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

Optimal. Leaf size=21 \[ \frac{2}{1-x}-\frac{3}{2 (1-x)^2} \]

[Out]

-3/(2*(1 - x)^2) + 2/(1 - x)

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Rubi [A]  time = 0.0192289, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2074} \[ \frac{2}{1-x}-\frac{3}{2 (1-x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3/(2*(1 - x)^2) + 2/(1 - x)

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

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

Mathematica [A]  time = 0.0033034, size = 14, normalized size = 0.67 \[ \frac{1-4 x}{2 (x-1)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 4*x)/(2*(-1 + x)^2)

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \begin{align*} -2\, \left ( x-1 \right ) ^{-1}-{\frac{3}{2\, \left ( x-1 \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.998447, size = 23, normalized size = 1.1 \begin{align*} -\frac{4 \, x - 1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*x - 1)/(x^2 - 2*x + 1)

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Fricas [A]  time = 1.45897, size = 43, normalized size = 2.05 \begin{align*} -\frac{4 \, x - 1}{2 \,{\left (x^{2} - 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*x - 1)/(x^2 - 2*x + 1)

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Sympy [A]  time = 0.084257, size = 15, normalized size = 0.71 \begin{align*} - \frac{4 x - 1}{2 x^{2} - 4 x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*x - 1)/(2*x**2 - 4*x + 2)

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Giac [A]  time = 1.13127, size = 16, normalized size = 0.76 \begin{align*} -\frac{4 \, x - 1}{2 \,{\left (x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*x - 1)/(x - 1)^2