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

Optimal. Leaf size=11 \[ \frac{x^2}{2}-x \]

[Out]

-x + x^2/2

________________________________________________________________________________________

Rubi [A]  time = 0.0074939, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1586} \[ \frac{x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)/(1 + x + x^2),x]

[Out]

-x + x^2/2

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

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

Mathematica [A]  time = 0.0004252, size = 11, normalized size = 1. \[ \frac{x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^3)/(1 + x + x^2),x]

[Out]

-x + x^2/2

________________________________________________________________________________________

Maple [A]  time = 0., size = 10, normalized size = 0.9 \begin{align*} -x+{\frac{{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x+1/2*x^2

________________________________________________________________________________________

Maxima [A]  time = 1.10546, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x

________________________________________________________________________________________

Fricas [A]  time = 1.36925, size = 18, normalized size = 1.64 \begin{align*} \frac{1}{2} \, x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x

________________________________________________________________________________________

Sympy [A]  time = 0.054409, size = 5, normalized size = 0.45 \begin{align*} \frac{x^{2}}{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - x

________________________________________________________________________________________

Giac [A]  time = 1.13676, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{2} \, x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - x