3.279 \(\int \frac{x^2+x^3}{-2+x+x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{x^2}{2}+\frac{2}{3} \log (1-x)+\frac{4}{3} \log (x+2) \]

[Out]

x^2/2 + (2*Log[1 - x])/3 + (4*Log[2 + x])/3

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Rubi [A]  time = 0.0245635, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1593, 800, 632, 31} \[ \frac{x^2}{2}+\frac{2}{3} \log (1-x)+\frac{4}{3} \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(x^2 + x^3)/(-2 + x + x^2),x]

[Out]

x^2/2 + (2*Log[1 - x])/3 + (4*Log[2 + x])/3

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0046166, size = 26, normalized size = 1. \[ \frac{x^2}{2}+\frac{2}{3} \log (1-x)+\frac{4}{3} \log (x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/2 + (2*Log[1 - x])/3 + (4*Log[2 + x])/3

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Maple [A]  time = 0.047, size = 19, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{4\,\ln \left ( 2+x \right ) }{3}}+{\frac{2\,\ln \left ( -1+x \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+x^2)/(x^2+x-2),x)

[Out]

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

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Maxima [A]  time = 0.998585, size = 24, normalized size = 0.92 \begin{align*} \frac{1}{2} \, x^{2} + \frac{4}{3} \, \log \left (x + 2\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.36524, size = 58, normalized size = 2.23 \begin{align*} \frac{1}{2} \, x^{2} + \frac{4}{3} \, \log \left (x + 2\right ) + \frac{2}{3} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.102618, size = 20, normalized size = 0.77 \begin{align*} \frac{x^{2}}{2} + \frac{2 \log{\left (x - 1 \right )}}{3} + \frac{4 \log{\left (x + 2 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19172, size = 27, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} + \frac{4}{3} \, \log \left ({\left | x + 2 \right |}\right ) + \frac{2}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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