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

Optimal. Leaf size=23 \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

[Out]

2*x + x^2/2 + 9*Log[3 - x] + Log[1 + x]

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Rubi [A]  time = 0.0267258, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1593, 1628, 632, 31} \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[(3*x + x^3)/(-3 - 2*x + x^2),x]

[Out]

2*x + x^2/2 + 9*Log[3 - x] + Log[1 + x]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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{3 x+x^3}{-3-2 x+x^2} \, dx &=\int \frac{x \left (3+x^2\right )}{-3-2 x+x^2} \, dx\\ &=\int \left (2+x+\frac{2 (3+5 x)}{-3-2 x+x^2}\right ) \, dx\\ &=2 x+\frac{x^2}{2}+2 \int \frac{3+5 x}{-3-2 x+x^2} \, dx\\ &=2 x+\frac{x^2}{2}+9 \int \frac{1}{-3+x} \, dx+\int \frac{1}{1+x} \, dx\\ &=2 x+\frac{x^2}{2}+9 \log (3-x)+\log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0039116, size = 23, normalized size = 1. \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*x + x^2/2 + 9*Log[3 - x] + Log[1 + x]

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Maple [A]  time = 0.004, size = 20, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+2\,x+\ln \left ( 1+x \right ) +9\,\ln \left ( -3+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2+2*x+ln(1+x)+9*ln(-3+x)

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Maxima [A]  time = 0.943727, size = 26, normalized size = 1.13 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*x + log(x + 1) + 9*log(x - 3)

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Fricas [A]  time = 0.727047, size = 58, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*x + log(x + 1) + 9*log(x - 3)

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Sympy [A]  time = 0.102927, size = 19, normalized size = 0.83 \begin{align*} \frac{x^{2}}{2} + 2 x + 9 \log{\left (x - 3 \right )} + \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 + 2*x + 9*log(x - 3) + log(x + 1)

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Giac [A]  time = 1.07347, size = 28, normalized size = 1.22 \begin{align*} \frac{1}{2} \, x^{2} + 2 \, x + \log \left ({\left | x + 1 \right |}\right ) + 9 \, \log \left ({\left | x - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 2*x + log(abs(x + 1)) + 9*log(abs(x - 3))