3.180 \(\int \frac{5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx\)

Optimal. Leaf size=38 \[ \frac{2}{1-x}+\frac{1}{x+1}-\frac{3}{2 (1-x)^2}+\log (1-x)-2 \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0619045, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {2074} \[ \frac{2}{1-x}+\frac{1}{x+1}-\frac{3}{2 (1-x)^2}+\log (1-x)-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(5 - 3*x + 6*x^2 + 5*x^3 - x^4)/(-1 + x + 2*x^2 - 2*x^3 - x^4 + x^5),x]

[Out]

-3/(2*(1 - x)^2) + 2/(1 - x) + (1 + x)^(-1) + Log[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{5-3 x+6 x^2+5 x^3-x^4}{-1+x+2 x^2-2 x^3-x^4+x^5} \, dx &=\int \left (\frac{3}{(-1+x)^3}+\frac{2}{(-1+x)^2}+\frac{1}{-1+x}-\frac{1}{(1+x)^2}-\frac{2}{1+x}\right ) \, dx\\ &=-\frac{3}{2 (1-x)^2}+\frac{2}{1-x}+\frac{1}{1+x}+\log (1-x)-2 \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0206963, size = 32, normalized size = 0.84 \[ -\frac{2}{x-1}+\frac{1}{x+1}-\frac{3}{2 (x-1)^2}+\log (x-1)-2 \log (x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - 3*x + 6*x^2 + 5*x^3 - x^4)/(-1 + x + 2*x^2 - 2*x^3 - x^4 + x^5),x]

[Out]

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

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Maple [A]  time = 0.01, size = 31, normalized size = 0.8 \begin{align*} \left ( 1+x \right ) ^{-1}-2\,\ln \left ( 1+x \right ) +\ln \left ( -1+x \right ) -{\frac{3}{2\, \left ( -1+x \right ) ^{2}}}-2\, \left ( -1+x \right ) ^{-1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^4+5*x^3+6*x^2-3*x+5)/(x^5-x^4-2*x^3+2*x^2+x-1),x)

[Out]

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

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Maxima [A]  time = 0.925671, size = 51, normalized size = 1.34 \begin{align*} -\frac{2 \, x^{2} + 7 \, x - 3}{2 \,{\left (x^{3} - x^{2} - x + 1\right )}} - 2 \, \log \left (x + 1\right ) + \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+5*x^3+6*x^2-3*x+5)/(x^5-x^4-2*x^3+2*x^2+x-1),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.79386, size = 154, normalized size = 4.05 \begin{align*} -\frac{2 \, x^{2} + 4 \,{\left (x^{3} - x^{2} - x + 1\right )} \log \left (x + 1\right ) - 2 \,{\left (x^{3} - x^{2} - x + 1\right )} \log \left (x - 1\right ) + 7 \, x - 3}{2 \,{\left (x^{3} - x^{2} - x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+5*x^3+6*x^2-3*x+5)/(x^5-x^4-2*x^3+2*x^2+x-1),x, algorithm="fricas")

[Out]

-1/2*(2*x^2 + 4*(x^3 - x^2 - x + 1)*log(x + 1) - 2*(x^3 - x^2 - x + 1)*log(x - 1) + 7*x - 3)/(x^3 - x^2 - x +
1)

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Sympy [A]  time = 0.126344, size = 36, normalized size = 0.95 \begin{align*} - \frac{2 x^{2} + 7 x - 3}{2 x^{3} - 2 x^{2} - 2 x + 2} + \log{\left (x - 1 \right )} - 2 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**4+5*x**3+6*x**2-3*x+5)/(x**5-x**4-2*x**3+2*x**2+x-1),x)

[Out]

-(2*x**2 + 7*x - 3)/(2*x**3 - 2*x**2 - 2*x + 2) + log(x - 1) - 2*log(x + 1)

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Giac [A]  time = 1.04539, size = 47, normalized size = 1.24 \begin{align*} -\frac{2 \, x^{2} + 7 \, x - 3}{2 \,{\left (x + 1\right )}{\left (x - 1\right )}^{2}} - 2 \, \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^4+5*x^3+6*x^2-3*x+5)/(x^5-x^4-2*x^3+2*x^2+x-1),x, algorithm="giac")

[Out]

-1/2*(2*x^2 + 7*x - 3)/((x + 1)*(x - 1)^2) - 2*log(abs(x + 1)) + log(abs(x - 1))