\(\int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+(600 x-150 x^2-8 x^3+2 x^4) \log (\frac {300-75 x-4 x^2+x^3}{x})} \, dx\) [2968]

Optimal result

Integrand size = 85, antiderivative size = 24 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (-3-\frac {x}{2}+\log \left ((4-x) \left (\frac {75}{x}-x\right )\right )\right ) \]

[Out]

ln(ln((75/x-x)*(-x+4))-1/2*x-3)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 6816} \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (-2 \log \left (x^2-4 x+\frac {300}{x}-75\right )+x+6\right ) \]

[In]

Int[(-600 - 300*x + 67*x^2 + 8*x^3 - x^4)/(-1800*x + 150*x^2 + 99*x^3 - 2*x^4 - x^5 + (600*x - 150*x^2 - 8*x^3
+ 2*x^4)*Log[(300 - 75*x - 4*x^2 + x^3)/x]),x]

[Out]

Log[6 + x - 2*Log[-75 + 300/x - 4*x + x^2]]

Rule 6816

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {600+300 x-67 x^2-8 x^3+x^4}{x \left (300-75 x-4 x^2+x^3\right ) \left (6+x-2 \log \left (-75+\frac {300}{x}-4 x+x^2\right )\right )} \, dx \\ & = \log \left (6+x-2 \log \left (-75+\frac {300}{x}-4 x+x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (6+x-2 \log \left (-75+\frac {300}{x}-4 x+x^2\right )\right ) \]

[In]

Integrate[(-600 - 300*x + 67*x^2 + 8*x^3 - x^4)/(-1800*x + 150*x^2 + 99*x^3 - 2*x^4 - x^5 + (600*x - 150*x^2 -
8*x^3 + 2*x^4)*Log[(300 - 75*x - 4*x^2 + x^3)/x]),x]

[Out]

Log[6 + x - 2*Log[-75 + 300/x - 4*x + x^2]]

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04

method	result	size
norman	\(\ln \left (x -2 \ln \left (\frac {x^{3}-4 x^{2}-75 x +300}{x}\right )+6\right )\)	\(25\)
risch	\(\ln \left (-\frac {x}{2}+\ln \left (\frac {x^{3}-4 x^{2}-75 x +300}{x}\right )-3\right )\)	\(25\)
parallelrisch	\(\ln \left (x -2 \ln \left (\frac {x^{3}-4 x^{2}-75 x +300}{x}\right )+6\right )\)	\(25\)

[In]

int((-x^4+8*x^3+67*x^2-300*x-600)/((2*x^4-8*x^3-150*x^2+600*x)*ln((x^3-4*x^2-75*x+300)/x)-x^5-2*x^4+99*x^3+150
*x^2-1800*x),x,method=_RETURNVERBOSE)

[Out]

ln(x-2*ln((x^3-4*x^2-75*x+300)/x)+6)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (-x + 2 \, \log \left (\frac {x^{3} - 4 \, x^{2} - 75 \, x + 300}{x}\right ) - 6\right ) \]

[In]

integrate((-x^4+8*x^3+67*x^2-300*x-600)/((2*x^4-8*x^3-150*x^2+600*x)*log((x^3-4*x^2-75*x+300)/x)-x^5-2*x^4+99*
x^3+150*x^2-1800*x),x, algorithm="fricas")

[Out]

log(-x + 2*log((x^3 - 4*x^2 - 75*x + 300)/x) - 6)

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log {\left (- \frac {x}{2} + \log {\left (\frac {x^{3} - 4 x^{2} - 75 x + 300}{x} \right )} - 3 \right )} \]

[In]

integrate((-x**4+8*x**3+67*x**2-300*x-600)/((2*x**4-8*x**3-150*x**2+600*x)*ln((x**3-4*x**2-75*x+300)/x)-x**5-2
*x**4+99*x**3+150*x**2-1800*x),x)

[Out]

log(-x/2 + log((x**3 - 4*x**2 - 75*x + 300)/x) - 3)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (-\frac {1}{2} \, x + \log \left (x^{2} - 75\right ) + \log \left (x - 4\right ) - \log \left (x\right ) - 3\right ) \]

[In]

integrate((-x^4+8*x^3+67*x^2-300*x-600)/((2*x^4-8*x^3-150*x^2+600*x)*log((x^3-4*x^2-75*x+300)/x)-x^5-2*x^4+99*
x^3+150*x^2-1800*x),x, algorithm="maxima")

[Out]

log(-1/2*x + log(x^2 - 75) + log(x - 4) - log(x) - 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\log \left (x - 2 \, \log \left (\frac {x^{3} - 4 \, x^{2} - 75 \, x + 300}{x}\right ) + 6\right ) \]

[In]

integrate((-x^4+8*x^3+67*x^2-300*x-600)/((2*x^4-8*x^3-150*x^2+600*x)*log((x^3-4*x^2-75*x+300)/x)-x^5-2*x^4+99*
x^3+150*x^2-1800*x),x, algorithm="giac")

[Out]

log(x - 2*log((x^3 - 4*x^2 - 75*x + 300)/x) + 6)

Mupad [B] (verification not implemented)

Time = 9.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {-600-300 x+67 x^2+8 x^3-x^4}{-1800 x+150 x^2+99 x^3-2 x^4-x^5+\left (600 x-150 x^2-8 x^3+2 x^4\right ) \log \left (\frac {300-75 x-4 x^2+x^3}{x}\right )} \, dx=\ln \left (\ln \left (-\frac {-x^3+4\,x^2+75\,x-300}{x}\right )-\frac {x}{2}-3\right ) \]

[In]

int((300*x - 67*x^2 - 8*x^3 + x^4 + 600)/(1800*x - log(-(75*x + 4*x^2 - x^3 - 300)/x)*(600*x - 150*x^2 - 8*x^3
+ 2*x^4) - 150*x^2 - 99*x^3 + 2*x^4 + x^5),x)

[Out]

log(log(-(75*x + 4*x^2 - x^3 - 300)/x) - x/2 - 3)