\(\int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx\) [2716]

Optimal result

Integrand size = 33, antiderivative size = 29 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\log \left (3 e^2 \left (\frac {1+\frac {5+\frac {-3+x}{x}}{x}}{x}+x^2\right )\right ) \]

[Out]

ln(3*(x^2+(1+1/x*((-3+x)/x+5))/x)*exp(2))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2099, 1601} \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\log \left (-x^5-x^2-6 x+3\right )-3 \log (x) \]

[In]

Int[(9 - 12*x - x^2 + 2*x^5)/(-3*x + 6*x^2 + x^3 + x^6),x]

[Out]

-3*Log[x] + Log[3 - 6*x - x^2 - x^5]

Rule 1601

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*(Log[RemoveConte
nt[Qq, x]]/(q*Coeff[Qq, x, q])), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]/(q*Coeff[Qq, x, q]))
*D[Qq, x]]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2099

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=-3 \log (x)+\log \left (3-6 x-x^2-x^5\right ) \]

[In]

Integrate[(9 - 12*x - x^2 + 2*x^5)/(-3*x + 6*x^2 + x^3 + x^6),x]

[Out]

-3*Log[x] + Log[3 - 6*x - x^2 - x^5]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62

method	result	size
default	\(\ln \left (x^{5}+x^{2}+6 x -3\right )-3 \ln \left (x \right )\)	\(18\)
norman	\(\ln \left (x^{5}+x^{2}+6 x -3\right )-3 \ln \left (x \right )\)	\(18\)
risch	\(\ln \left (x^{5}+x^{2}+6 x -3\right )-3 \ln \left (x \right )\)	\(18\)
parallelrisch	\(\ln \left (x^{5}+x^{2}+6 x -3\right )-3 \ln \left (x \right )\)	\(18\)

[In]

int((2*x^5-x^2-12*x+9)/(x^6+x^3+6*x^2-3*x),x,method=_RETURNVERBOSE)

[Out]

ln(x^5+x^2+6*x-3)-3*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\log \left (x^{5} + x^{2} + 6 \, x - 3\right ) - 3 \, \log \left (x\right ) \]

[In]

integrate((2*x^5-x^2-12*x+9)/(x^6+x^3+6*x^2-3*x),x, algorithm="fricas")

[Out]

log(x^5 + x^2 + 6*x - 3) - 3*log(x)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=- 3 \log {\left (x \right )} + \log {\left (x^{5} + x^{2} + 6 x - 3 \right )} \]

[In]

integrate((2*x**5-x**2-12*x+9)/(x**6+x**3+6*x**2-3*x),x)

[Out]

-3*log(x) + log(x**5 + x**2 + 6*x - 3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\log \left (x^{5} + x^{2} + 6 \, x - 3\right ) - 3 \, \log \left (x\right ) \]

[In]

integrate((2*x^5-x^2-12*x+9)/(x^6+x^3+6*x^2-3*x),x, algorithm="maxima")

[Out]

log(x^5 + x^2 + 6*x - 3) - 3*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\log \left ({\left | x^{5} + x^{2} + 6 \, x - 3 \right |}\right ) - 3 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((2*x^5-x^2-12*x+9)/(x^6+x^3+6*x^2-3*x),x, algorithm="giac")

[Out]

log(abs(x^5 + x^2 + 6*x - 3)) - 3*log(abs(x))

Mupad [B] (verification not implemented)

Time = 9.39 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {9-12 x-x^2+2 x^5}{-3 x+6 x^2+x^3+x^6} \, dx=\ln \left (x^5+x^2+6\,x-3\right )-3\,\ln \left (x\right ) \]

[In]

int(-(12*x + x^2 - 2*x^5 - 9)/(6*x^2 - 3*x + x^3 + x^6),x)

[Out]

log(6*x + x^2 + x^5 - 3) - 3*log(x)