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

Optimal. Leaf size=17 \[ -\frac {a}{5 x^5}-\frac {b}{3 x^3} \]

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \begin {gather*} -\frac {a}{5 x^5}-\frac {b}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^6,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {a}{5 x^5}-\frac {b}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^6,x]

[Out]

-1/5*a/x^5 - b/(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}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)/x^6,x]

[Out]

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

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fricas [A]  time = 0.91, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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giac [A]  time = 1.09, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \begin {gather*} -\frac {b}{3 x^{3}}-\frac {a}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5 \, b x^{2} + 3 \, a}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(5*b*x^2 + 3*a)/x^5

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mupad [B]  time = 0.03, size = 15, normalized size = 0.88 \begin {gather*} -\frac {5\,b\,x^2+3\,a}{15\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)/x^6,x)

[Out]

-(3*a + 5*b*x^2)/(15*x^5)

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sympy [A]  time = 0.13, size = 15, normalized size = 0.88 \begin {gather*} \frac {- 3 a - 5 b x^{2}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a - 5*b*x**2)/(15*x**5)

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