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

Optimal. Leaf size=23 \[ -\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {a}{5 x^5}-\frac {b}{3 x^3}-\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

-1/5*a/x^5 - b/(3*x^3) - c/x

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x^2+c x^4}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^6,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^6, x]

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fricas [A]  time = 1.92, size = 21, normalized size = 0.91 \begin {gather*} -\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

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giac [A]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

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maple [A]  time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} -\frac {c}{x}-\frac {b}{3 x^{3}}-\frac {a}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6,x)

[Out]

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

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maxima [A]  time = 1.36, size = 21, normalized size = 0.91 \begin {gather*} -\frac {15 \, c x^{4} + 5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(15*c*x^4 + 5*b*x^2 + 3*a)/x^5

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mupad [B]  time = 0.03, size = 20, normalized size = 0.87 \begin {gather*} -\frac {c\,x^4+\frac {b\,x^2}{3}+\frac {a}{5}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^6,x)

[Out]

-(a/5 + (b*x^2)/3 + c*x^4)/x^5

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sympy [A]  time = 0.26, size = 22, normalized size = 0.96 \begin {gather*} \frac {- 3 a - 5 b x^{2} - 15 c x^{4}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a - 5*b*x**2 - 15*c*x**4)/(15*x**5)

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