 1.12 \begin{align*} -\frac{6 \, c x^{4} + 3 \, b x^{2} + 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*c*x^4 + 3*b*x^2 + 2*a)/x^6