Integrand size = 95, antiderivative size = 30 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {5 x^2}{\frac {3}{5 x^2}-(-4+x) \left (1+x-x^2\right ) \log (2)} \]
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25 x^4}{3+20 x^2 \log (2)+15 x^3 \log (2)-25 x^4 \log (2)+x^5 \log (32)} \]
Integrate[(300*x^3 + (1000*x^5 + 375*x^6 - 125*x^8)*Log[2])/(9 + (120*x^2 + 90*x^3 - 150*x^4 + 30*x^5)*Log[2] + (400*x^4 + 600*x^5 - 775*x^6 - 550*x ^7 + 775*x^8 - 250*x^9 + 25*x^10)*Log[2]^2),x]
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.
\(\displaystyle \int \frac {300 x^3+\left (-125 x^8+375 x^6+1000 x^5\right ) \log (2)}{\left (30 x^5-150 x^4+90 x^3+120 x^2\right ) \log (2)+\left (25 x^{10}-250 x^9+775 x^8-550 x^7-775 x^6+600 x^5+400 x^4\right ) \log ^2(2)+9} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {25 \left (-1315 x^4 \log (2)+5 x^3 (3+280 \log (2))+5 x^2 (3+272 \log (2))+57 x+204\right )}{\left (x^5 \log (32)-25 x^4 \log (2)+15 x^3 \log (2)+20 x^2 \log (2)+3\right )^2}-\frac {25 \left (x^3+5 x^2+19 x+68\right )}{x^5 \log (32)-25 x^4 \log (2)+15 x^3 \log (2)+20 x^2 \log (2)+3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 25 \left (57+\frac {10520 \log ^2(2)}{\log (32)}\right ) \int \frac {x}{\left (\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3\right )^2}dx+125 \left (3+\frac {2367 \log ^2(2)}{\log (32)}+272 \log (2)\right ) \int \frac {x^2}{\left (\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3\right )^2}dx+125 \left (3-\frac {5260 \log ^2(2)}{\log (32)}+280 \log (2)\right ) \int \frac {x^3}{\left (\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3\right )^2}dx+5100 \int \frac {1}{\left (\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3\right )^2}dx-1700 \int \frac {1}{\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3}dx-475 \int \frac {x}{\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3}dx-125 \int \frac {x^2}{\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3}dx-25 \int \frac {x^3}{\log (32) x^5-25 \log (2) x^4+15 \log (2) x^3+20 \log (2) x^2+3}dx+\frac {6575 \log (2)}{\log (32) \left (x^5 \log (32)-25 x^4 \log (2)+15 x^3 \log (2)+20 x^2 \log (2)+3\right )}\) |
Int[(300*x^3 + (1000*x^5 + 375*x^6 - 125*x^8)*Log[2])/(9 + (120*x^2 + 90*x ^3 - 150*x^4 + 30*x^5)*Log[2] + (400*x^4 + 600*x^5 - 775*x^6 - 550*x^7 + 7 75*x^8 - 250*x^9 + 25*x^10)*Log[2]^2),x]
3.12.66.3.1 Defintions of rubi rules used
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23
method | result | size |
default | \(\frac {5 x^{4}}{x^{5} \ln \left (2\right )-5 x^{4} \ln \left (2\right )+3 x^{3} \ln \left (2\right )+4 x^{2} \ln \left (2\right )+\frac {3}{5}}\) | \(37\) |
risch | \(\frac {5 x^{4}}{x^{5} \ln \left (2\right )-5 x^{4} \ln \left (2\right )+3 x^{3} \ln \left (2\right )+4 x^{2} \ln \left (2\right )+\frac {3}{5}}\) | \(37\) |
gosper | \(\frac {25 x^{4}}{5 x^{5} \ln \left (2\right )-25 x^{4} \ln \left (2\right )+15 x^{3} \ln \left (2\right )+20 x^{2} \ln \left (2\right )+3}\) | \(38\) |
norman | \(\frac {25 x^{4}}{5 x^{5} \ln \left (2\right )-25 x^{4} \ln \left (2\right )+15 x^{3} \ln \left (2\right )+20 x^{2} \ln \left (2\right )+3}\) | \(38\) |
parallelrisch | \(\frac {25 x^{4}}{5 x^{5} \ln \left (2\right )-25 x^{4} \ln \left (2\right )+15 x^{3} \ln \left (2\right )+20 x^{2} \ln \left (2\right )+3}\) | \(38\) |
int(((-125*x^8+375*x^6+1000*x^5)*ln(2)+300*x^3)/((25*x^10-250*x^9+775*x^8- 550*x^7-775*x^6+600*x^5+400*x^4)*ln(2)^2+(30*x^5-150*x^4+90*x^3+120*x^2)*l n(2)+9),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25 \, x^{4}}{5 \, {\left (x^{5} - 5 \, x^{4} + 3 \, x^{3} + 4 \, x^{2}\right )} \log \left (2\right ) + 3} \]
integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+7 75*x^8-550*x^7-775*x^6+600*x^5+400*x^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+12 0*x^2)*log(2)+9),x, algorithm=\
Time = 2.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25 x^{4}}{5 x^{5} \log {\left (2 \right )} - 25 x^{4} \log {\left (2 \right )} + 15 x^{3} \log {\left (2 \right )} + 20 x^{2} \log {\left (2 \right )} + 3} \]
integrate(((-125*x**8+375*x**6+1000*x**5)*ln(2)+300*x**3)/((25*x**10-250*x **9+775*x**8-550*x**7-775*x**6+600*x**5+400*x**4)*ln(2)**2+(30*x**5-150*x* *4+90*x**3+120*x**2)*ln(2)+9),x)
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25 \, x^{4}}{5 \, x^{5} \log \left (2\right ) - 25 \, x^{4} \log \left (2\right ) + 15 \, x^{3} \log \left (2\right ) + 20 \, x^{2} \log \left (2\right ) + 3} \]
integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+7 75*x^8-550*x^7-775*x^6+600*x^5+400*x^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+12 0*x^2)*log(2)+9),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25 \, x^{4}}{5 \, x^{5} \log \left (2\right ) - 25 \, x^{4} \log \left (2\right ) + 15 \, x^{3} \log \left (2\right ) + 20 \, x^{2} \log \left (2\right ) + 3} \]
integrate(((-125*x^8+375*x^6+1000*x^5)*log(2)+300*x^3)/((25*x^10-250*x^9+7 75*x^8-550*x^7-775*x^6+600*x^5+400*x^4)*log(2)^2+(30*x^5-150*x^4+90*x^3+12 0*x^2)*log(2)+9),x, algorithm=\
Time = 0.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {300 x^3+\left (1000 x^5+375 x^6-125 x^8\right ) \log (2)}{9+\left (120 x^2+90 x^3-150 x^4+30 x^5\right ) \log (2)+\left (400 x^4+600 x^5-775 x^6-550 x^7+775 x^8-250 x^9+25 x^{10}\right ) \log ^2(2)} \, dx=\frac {25\,x^4}{5\,\ln \left (2\right )\,x^5-25\,\ln \left (2\right )\,x^4+15\,\ln \left (2\right )\,x^3+20\,\ln \left (2\right )\,x^2+3} \]