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

Optimal. Leaf size=23 \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

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Rubi [A]  time = 0.0540768, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1594, 1628, 628} \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Int[(3 - x + 3*x^2 - 2*x^3 + x^4)/(3*x - 2*x^2 + x^3),x]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0074805, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - x + 3*x^2 - 2*x^3 + x^4)/(3*x - 2*x^2 + x^3),x]

[Out]

x^2/2 + Log[x] - Log[3 - 2*x + x^2]/2

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Maple [A]  time = 0.004, size = 20, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}-2\,x+3 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.993679, size = 26, normalized size = 1.13 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4448, size = 58, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.098815, size = 19, normalized size = 0.83 \begin{align*} \frac{x^{2}}{2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} - 2 x + 3 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + log(x) - log(x**2 - 2*x + 3)/2

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Giac [A]  time = 1.38314, size = 27, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*log(x^2 - 2*x + 3) + log(abs(x))