3.301 \(\int \frac{-1+7 x+2 x^2}{-1-x+x^2+x^3} \, dx\)

Optimal. Leaf size=16 \[ 2 \log (1-x)-\frac{3}{x+1} \]

[Out]

-3/(1 + x) + 2*Log[1 - x]

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Rubi [A]  time = 0.0249667, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {2074} \[ 2 \log (1-x)-\frac{3}{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 7*x + 2*x^2)/(-1 - x + x^2 + x^3),x]

[Out]

-3/(1 + x) + 2*Log[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+7 x+2 x^2}{-1-x+x^2+x^3} \, dx &=\int \left (\frac{2}{-1+x}+\frac{3}{(1+x)^2}\right ) \, dx\\ &=-\frac{3}{1+x}+2 \log (1-x)\\ \end{align*}

Mathematica [A]  time = 0.0081222, size = 14, normalized size = 0.88 \[ 2 \log (x-1)-\frac{3}{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 7*x + 2*x^2)/(-1 - x + x^2 + x^3),x]

[Out]

-3/(1 + x) + 2*Log[-1 + x]

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Maple [A]  time = 0.007, size = 15, normalized size = 0.9 \begin{align*} 2\,\ln \left ( x-1 \right ) -3\, \left ( 1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+7*x-1)/(x^3+x^2-x-1),x)

[Out]

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

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Maxima [A]  time = 0.969146, size = 19, normalized size = 1.19 \begin{align*} -\frac{3}{x + 1} + 2 \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+7*x-1)/(x^3+x^2-x-1),x, algorithm="maxima")

[Out]

-3/(x + 1) + 2*log(x - 1)

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Fricas [A]  time = 1.54451, size = 49, normalized size = 3.06 \begin{align*} \frac{2 \,{\left (x + 1\right )} \log \left (x - 1\right ) - 3}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+7*x-1)/(x^3+x^2-x-1),x, algorithm="fricas")

[Out]

(2*(x + 1)*log(x - 1) - 3)/(x + 1)

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Sympy [A]  time = 0.08878, size = 10, normalized size = 0.62 \begin{align*} 2 \log{\left (x - 1 \right )} - \frac{3}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+7*x-1)/(x**3+x**2-x-1),x)

[Out]

2*log(x - 1) - 3/(x + 1)

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Giac [A]  time = 1.12265, size = 20, normalized size = 1.25 \begin{align*} -\frac{3}{x + 1} + 2 \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+7*x-1)/(x^3+x^2-x-1),x, algorithm="giac")

[Out]

-3/(x + 1) + 2*log(abs(x - 1))