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

Optimal. Leaf size=24 \[ -x+\frac {3+x}{x}+\log \left (\left (-1+\frac {1}{625 x^5}+x\right )^2\right ) \]

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Rubi [A]  time = 0.24, antiderivative size = 28, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.064, Rules used = {1594, 6742, 1587} \begin {gather*} 2 \log \left (625 x^6-625 x^5+1\right )-x+\frac {3}{x}-10 \log (x) \end {gather*}

Antiderivative was successfully verified.

[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 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 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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 1.17 \begin {gather*} \frac {3}{x}-x-10 \log (x)+2 \log \left (1-625 x^5+625 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

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fricas [A]  time = 0.69, size = 31, normalized size = 1.29 \begin {gather*} -\frac {x^{2} - 2 \, x \log \left (625 \, x^{6} - 625 \, x^{5} + 1\right ) + 10 \, x \log \relax (x) - 3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.13, size = 30, normalized size = 1.25 \begin {gather*} -x + \frac {3}{x} + 2 \, \log \left ({\left | 625 \, x^{6} - 625 \, x^{5} + 1 \right |}\right ) - 10 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.04, size = 29, normalized size = 1.21

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.35, size = 28, normalized size = 1.17 \begin {gather*} -x + \frac {3}{x} + 2 \, \log \left (625 \, x^{6} - 625 \, x^{5} + 1\right ) - 10 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.23, size = 26, normalized size = 1.08 \begin {gather*} 2\,\ln \left (x^6-x^5+\frac {1}{625}\right )-x-10\,\ln \relax (x)+\frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.12, size = 24, normalized size = 1.00 \begin {gather*} - x - 10 \log {\relax (x )} + 2 \log {\left (625 x^{6} - 625 x^{5} + 1 \right )} + \frac {3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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