3.5.6 \(\int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+(-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8) \log (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5})+(-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7) \log ^2(\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5})} \, dx\) [406]

3.5.6.1 Optimal result

Integrand size = 220, antiderivative size = 35 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=\frac {5}{\frac {2 x}{3}-\frac {5}{3} \left (4+\log \left (\frac {x^2}{x-\frac {1}{x^2 (2+x)^2}}\right )\right )} \end {dmath*}

output

5/(2/3*x-5/3*ln(x^2/(x-1/x^2/(2+x)^2))-20/3)
 

3.5.6.2 Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.09 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=-\frac {15}{20-2 x+5 \log \left (\frac {x^4 (2+x)^2}{-1+4 x^3+4 x^4+x^5}\right )} \end {dmath*}

input

Integrate[(-600 - 390*x + 30*x^2 + 600*x^3 + 660*x^4 + 90*x^5 - 105*x^6 - 
30*x^7)/(-800*x - 240*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 1472*x^6 - 32*x 
^7 - 56*x^8 + 4*x^9 + (-400*x - 160*x^2 + 20*x^3 + 1600*x^4 + 2240*x^5 + 9 
60*x^6 + 80*x^7 - 20*x^8)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + 
x^5)] + (-50*x - 25*x^2 + 200*x^4 + 300*x^5 + 150*x^6 + 25*x^7)*Log[(4*x^4 
 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)]^2),x]
 

output

-15/(20 - 2*x + 5*Log[(x^4*(2 + x)^2)/(-1 + 4*x^3 + 4*x^4 + x^5)])
 

3.5.6.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-30 x^7-105 x^6+90 x^5+660 x^4+600 x^3+30 x^2-390 x-600}{4 x^9-56 x^8-32 x^7+1472 x^6+4160 x^5+3196 x^4+72 x^3-240 x^2+\left (25 x^7+150 x^6+300 x^5+200 x^4-25 x^2-50 x\right ) \log ^2\left (\frac {x^6+4 x^5+4 x^4}{x^5+4 x^4+4 x^3-1}\right )+\left (-20 x^8+80 x^7+960 x^6+2240 x^5+1600 x^4+20 x^3-160 x^2-400 x\right ) \log \left (\frac {x^6+4 x^5+4 x^4}{x^5+4 x^4+4 x^3-1}\right )-800 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {15 \left (2 x^7+7 x^6-6 x^5-44 x^4-40 x^3-2 x^2+26 x+40\right )}{x \left (-x^6-6 x^5-12 x^4-8 x^3+x+2\right ) \left (5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-2 x+20\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 15 \int \frac {2 x^7+7 x^6-6 x^5-44 x^4-40 x^3-2 x^2+26 x+40}{x \left (-x^6-6 x^5-12 x^4-8 x^3+x+2\right ) \left (-2 x+5 \log \left (-\frac {x^4 (x+2)^2}{-x^5-4 x^4-4 x^3+1}\right )+20\right )^2}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle 15 \int \left (\frac {2 x^7+7 x^6-6 x^5-44 x^4-40 x^3-2 x^2+26 x+40}{x (x+2) \left (-2 x+5 \log \left (-\frac {x^4 (x+2)^2}{-x^5-4 x^4-4 x^3+1}\right )+20\right )^2}-\frac {x^2 (x+2) \left (2 x^7+7 x^6-6 x^5-44 x^4-40 x^3-2 x^2+26 x+40\right )}{\left (x^5+4 x^4+4 x^3-1\right ) \left (-2 x+5 \log \left (-\frac {x^4 (x+2)^2}{-x^5-4 x^4-4 x^3+1}\right )+20\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 15 \left (-2 \int \frac {1}{\left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx+20 \int \frac {1}{x \left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx+10 \int \frac {1}{(x+2) \left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx-80 \int \frac {x^3}{\left (x^5+4 x^4+4 x^3-1\right ) \left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx-25 \int \frac {x^4}{\left (x^5+4 x^4+4 x^3-1\right ) \left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx-60 \int \frac {x^2}{\left (x^5+4 x^4+4 x^3-1\right ) \left (2 x-5 \log \left (\frac {x^4 (x+2)^2}{x^5+4 x^4+4 x^3-1}\right )-20\right )^2}dx\right )\)

input

Int[(-600 - 390*x + 30*x^2 + 600*x^3 + 660*x^4 + 90*x^5 - 105*x^6 - 30*x^7 
)/(-800*x - 240*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 1472*x^6 - 32*x^7 - 5 
6*x^8 + 4*x^9 + (-400*x - 160*x^2 + 20*x^3 + 1600*x^4 + 2240*x^5 + 960*x^6 
 + 80*x^7 - 20*x^8)*Log[(4*x^4 + 4*x^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)] 
+ (-50*x - 25*x^2 + 200*x^4 + 300*x^5 + 150*x^6 + 25*x^7)*Log[(4*x^4 + 4*x 
^5 + x^6)/(-1 + 4*x^3 + 4*x^4 + x^5)]^2),x]
 

output

$Aborted
 

3.5.6.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 7239

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

3.5.6.4 Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20

method	result	size
parallelrisch	\(\frac {15}{2 x -5 \ln \left (\frac {x^{4} \left (x^{2}+4 x +4\right )}{x^{5}+4 x^{4}+4 x^{3}-1}\right )-20}\)	\(42\)
risch	\(\frac {15}{-20+2 x -5 \ln \left (\frac {x^{6}+4 x^{5}+4 x^{4}}{x^{5}+4 x^{4}+4 x^{3}-1}\right )}\)	\(45\)

input

int((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x^7+150 
*x^6+300*x^5+200*x^4-25*x^2-50*x)*ln((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1) 
)^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*x)*ln((x^ 
6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4160*x^5+ 
3196*x^4+72*x^3-240*x^2-800*x),x,method=_RETURNVERBOSE)
 

output

15/(2*x-5*ln(x^4*(x^2+4*x+4)/(x^5+4*x^4+4*x^3-1))-20)
 

3.5.6.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=\frac {15}{2 \, x - 5 \, \log \left (\frac {x^{6} + 4 \, x^{5} + 4 \, x^{4}}{x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1}\right ) - 20} \end {dmath*}

input

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x 
^7+150*x^6+300*x^5+200*x^4-25*x^2-50*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4 
*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*x) 
*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4 
160*x^5+3196*x^4+72*x^3-240*x^2-800*x),x, algorithm=\
 

output

15/(2*x - 5*log((x^6 + 4*x^5 + 4*x^4)/(x^5 + 4*x^4 + 4*x^3 - 1)) - 20)
 

3.5.6.6 Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.11 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=- \frac {3}{- \frac {2 x}{5} + \log {\left (\frac {x^{6} + 4 x^{5} + 4 x^{4}}{x^{5} + 4 x^{4} + 4 x^{3} - 1} \right )} + 4} \end {dmath*}

input

integrate((-30*x**7-105*x**6+90*x**5+660*x**4+600*x**3+30*x**2-390*x-600)/ 
((25*x**7+150*x**6+300*x**5+200*x**4-25*x**2-50*x)*ln((x**6+4*x**5+4*x**4) 
/(x**5+4*x**4+4*x**3-1))**2+(-20*x**8+80*x**7+960*x**6+2240*x**5+1600*x**4 
+20*x**3-160*x**2-400*x)*ln((x**6+4*x**5+4*x**4)/(x**5+4*x**4+4*x**3-1))+4 
*x**9-56*x**8-32*x**7+1472*x**6+4160*x**5+3196*x**4+72*x**3-240*x**2-800*x 
),x)
 

output

-3/(-2*x/5 + log((x**6 + 4*x**5 + 4*x**4)/(x**5 + 4*x**4 + 4*x**3 - 1)) + 
4)
 

3.5.6.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=\frac {15}{2 \, x + 5 \, \log \left (x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1\right ) - 10 \, \log \left (x + 2\right ) - 20 \, \log \left (x\right ) - 20} \end {dmath*}

input

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x 
^7+150*x^6+300*x^5+200*x^4-25*x^2-50*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4 
*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*x) 
*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4 
160*x^5+3196*x^4+72*x^3-240*x^2-800*x),x, algorithm=\
 

output

15/(2*x + 5*log(x^5 + 4*x^4 + 4*x^3 - 1) - 10*log(x + 2) - 20*log(x) - 20)
 

3.5.6.8 Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=\frac {15}{2 \, x - 5 \, \log \left (\frac {x^{6} + 4 \, x^{5} + 4 \, x^{4}}{x^{5} + 4 \, x^{4} + 4 \, x^{3} - 1}\right ) - 20} \end {dmath*}

input

integrate((-30*x^7-105*x^6+90*x^5+660*x^4+600*x^3+30*x^2-390*x-600)/((25*x 
^7+150*x^6+300*x^5+200*x^4-25*x^2-50*x)*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4 
*x^3-1))^2+(-20*x^8+80*x^7+960*x^6+2240*x^5+1600*x^4+20*x^3-160*x^2-400*x) 
*log((x^6+4*x^5+4*x^4)/(x^5+4*x^4+4*x^3-1))+4*x^9-56*x^8-32*x^7+1472*x^6+4 
160*x^5+3196*x^4+72*x^3-240*x^2-800*x),x, algorithm=\
 

output

15/(2*x - 5*log((x^6 + 4*x^5 + 4*x^4)/(x^5 + 4*x^4 + 4*x^3 - 1)) - 20)
 

3.5.6.9 Mupad [F(-1)]

Timed out. \begin {dmath*} \int \frac {-600-390 x+30 x^2+600 x^3+660 x^4+90 x^5-105 x^6-30 x^7}{-800 x-240 x^2+72 x^3+3196 x^4+4160 x^5+1472 x^6-32 x^7-56 x^8+4 x^9+\left (-400 x-160 x^2+20 x^3+1600 x^4+2240 x^5+960 x^6+80 x^7-20 x^8\right ) \log \left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )+\left (-50 x-25 x^2+200 x^4+300 x^5+150 x^6+25 x^7\right ) \log ^2\left (\frac {4 x^4+4 x^5+x^6}{-1+4 x^3+4 x^4+x^5}\right )} \, dx=\int -\frac {30\,x^7+105\,x^6-90\,x^5-660\,x^4-600\,x^3-30\,x^2+390\,x+600}{{\ln \left (\frac {x^6+4\,x^5+4\,x^4}{x^5+4\,x^4+4\,x^3-1}\right )}^2\,\left (25\,x^7+150\,x^6+300\,x^5+200\,x^4-25\,x^2-50\,x\right )-800\,x+\ln \left (\frac {x^6+4\,x^5+4\,x^4}{x^5+4\,x^4+4\,x^3-1}\right )\,\left (-20\,x^8+80\,x^7+960\,x^6+2240\,x^5+1600\,x^4+20\,x^3-160\,x^2-400\,x\right )-240\,x^2+72\,x^3+3196\,x^4+4160\,x^5+1472\,x^6-32\,x^7-56\,x^8+4\,x^9} \,d x \end {dmath*}

input

int(-(390*x - 30*x^2 - 600*x^3 - 660*x^4 - 90*x^5 + 105*x^6 + 30*x^7 + 600 
)/(log((4*x^4 + 4*x^5 + x^6)/(4*x^3 + 4*x^4 + x^5 - 1))^2*(200*x^4 - 25*x^ 
2 - 50*x + 300*x^5 + 150*x^6 + 25*x^7) - 800*x + log((4*x^4 + 4*x^5 + x^6) 
/(4*x^3 + 4*x^4 + x^5 - 1))*(20*x^3 - 160*x^2 - 400*x + 1600*x^4 + 2240*x^ 
5 + 960*x^6 + 80*x^7 - 20*x^8) - 240*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 
1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9),x)
 

output

int(-(390*x - 30*x^2 - 600*x^3 - 660*x^4 - 90*x^5 + 105*x^6 + 30*x^7 + 600 
)/(log((4*x^4 + 4*x^5 + x^6)/(4*x^3 + 4*x^4 + x^5 - 1))^2*(200*x^4 - 25*x^ 
2 - 50*x + 300*x^5 + 150*x^6 + 25*x^7) - 800*x + log((4*x^4 + 4*x^5 + x^6) 
/(4*x^3 + 4*x^4 + x^5 - 1))*(20*x^3 - 160*x^2 - 400*x + 1600*x^4 + 2240*x^ 
5 + 960*x^6 + 80*x^7 - 20*x^8) - 240*x^2 + 72*x^3 + 3196*x^4 + 4160*x^5 + 
1472*x^6 - 32*x^7 - 56*x^8 + 4*x^9), x)