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

Optimal. Leaf size=17 \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1594, 1628} \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 - x] + Log[x] + 3*Log[3 + x]

Rule 1594

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ 2 \log (1-x)+\log (x)+3 \log (x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Log[1 - x] + Log[x] + 3*Log[3 + x]

________________________________________________________________________________________

fricas [A]  time = 1.13, size = 15, normalized size = 0.88 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x + 3) + 2*log(x - 1) + log(x)

________________________________________________________________________________________

giac [A]  time = 0.38, size = 18, normalized size = 1.06 \[ 3 \, \log \left ({\left | x + 3 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(abs(x + 3)) + 2*log(abs(x - 1)) + log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 16, normalized size = 0.94 \[ \ln \relax (x )+2 \ln \left (x -1\right )+3 \ln \left (x +3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.85, size = 15, normalized size = 0.88 \[ 3 \, \log \left (x + 3\right ) + 2 \, \log \left (x - 1\right ) + \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x + 3) + 2*log(x - 1) + log(x)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 15, normalized size = 0.88 \[ 2\,\ln \left (x-1\right )+3\,\ln \left (x+3\right )+\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x - 1) + 3*log(x + 3) + log(x)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 15, normalized size = 0.88 \[ \log {\relax (x )} + 2 \log {\left (x - 1 \right )} + 3 \log {\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x) + 2*log(x - 1) + 3*log(x + 3)

________________________________________________________________________________________