3.7.27 \(\int \frac {a+b x^2+c x^4}{x^8} \, dx\)

Optimal. Leaf size=25 \[ -\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, 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}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-a/(7*x^7) - b/(5*x^5) - c/(3*x^3)

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^8} \, dx &=\int \left (\frac {a}{x^8}+\frac {b}{x^6}+\frac {c}{x^4}\right ) \, dx\\ &=-\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {a}{7 x^7}-\frac {b}{5 x^5}-\frac {c}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

-1/7*a/x^7 - b/(5*x^5) - c/(3*x^3)

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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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^8,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^8, x]

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fricas [A]  time = 0.67, size = 21, normalized size = 0.84 \begin {gather*} -\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="fricas")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

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giac [A]  time = 0.17, size = 21, normalized size = 0.84 \begin {gather*} -\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="giac")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} -\frac {c}{3 x^{3}}-\frac {b}{5 x^{5}}-\frac {a}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^8,x)

[Out]

-1/7*a/x^7-1/5*b/x^5-1/3*c/x^3

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maxima [A]  time = 1.34, size = 21, normalized size = 0.84 \begin {gather*} -\frac {35 \, c x^{4} + 21 \, b x^{2} + 15 \, a}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8,x, algorithm="maxima")

[Out]

-1/105*(35*c*x^4 + 21*b*x^2 + 15*a)/x^7

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mupad [B]  time = 0.03, size = 21, normalized size = 0.84 \begin {gather*} -\frac {\frac {c\,x^4}{3}+\frac {b\,x^2}{5}+\frac {a}{7}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^8,x)

[Out]

-(a/7 + (b*x^2)/5 + (c*x^4)/3)/x^7

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sympy [A]  time = 0.32, size = 22, normalized size = 0.88 \begin {gather*} \frac {- 15 a - 21 b x^{2} - 35 c x^{4}}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**8,x)

[Out]

(-15*a - 21*b*x**2 - 35*c*x**4)/(105*x**7)

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