\(\int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx\) [6053]

Optimal result

Integrand size = 47, antiderivative size = 24 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=-x+\frac {3+x}{x}+\log \left (\left (-1+\frac {1}{625 x^5}+x\right )^2\right ) \]

[Out]

ln((x+1/625/x^5-1)^2)-x+(3+x)/x

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {1608, 6874, 1601} \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=2 \log \left (625 x^6-625 x^5+1\right )-x+\frac {3}{x}-10 \log (x) \]

[In]

Int[(-3 - 10*x - x^2 + 1875*x^5 - 1875*x^6 + 1875*x^7 - 625*x^8)/(x^2 - 625*x^7 + 625*x^8),x]

[Out]

3/x - x - 10*Log[x] + 2*Log[1 - 625*x^5 + 625*x^6]

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 1608

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2 \left (1-625 x^5+625 x^6\right )} \, dx \\ & = \int \left (-1-\frac {3}{x^2}-\frac {10}{x}+\frac {1250 x^4 (-5+6 x)}{1-625 x^5+625 x^6}\right ) \, dx \\ & = \frac {3}{x}-x-10 \log (x)+1250 \int \frac {x^4 (-5+6 x)}{1-625 x^5+625 x^6} \, dx \\ & = \frac {3}{x}-x-10 \log (x)+2 \log \left (1-625 x^5+625 x^6\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=\frac {3}{x}-x-10 \log (x)+2 \log \left (1-625 x^5+625 x^6\right ) \]

[In]

Integrate[(-3 - 10*x - x^2 + 1875*x^5 - 1875*x^6 + 1875*x^7 - 625*x^8)/(x^2 - 625*x^7 + 625*x^8),x]

[Out]

3/x - x - 10*Log[x] + 2*Log[1 - 625*x^5 + 625*x^6]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21

method	result	size
default	\(-x +\frac {3}{x}-10 \ln \left (x \right )+2 \ln \left (625 x^{6}-625 x^{5}+1\right )\)	\(29\)
risch	\(-x +\frac {3}{x}-10 \ln \left (x \right )+2 \ln \left (625 x^{6}-625 x^{5}+1\right )\)	\(29\)
parallelrisch	\(-\frac {10 x \ln \left (x \right )-2 \ln \left (x^{6}+\frac {1}{625}-x^{5}\right ) x +x^{2}-3}{x}\)	\(30\)
norman	\(\frac {-x^{2}+3}{x}-10 \ln \left (x \right )+2 \ln \left (625 x^{6}-625 x^{5}+1\right )\)	\(32\)

[In]

int((-625*x^8+1875*x^7-1875*x^6+1875*x^5-x^2-10*x-3)/(625*x^8-625*x^7+x^2),x,method=_RETURNVERBOSE)

[Out]

-x+3/x-10*ln(x)+2*ln(625*x^6-625*x^5+1)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=-\frac {x^{2} - 2 \, x \log \left (625 \, x^{6} - 625 \, x^{5} + 1\right ) + 10 \, x \log \left (x\right ) - 3}{x} \]

[In]

integrate((-625*x^8+1875*x^7-1875*x^6+1875*x^5-x^2-10*x-3)/(625*x^8-625*x^7+x^2),x, algorithm="fricas")

[Out]

-(x^2 - 2*x*log(625*x^6 - 625*x^5 + 1) + 10*x*log(x) - 3)/x

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=- x - 10 \log {\left (x \right )} + 2 \log {\left (625 x^{6} - 625 x^{5} + 1 \right )} + \frac {3}{x} \]

[In]

integrate((-625*x**8+1875*x**7-1875*x**6+1875*x**5-x**2-10*x-3)/(625*x**8-625*x**7+x**2),x)

[Out]

-x - 10*log(x) + 2*log(625*x**6 - 625*x**5 + 1) + 3/x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=-x + \frac {3}{x} + 2 \, \log \left (625 \, x^{6} - 625 \, x^{5} + 1\right ) - 10 \, \log \left (x\right ) \]

[In]

integrate((-625*x^8+1875*x^7-1875*x^6+1875*x^5-x^2-10*x-3)/(625*x^8-625*x^7+x^2),x, algorithm="maxima")

[Out]

-x + 3/x + 2*log(625*x^6 - 625*x^5 + 1) - 10*log(x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=-x + \frac {3}{x} + 2 \, \log \left ({\left | 625 \, x^{6} - 625 \, x^{5} + 1 \right |}\right ) - 10 \, \log \left ({\left | x \right |}\right ) \]

[In]

integrate((-625*x^8+1875*x^7-1875*x^6+1875*x^5-x^2-10*x-3)/(625*x^8-625*x^7+x^2),x, algorithm="giac")

[Out]

-x + 3/x + 2*log(abs(625*x^6 - 625*x^5 + 1)) - 10*log(abs(x))

Mupad [B] (verification not implemented)

Time = 11.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-3-10 x-x^2+1875 x^5-1875 x^6+1875 x^7-625 x^8}{x^2-625 x^7+625 x^8} \, dx=2\,\ln \left (x^6-x^5+\frac {1}{625}\right )-x-10\,\ln \left (x\right )+\frac {3}{x} \]

[In]

int(-(10*x + x^2 - 1875*x^5 + 1875*x^6 - 1875*x^7 + 625*x^8 + 3)/(x^2 - 625*x^7 + 625*x^8),x)

[Out]

2*log(x^6 - x^5 + 1/625) - x - 10*log(x) + 3/x