\(\int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx\) [5553]

Optimal result

Integrand size = 47, antiderivative size = 23 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log \left (x-12 \left (1+\frac {2 x}{5}\right ) x^2-\frac {x}{8+x}\right ) \]

[Out]

ln(x-x/(x+8)-3*x^2*(4+8/5*x))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2099, 1601} \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log \left (-24 x^3-252 x^2-475 x+35\right )+\log (x)-\log (x+8) \]

[In]

Int[(-280 + 7600*x + 6523*x^2 + 1272*x^3 + 72*x^4)/(-280*x + 3765*x^2 + 2491*x^3 + 444*x^4 + 24*x^5),x]

[Out]

Log[x] - Log[8 + x] + Log[35 - 475*x - 252*x^2 - 24*x^3]

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}{-8-x}+\frac {1}{x}+\frac {475+504 x+72 x^2}{-35+475 x+252 x^2+24 x^3}\right ) \, dx \\ & = \log (x)-\log (8+x)+\int \frac {475+504 x+72 x^2}{-35+475 x+252 x^2+24 x^3} \, dx \\ & = \log (x)-\log (8+x)+\log \left (35-475 x-252 x^2-24 x^3\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log (x)-\log (8+x)+\log \left (35-475 x-252 x^2-24 x^3\right ) \]

[In]

Integrate[(-280 + 7600*x + 6523*x^2 + 1272*x^3 + 72*x^4)/(-280*x + 3765*x^2 + 2491*x^3 + 444*x^4 + 24*x^5),x]

[Out]

Log[x] - Log[8 + x] + Log[35 - 475*x - 252*x^2 - 24*x^3]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

method	result	size
parallelrisch	\(\ln \left (x \right )-\ln \left (x +8\right )+\ln \left (x^{3}+\frac {21}{2} x^{2}+\frac {475}{24} x -\frac {35}{24}\right )\)	\(24\)
default	\(\ln \left (24 x^{3}+252 x^{2}+475 x -35\right )+\ln \left (x \right )-\ln \left (x +8\right )\)	\(26\)
norman	\(\ln \left (24 x^{3}+252 x^{2}+475 x -35\right )+\ln \left (x \right )-\ln \left (x +8\right )\)	\(26\)
risch	\(-\ln \left (x +8\right )+\ln \left (24 x^{4}+252 x^{3}+475 x^{2}-35 x \right )\)	\(28\)

[In]

int((72*x^4+1272*x^3+6523*x^2+7600*x-280)/(24*x^5+444*x^4+2491*x^3+3765*x^2-280*x),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(x+8)+ln(x^3+21/2*x^2+475/24*x-35/24)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log \left (24 \, x^{4} + 252 \, x^{3} + 475 \, x^{2} - 35 \, x\right ) - \log \left (x + 8\right ) \]

[In]

integrate((72*x^4+1272*x^3+6523*x^2+7600*x-280)/(24*x^5+444*x^4+2491*x^3+3765*x^2-280*x),x, algorithm="fricas"
)

[Out]

log(24*x^4 + 252*x^3 + 475*x^2 - 35*x) - log(x + 8)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=- \log {\left (x + 8 \right )} + \log {\left (24 x^{4} + 252 x^{3} + 475 x^{2} - 35 x \right )} \]

[In]

integrate((72*x**4+1272*x**3+6523*x**2+7600*x-280)/(24*x**5+444*x**4+2491*x**3+3765*x**2-280*x),x)

[Out]

-log(x + 8) + log(24*x**4 + 252*x**3 + 475*x**2 - 35*x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log \left (24 \, x^{3} + 252 \, x^{2} + 475 \, x - 35\right ) - \log \left (x + 8\right ) + \log \left (x\right ) \]

[In]

integrate((72*x^4+1272*x^3+6523*x^2+7600*x-280)/(24*x^5+444*x^4+2491*x^3+3765*x^2-280*x),x, algorithm="maxima"
)

[Out]

log(24*x^3 + 252*x^2 + 475*x - 35) - log(x + 8) + log(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\log \left ({\left | 24 \, x^{3} + 252 \, x^{2} + 475 \, x - 35 \right |}\right ) - \log \left ({\left | x + 8 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]

[In]

integrate((72*x^4+1272*x^3+6523*x^2+7600*x-280)/(24*x^5+444*x^4+2491*x^3+3765*x^2-280*x),x, algorithm="giac")

[Out]

log(abs(24*x^3 + 252*x^2 + 475*x - 35)) - log(abs(x + 8)) + log(abs(x))

Mupad [B] (verification not implemented)

Time = 10.77 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-280+7600 x+6523 x^2+1272 x^3+72 x^4}{-280 x+3765 x^2+2491 x^3+444 x^4+24 x^5} \, dx=\ln \left (x\,\left (24\,x^3+252\,x^2+475\,x-35\right )\right )-\ln \left (x+8\right ) \]

[In]

int((7600*x + 6523*x^2 + 1272*x^3 + 72*x^4 - 280)/(3765*x^2 - 280*x + 2491*x^3 + 444*x^4 + 24*x^5),x)

[Out]

log(x*(475*x + 252*x^2 + 24*x^3 - 35)) - log(x + 8)