3.7.24 \(\int \frac {a+b x^2+c x^4}{x^5} \, dx\)

Optimal. Leaf size=21 \[ -\frac {a}{4 x^4}-\frac {b}{2 x^2}+c \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 21, 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}{4 x^4}-\frac {b}{2 x^2}+c \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {a}{4 x^4}-\frac {b}{2 x^2}+c \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

-1/4*a/x^4 - b/(2*x^2) + c*Log[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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^5, x]

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fricas [A]  time = 0.81, size = 23, normalized size = 1.10 \begin {gather*} \frac {4 \, c x^{4} \log \relax (x) - 2 \, b x^{2} - a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^5,x, algorithm="fricas")

[Out]

1/4*(4*c*x^4*log(x) - 2*b*x^2 - a)/x^4

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giac [A]  time = 0.16, size = 27, normalized size = 1.29 \begin {gather*} \frac {1}{2} \, c \log \left (x^{2}\right ) - \frac {3 \, c x^{4} + 2 \, b x^{2} + a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^5,x, algorithm="giac")

[Out]

1/2*c*log(x^2) - 1/4*(3*c*x^4 + 2*b*x^2 + a)/x^4

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \begin {gather*} c \ln \relax (x )-\frac {b}{2 x^{2}}-\frac {a}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^5,x)

[Out]

-1/4*a/x^4-1/2*b/x^2+c*ln(x)

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maxima [A]  time = 1.31, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{2} \, c \log \left (x^{2}\right ) - \frac {2 \, b x^{2} + a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^5,x, algorithm="maxima")

[Out]

1/2*c*log(x^2) - 1/4*(2*b*x^2 + a)/x^4

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mupad [B]  time = 0.04, size = 20, normalized size = 0.95 \begin {gather*} c\,\ln \relax (x)-\frac {\frac {b\,x^2}{2}+\frac {a}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^5,x)

[Out]

c*log(x) - (a/4 + (b*x^2)/2)/x^4

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sympy [A]  time = 0.24, size = 19, normalized size = 0.90 \begin {gather*} c \log {\relax (x )} + \frac {- a - 2 b x^{2}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**5,x)

[Out]

c*log(x) + (-a - 2*b*x**2)/(4*x**4)

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