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

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

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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Rubi [A]  time = 0.005449, 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} \[ a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x,x]

[Out]

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

Mathematica [A]  time = 0.0015812, size = 21, normalized size = 1. \[ a \log (x)+\frac{b x^2}{2}+\frac{c x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x,x]

[Out]

(b*x^2)/2 + (c*x^4)/4 + a*Log[x]

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Maple [A]  time = 0.043, size = 18, normalized size = 0.9 \begin{align*}{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{4}}{4}}+a\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,x)

[Out]

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

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Maxima [A]  time = 0.995727, size = 27, normalized size = 1.29 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + \frac{1}{2} \, a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48111, size = 46, normalized size = 2.19 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*c*x^4 + 1/2*b*x^2 + a*log(x)

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Sympy [A]  time = 0.096976, size = 17, normalized size = 0.81 \begin{align*} a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) + b*x**2/2 + c*x**4/4

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Giac [A]  time = 1.32177, size = 27, normalized size = 1.29 \begin{align*} \frac{1}{4} \, c x^{4} + \frac{1}{2} \, b x^{2} + \frac{1}{2} \, a \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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