3.819 \(\int \frac{a+b x^2+c x^4}{x^3} \, dx\)

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

[Out]

-a/(2*x^2) + (c*x^2)/2 + b*Log[x]

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Rubi [A]  time = 0.0069357, 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} \[ -\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0019273, size = 21, normalized size = 1. \[ -\frac{a}{2 x^2}+b \log (x)+\frac{c x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^3,x]

[Out]

-a/(2*x^2) + (c*x^2)/2 + b*Log[x]

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Maple [A]  time = 0.046, size = 18, normalized size = 0.9 \begin{align*} -{\frac{a}{2\,{x}^{2}}}+{\frac{c{x}^{2}}{2}}+b\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^3,x)

[Out]

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

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Maxima [A]  time = 0.989109, size = 27, normalized size = 1.29 \begin{align*} \frac{1}{2} \, c x^{2} + \frac{1}{2} \, b \log \left (x^{2}\right ) - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45976, size = 51, normalized size = 2.43 \begin{align*} \frac{c x^{4} + 2 \, b x^{2} \log \left (x\right ) - a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="fricas")

[Out]

1/2*(c*x^4 + 2*b*x^2*log(x) - a)/x^2

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Sympy [A]  time = 0.292389, size = 17, normalized size = 0.81 \begin{align*} - \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**3,x)

[Out]

-a/(2*x**2) + b*log(x) + c*x**2/2

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Giac [A]  time = 1.27056, size = 35, normalized size = 1.67 \begin{align*} \frac{1}{2} \, c x^{2} + \frac{1}{2} \, b \log \left (x^{2}\right ) - \frac{b x^{2} + a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^3,x, algorithm="giac")

[Out]

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