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

Optimal. Leaf size=29 \[ \log \left (3 e^2 \left (\frac {1+\frac {5+\frac {-3+x}{x}}{x}}{x}+x^2\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 0.72, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2074, 1587} \begin {gather*} \log \left (-x^5-x^2-6 x+3\right )-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 1587

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]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, 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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.72 \begin {gather*} -3 \log (x)+\log \left (3-6 x-x^2-x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.49, size = 17, normalized size = 0.59 \begin {gather*} \log \left (x^{5} + x^{2} + 6 \, x - 3\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.23, size = 19, normalized size = 0.66 \begin {gather*} \log \left ({\left | x^{5} + x^{2} + 6 \, x - 3 \right |}\right ) - 3 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.02, size = 18, normalized size = 0.62

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.71, size = 17, normalized size = 0.59 \begin {gather*} \log \left (x^{5} + x^{2} + 6 \, x - 3\right ) - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 1.49, size = 17, normalized size = 0.59 \begin {gather*} \ln \left (x^5+x^2+6\,x-3\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.12, size = 17, normalized size = 0.59 \begin {gather*} - 3 \log {\relax (x )} + \log {\left (x^{5} + x^{2} + 6 x - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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