3.295 \(\int \frac{2-7 x+x^2-x^3+x^4}{-24-14 x+x^2+x^3} \, dx\)

Optimal. Leaf size=35 \[ \frac{x^2}{2}-2 x+\frac{13}{3} \log (4-x)-\frac{22}{3} \log (x+2)+20 \log (x+3) \]

[Out]

-2*x + x^2/2 + (13*Log[4 - x])/3 - (22*Log[2 + x])/3 + 20*Log[3 + x]

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Rubi [A]  time = 0.0397596, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2074} \[ \frac{x^2}{2}-2 x+\frac{13}{3} \log (4-x)-\frac{22}{3} \log (x+2)+20 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[(2 - 7*x + x^2 - x^3 + x^4)/(-24 - 14*x + x^2 + x^3),x]

[Out]

-2*x + x^2/2 + (13*Log[4 - x])/3 - (22*Log[2 + x])/3 + 20*Log[3 + 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{2-7 x+x^2-x^3+x^4}{-24-14 x+x^2+x^3} \, dx &=\int \left (-2+\frac{13}{3 (-4+x)}+x-\frac{22}{3 (2+x)}+\frac{20}{3+x}\right ) \, dx\\ &=-2 x+\frac{x^2}{2}+\frac{13}{3} \log (4-x)-\frac{22}{3} \log (2+x)+20 \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.0097404, size = 35, normalized size = 1. \[ \frac{x^2}{2}-2 x+\frac{13}{3} \log (4-x)-\frac{22}{3} \log (x+2)+20 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 7*x + x^2 - x^3 + x^4)/(-24 - 14*x + x^2 + x^3),x]

[Out]

-2*x + x^2/2 + (13*Log[4 - x])/3 - (22*Log[2 + x])/3 + 20*Log[3 + x]

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Maple [A]  time = 0.008, size = 28, normalized size = 0.8 \begin{align*} -2\,x+{\frac{{x}^{2}}{2}}-{\frac{22\,\ln \left ( 2+x \right ) }{3}}+{\frac{13\,\ln \left ( x-4 \right ) }{3}}+20\,\ln \left ( 3+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x^3+x^2-7*x+2)/(x^3+x^2-14*x-24),x)

[Out]

-2*x+1/2*x^2-22/3*ln(2+x)+13/3*ln(x-4)+20*ln(3+x)

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Maxima [A]  time = 0.962659, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{2} \, x^{2} - 2 \, x + 20 \, \log \left (x + 3\right ) - \frac{22}{3} \, \log \left (x + 2\right ) + \frac{13}{3} \, \log \left (x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+x^2-7*x+2)/(x^3+x^2-14*x-24),x, algorithm="maxima")

[Out]

1/2*x^2 - 2*x + 20*log(x + 3) - 22/3*log(x + 2) + 13/3*log(x - 4)

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Fricas [A]  time = 1.53091, size = 90, normalized size = 2.57 \begin{align*} \frac{1}{2} \, x^{2} - 2 \, x + 20 \, \log \left (x + 3\right ) - \frac{22}{3} \, \log \left (x + 2\right ) + \frac{13}{3} \, \log \left (x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+x^2-7*x+2)/(x^3+x^2-14*x-24),x, algorithm="fricas")

[Out]

1/2*x^2 - 2*x + 20*log(x + 3) - 22/3*log(x + 2) + 13/3*log(x - 4)

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Sympy [A]  time = 0.132018, size = 31, normalized size = 0.89 \begin{align*} \frac{x^{2}}{2} - 2 x + \frac{13 \log{\left (x - 4 \right )}}{3} - \frac{22 \log{\left (x + 2 \right )}}{3} + 20 \log{\left (x + 3 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x**3+x**2-7*x+2)/(x**3+x**2-14*x-24),x)

[Out]

x**2/2 - 2*x + 13*log(x - 4)/3 - 22*log(x + 2)/3 + 20*log(x + 3)

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Giac [A]  time = 1.12382, size = 41, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} - 2 \, x + 20 \, \log \left ({\left | x + 3 \right |}\right ) - \frac{22}{3} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{13}{3} \, \log \left ({\left | x - 4 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x^3+x^2-7*x+2)/(x^3+x^2-14*x-24),x, algorithm="giac")

[Out]

1/2*x^2 - 2*x + 20*log(abs(x + 3)) - 22/3*log(abs(x + 2)) + 13/3*log(abs(x - 4))