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

Optimal. Leaf size=16 \[ \frac{x^3}{3}+\frac{x^2}{2}+x \]

[Out]

x + x^2/2 + x^3/3

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Rubi [A]  time = 0.0060318, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1586} \[ \frac{x^3}{3}+\frac{x^2}{2}+x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x^3)/(-1 + x),x]

[Out]

x + x^2/2 + x^3/3

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

Mathematica [A]  time = 0.0004014, size = 16, normalized size = 1. \[ \frac{x^3}{3}+\frac{x^2}{2}+x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^3)/(-1 + x),x]

[Out]

x + x^2/2 + x^3/3

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*} x+{\frac{{x}^{2}}{2}}+{\frac{{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/2*x^2+1/3*x^3

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Maxima [A]  time = 1.26357, size = 16, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*x^2 + x

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Fricas [A]  time = 0.944745, size = 31, normalized size = 1.94 \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*x^2 + x

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Sympy [A]  time = 0.055744, size = 10, normalized size = 0.62 \begin{align*} \frac{x^{3}}{3} + \frac{x^{2}}{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + x**2/2 + x

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Giac [A]  time = 1.0754, size = 16, normalized size = 1. \begin{align*} \frac{1}{3} \, x^{3} + \frac{1}{2} \, x^{2} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 1/2*x^2 + x