\(\int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx\) [9711]

Optimal result

Integrand size = 55, antiderivative size = 23 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\log \left (16 x \left (-3+\frac {3 \left (3+x+x^2\right )^2}{12-x}\right )\right ) \]

[Out]

ln(16*x*(3/(12-x)*(x^2+x+3)^2-3))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2099, 1601} \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\log \left (-x^4-2 x^3-7 x^2-7 x+3\right )-\log (12-x)+\log (x) \]

[In]

Int[(36 - 168*x - 245*x^2 - 82*x^3 - 54*x^4 + 4*x^5)/(36*x - 87*x^2 - 77*x^3 - 17*x^4 - 10*x^5 + x^6),x]

[Out]

-Log[12 - x] + Log[x] + Log[3 - 7*x - 7*x^2 - 2*x^3 - x^4]

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 {1}{12-x}+\frac {1}{x}+\frac {7+14 x+6 x^2+4 x^3}{-3+7 x+7 x^2+2 x^3+x^4}\right ) \, dx \\ & = -\log (12-x)+\log (x)+\int \frac {7+14 x+6 x^2+4 x^3}{-3+7 x+7 x^2+2 x^3+x^4} \, dx \\ & = -\log (12-x)+\log (x)+\log \left (3-7 x-7 x^2-2 x^3-x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=-\log (12-x)+\log (x)+\log \left (3-7 x-7 x^2-2 x^3-x^4\right ) \]

[In]

Integrate[(36 - 168*x - 245*x^2 - 82*x^3 - 54*x^4 + 4*x^5)/(36*x - 87*x^2 - 77*x^3 - 17*x^4 - 10*x^5 + x^6),x]

[Out]

-Log[12 - x] + Log[x] + Log[3 - 7*x - 7*x^2 - 2*x^3 - x^4]

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26

method	result	size
default	\(\ln \left (x \right )-\ln \left (x -12\right )+\ln \left (x^{4}+2 x^{3}+7 x^{2}+7 x -3\right )\)	\(29\)
norman	\(\ln \left (x \right )-\ln \left (x -12\right )+\ln \left (x^{4}+2 x^{3}+7 x^{2}+7 x -3\right )\)	\(29\)
parallelrisch	\(\ln \left (x \right )-\ln \left (x -12\right )+\ln \left (x^{4}+2 x^{3}+7 x^{2}+7 x -3\right )\)	\(29\)
risch	\(-\ln \left (x -12\right )+\ln \left (x^{5}+2 x^{4}+7 x^{3}+7 x^{2}-3 x \right )\)	\(31\)

[In]

int((4*x^5-54*x^4-82*x^3-245*x^2-168*x+36)/(x^6-10*x^5-17*x^4-77*x^3-87*x^2+36*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(x-12)+ln(x^4+2*x^3+7*x^2+7*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\log \left (x^{5} + 2 \, x^{4} + 7 \, x^{3} + 7 \, x^{2} - 3 \, x\right ) - \log \left (x - 12\right ) \]

[In]

integrate((4*x^5-54*x^4-82*x^3-245*x^2-168*x+36)/(x^6-10*x^5-17*x^4-77*x^3-87*x^2+36*x),x, algorithm="fricas")

[Out]

log(x^5 + 2*x^4 + 7*x^3 + 7*x^2 - 3*x) - log(x - 12)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=- \log {\left (x - 12 \right )} + \log {\left (x^{5} + 2 x^{4} + 7 x^{3} + 7 x^{2} - 3 x \right )} \]

[In]

integrate((4*x**5-54*x**4-82*x**3-245*x**2-168*x+36)/(x**6-10*x**5-17*x**4-77*x**3-87*x**2+36*x),x)

[Out]

-log(x - 12) + log(x**5 + 2*x**4 + 7*x**3 + 7*x**2 - 3*x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\log \left (x^{4} + 2 \, x^{3} + 7 \, x^{2} + 7 \, x - 3\right ) - \log \left (x - 12\right ) + \log \left (x\right ) \]

[In]

integrate((4*x^5-54*x^4-82*x^3-245*x^2-168*x+36)/(x^6-10*x^5-17*x^4-77*x^3-87*x^2+36*x),x, algorithm="maxima")

[Out]

log(x^4 + 2*x^3 + 7*x^2 + 7*x - 3) - log(x - 12) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\log \left ({\left | x^{4} + 2 \, x^{3} + 7 \, x^{2} + 7 \, x - 3 \right |}\right ) - \log \left ({\left | x - 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((4*x^5-54*x^4-82*x^3-245*x^2-168*x+36)/(x^6-10*x^5-17*x^4-77*x^3-87*x^2+36*x),x, algorithm="giac")

[Out]

log(abs(x^4 + 2*x^3 + 7*x^2 + 7*x - 3)) - log(abs(x - 12)) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {36-168 x-245 x^2-82 x^3-54 x^4+4 x^5}{36 x-87 x^2-77 x^3-17 x^4-10 x^5+x^6} \, dx=\ln \left (x\,\left (x^4+2\,x^3+7\,x^2+7\,x-3\right )\right )-\ln \left (x-12\right ) \]

[In]

int((168*x + 245*x^2 + 82*x^3 + 54*x^4 - 4*x^5 - 36)/(87*x^2 - 36*x + 77*x^3 + 17*x^4 + 10*x^5 - x^6),x)

[Out]

log(x*(7*x + 7*x^2 + 2*x^3 + x^4 - 3)) - log(x - 12)