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

Optimal. Leaf size=15 \[ -\frac {b}{3 x^3}-\frac {c}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \begin {gather*} -\frac {b}{3 x^3}-\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^6,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b}{3 x^3}-\frac {c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^6,x]

[Out]

-1/3*b/x^3 - c/x

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, c x^{2} + b}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3

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giac [A]  time = 0.18, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, c x^{2} + b}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \begin {gather*} -\frac {c}{x}-\frac {b}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3 \, c x^{2} + b}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*c*x^2 + b)/x^3

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mupad [B]  time = 0.03, size = 13, normalized size = 0.87 \begin {gather*} -\frac {3\,c\,x^2+b}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^6,x)

[Out]

-(b + 3*c*x^2)/(3*x^3)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.93 \begin {gather*} \frac {- b - 3 c x^{2}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b - 3*c*x**2)/(3*x**3)

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