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

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

[Out]

-a/(4*x^4) - b/(2*x^2) + c*Log[x]

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Rubi [A]  time = 0.0069292, antiderivative size = 21, normalized size of antiderivative = 1., 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} \[ -\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x) \]

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*}

Mathematica [A]  time = 0.0028227, size = 21, normalized size = 1. \[ -\frac{a}{4 x^4}-\frac{b}{2 x^2}+c \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^5,x]

[Out]

-a/(4*x^4) - b/(2*x^2) + c*Log[x]

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Maple [A]  time = 0.047, size = 18, normalized size = 0.9 \begin{align*} -{\frac{a}{4\,{x}^{4}}}-{\frac{b}{2\,{x}^{2}}}+c\ln \left ( x \right ) \end{align*}

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/x^2*b+c*ln(x)

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Maxima [A]  time = 0.957028, size = 28, normalized size = 1.33 \begin{align*} \frac{1}{2} \, c \log \left (x^{2}\right ) - \frac{2 \, b x^{2} + a}{4 \, x^{4}} \end{align*}

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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Fricas [A]  time = 1.51292, size = 54, normalized size = 2.57 \begin{align*} \frac{4 \, c x^{4} \log \left (x\right ) - 2 \, b x^{2} - a}{4 \, x^{4}} \end{align*}

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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Sympy [A]  time = 0.40308, size = 17, normalized size = 0.81 \begin{align*} c \log{\left (x \right )} - \frac{a + 2 b x^{2}}{4 x^{4}} \end{align*}

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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Giac [A]  time = 1.30952, size = 36, normalized size = 1.71 \begin{align*} \frac{1}{2} \, c \log \left (x^{2}\right ) - \frac{3 \, c x^{4} + 2 \, b x^{2} + a}{4 \, x^{4}} \end{align*}

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