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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ \frac {x^3}{3}+\frac {x^2}{2}+x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^3)/(-1 + x),x]

[Out]

x + x^2/2 + x^3/3

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fricas [A]  time = 0.63, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} + x \]

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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giac [A]  time = 0.35, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} + x \]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \[ \frac {1}{3} x^{3}+\frac {1}{2} x^{2}+x \]

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 = 0.89, size = 12, normalized size = 0.75 \[ \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} + x \]

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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mupad [B]  time = 0.02, size = 13, normalized size = 0.81 \[ \frac {x\,\left (2\,x^2+3\,x+6\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - 1)/(x - 1),x)

[Out]

(x*(3*x + 2*x^2 + 6))/6

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sympy [A]  time = 0.06, size = 10, normalized size = 0.62 \[ \frac {x^{3}}{3} + \frac {x^{2}}{2} + x \]

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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