3.473 \(\int \frac{-2+x^2+x^3}{x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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Rubi [A]  time = 0.0039203, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {14} \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + x^2 + x^3)/x^4,x]

[Out]

2/(3*x^3) - x^(-1) + 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{-2+x^2+x^3}{x^4} \, dx &=\int \left (-\frac{2}{x^4}+\frac{1}{x^2}+\frac{1}{x}\right ) \, dx\\ &=\frac{2}{3 x^3}-\frac{1}{x}+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0018698, size = 15, normalized size = 1. \[ \frac{2}{3 x^3}-\frac{1}{x}+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + x^2 + x^3)/x^4,x]

[Out]

2/(3*x^3) - x^(-1) + Log[x]

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \begin{align*}{\frac{2}{3\,{x}^{3}}}-{x}^{-1}+\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+x^2-2)/x^4,x)

[Out]

2/3/x^3-1/x+ln(x)

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Maxima [A]  time = 1.10812, size = 20, normalized size = 1.33 \begin{align*} -\frac{3 \, x^{2} - 2}{3 \, x^{3}} + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2-2)/x^4,x, algorithm="maxima")

[Out]

-1/3*(3*x^2 - 2)/x^3 + log(x)

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Fricas [A]  time = 1.1896, size = 49, normalized size = 3.27 \begin{align*} \frac{3 \, x^{3} \log \left (x\right ) - 3 \, x^{2} + 2}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2-2)/x^4,x, algorithm="fricas")

[Out]

1/3*(3*x^3*log(x) - 3*x^2 + 2)/x^3

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Sympy [A]  time = 0.084129, size = 14, normalized size = 0.93 \begin{align*} \log{\left (x \right )} - \frac{3 x^{2} - 2}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+x**2-2)/x**4,x)

[Out]

log(x) - (3*x**2 - 2)/(3*x**3)

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Giac [A]  time = 1.16263, size = 22, normalized size = 1.47 \begin{align*} -\frac{3 \, x^{2} - 2}{3 \, x^{3}} + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x^2-2)/x^4,x, algorithm="giac")

[Out]

-1/3*(3*x^2 - 2)/x^3 + log(abs(x))