Integrand size = 92, antiderivative size = 31 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {(-4+x)^2+x}{x}-\frac {4 \log ^2(x)}{5 x+\frac {1+x}{x}} \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.08 (sec) , antiderivative size = 380, normalized size of antiderivative = 12.26 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {16 \sqrt {19}+16 \sqrt {19} x+81 \sqrt {19} x^2+\sqrt {19} x^3+5 \sqrt {19} x^4-8 i x \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-8 i x^2 \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-40 i x^3 \log \left (\frac {-i+\sqrt {19}-10 i x}{-i+\sqrt {19}}\right ) \log (x)-4 \sqrt {19} x^2 \log ^2(x)+8 i x \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )+8 i x^2 \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )+40 i x^3 \log (x) \log \left (\frac {9 i+\sqrt {19}+5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )-8 i x \left (1+x+5 x^2\right ) \operatorname {PolyLog}\left (2,\frac {10 i x}{-i+\sqrt {19}}\right )+8 i x \left (1+x+5 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {5 \left (-i+\sqrt {19}\right ) x}{9 i+\sqrt {19}}\right )}{\sqrt {19} x \left (1+x+5 x^2\right )} \]
Integrate[(-16 - 32*x - 175*x^2 - 158*x^3 - 389*x^4 + 10*x^5 + 25*x^6 + (- 8*x^2 - 8*x^3 - 40*x^4)*Log[x] + (-4*x^2 + 20*x^4)*Log[x]^2)/(x^2 + 2*x^3 + 11*x^4 + 10*x^5 + 25*x^6),x]
(16*Sqrt[19] + 16*Sqrt[19]*x + 81*Sqrt[19]*x^2 + Sqrt[19]*x^3 + 5*Sqrt[19] *x^4 - (8*I)*x*Log[(-I + Sqrt[19] - (10*I)*x)/(-I + Sqrt[19])]*Log[x] - (8 *I)*x^2*Log[(-I + Sqrt[19] - (10*I)*x)/(-I + Sqrt[19])]*Log[x] - (40*I)*x^ 3*Log[(-I + Sqrt[19] - (10*I)*x)/(-I + Sqrt[19])]*Log[x] - 4*Sqrt[19]*x^2* Log[x]^2 + (8*I)*x*Log[x]*Log[(9*I + Sqrt[19] + 5*(-I + Sqrt[19])*x)/(9*I + Sqrt[19])] + (8*I)*x^2*Log[x]*Log[(9*I + Sqrt[19] + 5*(-I + Sqrt[19])*x) /(9*I + Sqrt[19])] + (40*I)*x^3*Log[x]*Log[(9*I + Sqrt[19] + 5*(-I + Sqrt[ 19])*x)/(9*I + Sqrt[19])] - (8*I)*x*(1 + x + 5*x^2)*PolyLog[2, ((10*I)*x)/ (-I + Sqrt[19])] + (8*I)*x*(1 + x + 5*x^2)*PolyLog[2, (-5*(-I + Sqrt[19])* x)/(9*I + Sqrt[19])])/(Sqrt[19]*x*(1 + x + 5*x^2))
Result contains complex when optimal does not.
Time = 4.32 (sec) , antiderivative size = 2189, normalized size of antiderivative = 70.61, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2026, 2463, 2009}
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.
| \(\displaystyle \int \frac {25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16}{25 x^6+10 x^5+11 x^4+2 x^3+x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16}{x^2 \left (25 x^4+10 x^3+11 x^2+2 x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {100 i \left (25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16\right )}{19 \sqrt {19} x^2 \left (10 x+i \sqrt {19}+1\right )}+\frac {100 i \left (25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16\right )}{19 \sqrt {19} \left (-10 x+i \sqrt {19}-1\right ) x^2}-\frac {100 \left (25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16\right )}{19 \left (-10 x+i \sqrt {19}-1\right )^2 x^2}-\frac {100 \left (25 x^6+10 x^5-389 x^4-158 x^3-175 x^2+\left (20 x^4-4 x^2\right ) \log ^2(x)+\left (-40 x^4-8 x^3-8 x^2\right ) \log (x)-32 x-16\right )}{19 x^2 \left (10 x+i \sqrt {19}+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {25 \left (19+i \sqrt {19}\right ) x^3}{1083}+\frac {25 \left (19-i \sqrt {19}\right ) x^3}{1083}-\frac {50 x^3}{57}-\frac {5}{722} \left (19+9 i \sqrt {19}\right ) x^2-\frac {5}{722} \left (19-9 i \sqrt {19}\right ) x^2+\frac {5}{361} \left (19+i \sqrt {19}\right ) x^2+\frac {5}{38} \left (1+i \sqrt {19}\right ) x^2+\frac {5}{361} \left (19-i \sqrt {19}\right ) x^2+\frac {5}{38} \left (1-i \sqrt {19}\right ) x^2-\frac {10 x^2}{19}+\frac {40 \log ^2(x) x}{\sqrt {19} \left (10 i x-\sqrt {19}+i\right )}-\frac {40 \log ^2(x) x}{\sqrt {19} \left (10 i x+\sqrt {19}+i\right )}+\frac {20}{361} \left (19+i \sqrt {19}\right ) \log ^2(x) x+\frac {20}{361} \left (19-i \sqrt {19}\right ) \log ^2(x) x-\frac {40}{19} \log ^2(x) x-\frac {40}{361} \left (19+i \sqrt {19}\right ) \log (x) x-\frac {40}{361} \left (19-i \sqrt {19}\right ) \log (x) x+\frac {160}{19} \log (x) x-\frac {\left (i+\sqrt {19}\right )^3 x}{76 \sqrt {19}}+\frac {800 x}{19 \sqrt {19} \left (i+\sqrt {19}\right )}-\frac {2}{361} \left (19+9 i \sqrt {19}\right ) x-\frac {2}{361} \left (19-9 i \sqrt {19}\right ) x-\frac {2}{361} \left (38-7 i \sqrt {19}\right ) x-\frac {349}{361} \left (19+i \sqrt {19}\right ) x+\frac {3}{38} \left (9+i \sqrt {19}\right ) x+\frac {2}{19} \left (1+i \sqrt {19}\right ) x-\frac {349}{361} \left (19-i \sqrt {19}\right ) x+\frac {3}{38} \left (9-i \sqrt {19}\right ) x+\frac {2}{19} \left (1-i \sqrt {19}\right ) x-\frac {800 x}{19 \sqrt {19} \left (i-\sqrt {19}\right )}+\frac {618 x}{19}+\frac {\left (171+31 i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )}{1805}+\frac {389 \left (19+9 i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )}{1805}-\frac {4 \left (38-7 i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )}{1805}-\frac {4}{95} \left (7+2 i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )-\frac {389}{95} \left (1+i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )-\frac {158}{361} \left (19-i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )+\frac {3}{95} \left (9-i \sqrt {19}\right ) \log \left (10 i x-\sqrt {19}+i\right )-\frac {1750 i \log \left (10 i x-\sqrt {19}+i\right )}{19 \sqrt {19}}+\frac {158}{19} \log \left (10 i x-\sqrt {19}+i\right )+\frac {3200 \log \left (10 i x+\sqrt {19}+i\right )}{19 \sqrt {19} \left (i+\sqrt {19}\right )}-\frac {16000 i \log \left (10 i x+\sqrt {19}+i\right )}{19 \sqrt {19} \left (i+\sqrt {19}\right )^2}+\frac {\left (171-31 i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )}{1805}+\frac {389 \left (19-9 i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )}{1805}-\frac {4 \left (38+7 i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )}{1805}-\frac {4}{95} \left (7-2 i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )-\frac {158}{361} \left (19+i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )+\frac {3}{95} \left (9+i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )-\frac {389}{95} \left (1-i \sqrt {19}\right ) \log \left (10 i x+\sqrt {19}+i\right )+\frac {1750 i \log \left (10 i x+\sqrt {19}+i\right )}{19 \sqrt {19}}+\frac {158}{19} \log \left (10 i x+\sqrt {19}+i\right )+\frac {200 i \left (\frac {2}{i-\sqrt {19}}-i x\right )^2 \log (x)}{19 \sqrt {19}}-\frac {200 i \left (\frac {2}{i+\sqrt {19}}-i x\right )^2 \log (x)}{19 \sqrt {19}}+\frac {8000 \log (x)}{19 \sqrt {19} \left (9 i+\sqrt {19}\right )}-\frac {3200 \log (x)}{19 \sqrt {19} \left (i+\sqrt {19}\right )}+\frac {800 i \log (x)}{19 \sqrt {19} \left (i+\sqrt {19}\right )^2}-\frac {3200 \log (x)}{19 \left (i+\sqrt {19}\right )^2}+\frac {4000 \log (x)}{19 \left (7+2 i \sqrt {19}\right )}-\frac {32000 \log (x)}{19 \left (1-i \sqrt {19}\right )^3}-\frac {8000 \log (x)}{19 \sqrt {19} \left (9 i-\sqrt {19}\right )}+\frac {3200 \log (x)}{19 \sqrt {19} \left (i-\sqrt {19}\right )}-\frac {800 i \log (x)}{19 \sqrt {19} \left (i-\sqrt {19}\right )^2}-\frac {3200 \log (x)}{19 \left (i-\sqrt {19}\right )^2}-\frac {3200 \log \left (10 x+i \sqrt {19}+1\right )}{19 \sqrt {19} \left (i-\sqrt {19}\right )}-\frac {4000 \log \left (-10 \left (7 i-2 \sqrt {19}\right ) x+9 \sqrt {19}+31 i\right )}{19 \left (7+2 i \sqrt {19}\right )}+\frac {1600 \log \left (5 \left (9 i-\sqrt {19}\right ) x+2 \left (7 i+2 \sqrt {19}\right )\right )}{19 \left (9+i \sqrt {19}\right )}-\frac {8000 \log \left (-5 \left (9 i+\sqrt {19}\right ) x-2 \left (7 i-2 \sqrt {19}\right )\right )}{19 \sqrt {19} \left (9 i+\sqrt {19}\right )}+\frac {1600 \log \left (-5 \left (9 i+\sqrt {19}\right ) x-2 \left (7 i-2 \sqrt {19}\right )\right )}{19 \left (9-i \sqrt {19}\right )}-\frac {4000 \log \left (-10 \left (7 i+2 \sqrt {19}\right ) x-9 \sqrt {19}+31 i\right )}{19 \left (7-2 i \sqrt {19}\right )}+\frac {31 i+9 \sqrt {19}}{95 \left (10 i x-\sqrt {19}+i\right )}+\frac {389 \left (9 i+\sqrt {19}\right )}{95 \left (10 i x-\sqrt {19}+i\right )}+\frac {158 \left (i-\sqrt {19}\right )}{19 \left (10 i x-\sqrt {19}+i\right )}-\frac {3200}{19 \left (i-\sqrt {19}\right ) \left (10 i x-\sqrt {19}+i\right )}+\frac {16000 i}{19 \left (i-\sqrt {19}\right )^2 \left (10 i x-\sqrt {19}+i\right )}+\frac {4 \left (7 i-2 \sqrt {19}\right )}{95 \left (10 i x-\sqrt {19}+i\right )}-\frac {1750 i}{19 \left (10 i x-\sqrt {19}+i\right )}+\frac {4 \left (7 i+2 \sqrt {19}\right )}{95 \left (10 i x+\sqrt {19}+i\right )}+\frac {158 \left (i+\sqrt {19}\right )}{19 \left (10 i x+\sqrt {19}+i\right )}-\frac {3200}{19 \left (i+\sqrt {19}\right ) \left (10 i x+\sqrt {19}+i\right )}+\frac {16000 i}{19 \left (i+\sqrt {19}\right )^2 \left (10 i x+\sqrt {19}+i\right )}+\frac {389 \left (9 i-\sqrt {19}\right )}{95 \left (10 i x+\sqrt {19}+i\right )}+\frac {31 i-9 \sqrt {19}}{95 \left (10 i x+\sqrt {19}+i\right )}-\frac {1750 i}{19 \left (10 i x+\sqrt {19}+i\right )}+\frac {1600}{19 \sqrt {19} \left (i+\sqrt {19}\right ) x}+\frac {1600}{19 \left (i+\sqrt {19}\right )^2 x}-\frac {1600}{19 \sqrt {19} \left (i-\sqrt {19}\right ) x}+\frac {1600}{19 \left (i-\sqrt {19}\right )^2 x}\) |
Int[(-16 - 32*x - 175*x^2 - 158*x^3 - 389*x^4 + 10*x^5 + 25*x^6 + (-8*x^2 - 8*x^3 - 40*x^4)*Log[x] + (-4*x^2 + 20*x^4)*Log[x]^2)/(x^2 + 2*x^3 + 11*x ^4 + 10*x^5 + 25*x^6),x]
((-1750*I)/19)/(I - Sqrt[19] + (10*I)*x) + (4*(7*I - 2*Sqrt[19]))/(95*(I - Sqrt[19] + (10*I)*x)) + ((16000*I)/19)/((I - Sqrt[19])^2*(I - Sqrt[19] + (10*I)*x)) - 3200/(19*(I - Sqrt[19])*(I - Sqrt[19] + (10*I)*x)) + (158*(I - Sqrt[19]))/(19*(I - Sqrt[19] + (10*I)*x)) + (389*(9*I + Sqrt[19]))/(95*( I - Sqrt[19] + (10*I)*x)) + (31*I + 9*Sqrt[19])/(95*(I - Sqrt[19] + (10*I) *x)) - ((1750*I)/19)/(I + Sqrt[19] + (10*I)*x) + (31*I - 9*Sqrt[19])/(95*( I + Sqrt[19] + (10*I)*x)) + (389*(9*I - Sqrt[19]))/(95*(I + Sqrt[19] + (10 *I)*x)) + ((16000*I)/19)/((I + Sqrt[19])^2*(I + Sqrt[19] + (10*I)*x)) - 32 00/(19*(I + Sqrt[19])*(I + Sqrt[19] + (10*I)*x)) + (158*(I + Sqrt[19]))/(1 9*(I + Sqrt[19] + (10*I)*x)) + (4*(7*I + 2*Sqrt[19]))/(95*(I + Sqrt[19] + (10*I)*x)) + 1600/(19*(I - Sqrt[19])^2*x) - 1600/(19*Sqrt[19]*(I - Sqrt[19 ])*x) + 1600/(19*(I + Sqrt[19])^2*x) + 1600/(19*Sqrt[19]*(I + Sqrt[19])*x) + (618*x)/19 - (800*x)/(19*Sqrt[19]*(I - Sqrt[19])) + (2*(1 - I*Sqrt[19]) *x)/19 + (3*(9 - I*Sqrt[19])*x)/38 - (349*(19 - I*Sqrt[19])*x)/361 + (2*(1 + I*Sqrt[19])*x)/19 + (3*(9 + I*Sqrt[19])*x)/38 - (349*(19 + I*Sqrt[19])* x)/361 - (2*(38 - (7*I)*Sqrt[19])*x)/361 - (2*(19 - (9*I)*Sqrt[19])*x)/361 - (2*(19 + (9*I)*Sqrt[19])*x)/361 + (800*x)/(19*Sqrt[19]*(I + Sqrt[19])) - ((I + Sqrt[19])^3*x)/(76*Sqrt[19]) - (10*x^2)/19 + (5*(1 - I*Sqrt[19])*x ^2)/38 + (5*(19 - I*Sqrt[19])*x^2)/361 + (5*(1 + I*Sqrt[19])*x^2)/38 + (5* (19 + I*Sqrt[19])*x^2)/361 - (5*(19 - (9*I)*Sqrt[19])*x^2)/722 - (5*(19...
3.7.37.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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]
Time = 0.37 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(-\frac {4 x \ln \left (x \right )^{2}}{5 x^{2}+x +1}+\frac {x^{2}+16}{x}\) | \(28\) |
| norman | \(\frac {16+\frac {79 x}{5}+\frac {404 x^{2}}{5}+5 x^{4}-4 x^{2} \ln \left (x \right )^{2}}{x \left (5 x^{2}+x +1\right )}\) | \(39\) |
| parallelrisch | \(\frac {25 x^{4}+80-20 x^{2} \ln \left (x \right )^{2}+404 x^{2}+79 x}{5 x \left (5 x^{2}+x +1\right )}\) | \(40\) |
int(((20*x^4-4*x^2)*ln(x)^2+(-40*x^4-8*x^3-8*x^2)*ln(x)+25*x^6+10*x^5-389* x^4-158*x^3-175*x^2-32*x-16)/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x,method=_RE TURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=\frac {5 \, x^{4} - 4 \, x^{2} \log \left (x\right )^{2} + x^{3} + 81 \, x^{2} + 16 \, x + 16}{5 \, x^{3} + x^{2} + x} \]
integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10* x^5-389*x^4-158*x^3-175*x^2-32*x-16)/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, a lgorithm=\
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=x - \frac {4 x \log {\left (x \right )}^{2}}{5 x^{2} + x + 1} + \frac {16}{x} \]
integrate(((20*x**4-4*x**2)*ln(x)**2+(-40*x**4-8*x**3-8*x**2)*ln(x)+25*x** 6+10*x**5-389*x**4-158*x**3-175*x**2-32*x-16)/(25*x**6+10*x**5+11*x**4+2*x **3+x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (31) = 62\).
Time = 0.47 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.61 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=-\frac {4 \, x \log \left (x\right )^{2}}{5 \, x^{2} + x + 1} + x + \frac {16 \, {\left (140 \, x^{2} + 33 \, x + 19\right )}}{19 \, {\left (5 \, x^{3} + x^{2} + x\right )}} + \frac {31 \, x - 14}{95 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {2 \, {\left (14 \, x + 9\right )}}{95 \, {\left (5 \, x^{2} + x + 1\right )}} - \frac {175 \, {\left (10 \, x + 1\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {389 \, {\left (9 \, x - 1\right )}}{95 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {32 \, {\left (5 \, x - 9\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} + \frac {158 \, {\left (x + 2\right )}}{19 \, {\left (5 \, x^{2} + x + 1\right )}} \]
integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10* x^5-389*x^4-158*x^3-175*x^2-32*x-16)/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, a lgorithm=\
-4*x*log(x)^2/(5*x^2 + x + 1) + x + 16/19*(140*x^2 + 33*x + 19)/(5*x^3 + x ^2 + x) + 1/95*(31*x - 14)/(5*x^2 + x + 1) + 2/95*(14*x + 9)/(5*x^2 + x + 1) - 175/19*(10*x + 1)/(5*x^2 + x + 1) + 389/95*(9*x - 1)/(5*x^2 + x + 1) + 32/19*(5*x - 9)/(5*x^2 + x + 1) + 158/19*(x + 2)/(5*x^2 + x + 1)
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=-\frac {4 \, x \log \left (x\right )^{2}}{5 \, x^{2} + x + 1} + x + \frac {16}{x} \]
integrate(((20*x^4-4*x^2)*log(x)^2+(-40*x^4-8*x^3-8*x^2)*log(x)+25*x^6+10* x^5-389*x^4-158*x^3-175*x^2-32*x-16)/(25*x^6+10*x^5+11*x^4+2*x^3+x^2),x, a lgorithm=\
Time = 9.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-16-32 x-175 x^2-158 x^3-389 x^4+10 x^5+25 x^6+\left (-8 x^2-8 x^3-40 x^4\right ) \log (x)+\left (-4 x^2+20 x^4\right ) \log ^2(x)}{x^2+2 x^3+11 x^4+10 x^5+25 x^6} \, dx=x+\frac {16\,x-x^2\,\left (4\,{\ln \left (x\right )}^2-80\right )+16}{x\,\left (5\,x^2+x+1\right )} \]