3.482 \(\int \frac {-1+3 x-3 x^2+x^3}{x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {x^2}{2}-3 x+\frac {1}{x}+3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {14} \[ \frac {x^2}{2}-3 x+\frac {1}{x}+3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x - 3*x^2 + x^3)/x^2,x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {x^2}{2}-3 x+\frac {1}{x}+3 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 3*x - 3*x^2 + x^3)/x^2,x]

[Out]

x^(-1) - 3*x + x^2/2 + 3*Log[x]

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fricas [A]  time = 0.62, size = 20, normalized size = 1.11 \[ \frac {x^{3} - 6 \, x^{2} + 6 \, x \log \relax (x) + 2}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-3*x^2+3*x-1)/x^2,x, algorithm="fricas")

[Out]

1/2*(x^3 - 6*x^2 + 6*x*log(x) + 2)/x

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giac [A]  time = 0.37, size = 17, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} - 3 \, x + \frac {1}{x} + 3 \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-3*x^2+3*x-1)/x^2,x, algorithm="giac")

[Out]

1/2*x^2 - 3*x + 1/x + 3*log(abs(x))

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maple [A]  time = 0.00, size = 17, normalized size = 0.94 \[ \frac {x^{2}}{2}-3 x +3 \ln \relax (x )+\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-3*x^2+3*x-1)/x^2,x)

[Out]

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

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maxima [A]  time = 0.73, size = 16, normalized size = 0.89 \[ \frac {1}{2} \, x^{2} - 3 \, x + \frac {1}{x} + 3 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-3*x^2+3*x-1)/x^2,x, algorithm="maxima")

[Out]

1/2*x^2 - 3*x + 1/x + 3*log(x)

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mupad [B]  time = 0.03, size = 16, normalized size = 0.89 \[ 3\,\ln \relax (x)-3\,x+\frac {1}{x}+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 3*x^2 + x^3 - 1)/x^2,x)

[Out]

3*log(x) - 3*x + 1/x + x^2/2

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sympy [A]  time = 0.08, size = 15, normalized size = 0.83 \[ \frac {x^{2}}{2} - 3 x + 3 \log {\relax (x )} + \frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-3*x**2+3*x-1)/x**2,x)

[Out]

x**2/2 - 3*x + 3*log(x) + 1/x

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