Integrand size = 127, antiderivative size = 32 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {-\frac {x}{5}+\log (3+x)}{2 \left (-1+\frac {3}{x}-x^2-3 \log (2)\right )} \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(32)=64\).
Time = 5.42 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.53 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x \left (x \left (2717+68 \log ^3(8)+4 \log ^4(8)+\log (8) (319-3 \log (64))-6 \log ^2(8) (-25+\log (64))+30 \log (64)\right )-5 \left (2717+379 \log (8)+144 \log ^2(8)+56 \log ^3(8)+4 \log ^4(8)\right ) \log (3+x)\right )}{10 (11+\log (8)) \left (-3+x+x^3+x \log (8)\right ) \left (247+12 \log (8)+12 \log ^2(8)+4 \log ^3(8)\right )} \end {dmath*}
Integrate[(-3*x - 8*x^2 + x^3 - 8*x^4 - x^5 + (-6*x^2 + 3*x^3)*Log[2] + (4 5 + 15*x + 30*x^3 + 10*x^4)*Log[3 + x])/(270 - 90*x - 30*x^2 - 170*x^3 + 2 0*x^5 + 30*x^6 + 10*x^7 + (-540*x + 60*x^3 + 180*x^4 + 60*x^5)*Log[2] + (2 70*x^2 + 90*x^3)*Log[2]^2),x]
(x*(x*(2717 + 68*Log[8]^3 + 4*Log[8]^4 + Log[8]*(319 - 3*Log[64]) - 6*Log[ 8]^2*(-25 + Log[64]) + 30*Log[64]) - 5*(2717 + 379*Log[8] + 144*Log[8]^2 + 56*Log[8]^3 + 4*Log[8]^4)*Log[3 + x]))/(10*(11 + Log[8])*(-3 + x + x^3 + x*Log[8])*(247 + 12*Log[8] + 12*Log[8]^2 + 4*Log[8]^3))
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.
| \(\displaystyle \int \frac {-x^5-8 x^4+x^3-8 x^2+\left (10 x^4+30 x^3+15 x+45\right ) \log (x+3)+\left (3 x^3-6 x^2\right ) \log (2)-3 x}{10 x^7+30 x^6+20 x^5-170 x^3-30 x^2+\left (90 x^3+270 x^2\right ) \log ^2(2)+\left (60 x^5+180 x^4+60 x^3-540 x\right ) \log (2)-90 x+270} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (x^2-3 x+10+\log (8)\right ) \left (-x^5-8 x^4+x^3-8 x^2+\left (10 x^4+30 x^3+15 x+45\right ) \log (x+3)+\left (3 x^3-6 x^2\right ) \log (2)-3 x\right )}{90 (11+\log (8))^2 \left (-x^3-x (1+\log (8))+3\right )}+\frac {\left (x^2-3 x+10+\log (8)\right ) \left (-x^5-8 x^4+x^3-8 x^2+\left (10 x^4+30 x^3+15 x+45\right ) \log (x+3)+\left (3 x^3-6 x^2\right ) \log (2)-3 x\right )}{30 (11+\log (8)) \left (-x^3-x (1+\log (8))+3\right )^2}+\frac {-x^5-8 x^4+x^3-8 x^2+\left (10 x^4+30 x^3+15 x+45\right ) \log (x+3)+\left (3 x^3-6 x^2\right ) \log (2)-3 x}{90 (x+3) (11+\log (8))^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (2 x^4+6 x^3+3 x+9\right ) \log (x+3)-x \left (x^4+8 x^3-x^2 (1+\log (8))+x (8+\log (64))+3\right )}{10 (x+3) \left (-x^3-x (1+\log (8))+3\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{10} \int -\frac {x \left (x^4+8 x^3-(1+\log (8)) x^2+(8+\log (64)) x+3\right )-5 \left (2 x^4+6 x^3+3 x+9\right ) \log (x+3)}{(x+3) \left (-x^3-(1+\log (8)) x+3\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{10} \int \frac {x \left (x^4+8 x^3-(1+\log (8)) x^2+(8+\log (64)) x+3\right )-5 \left (2 x^4+6 x^3+3 x+9\right ) \log (x+3)}{(x+3) \left (-x^3-(1+\log (8)) x+3\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{10} \int \left (\frac {x \left (x^4+8 x^3-(1+\log (8)) x^2+(8+\log (64)) x+3\right )}{(x+3) \left (-x^3-(1+\log (8)) x+3\right )^2}+\frac {5 \left (-2 x^3-3\right ) \log (x+3)}{\left (-x^3-(1+\log (8)) x+3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{10} \int \left (\frac {x \left (x^4+8 x^3-(1+\log (8)) x^2+(8+\log (64)) x+3\right )}{(x+3) \left (-x^3-(1+\log (8)) x+3\right )^2}+\frac {5 \left (-2 x^3-3\right ) \log (x+3)}{\left (-x^3-(1+\log (8)) x+3\right )^2}\right )dx\) |
Int[(-3*x - 8*x^2 + x^3 - 8*x^4 - x^5 + (-6*x^2 + 3*x^3)*Log[2] + (45 + 15 *x + 30*x^3 + 10*x^4)*Log[3 + x])/(270 - 90*x - 30*x^2 - 170*x^3 + 20*x^5 + 30*x^6 + 10*x^7 + (-540*x + 60*x^3 + 180*x^4 + 60*x^5)*Log[2] + (270*x^2 + 90*x^3)*Log[2]^2),x]
3.1.72.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(\frac {x^{2}-5 x \ln \left (3+x \right )}{10 x^{3}+30 x \ln \left (2\right )+10 x -30}\) | \(27\) |
| norman | \(\frac {-\frac {x \ln \left (3+x \right )}{2}+\frac {x^{2}}{10}}{x^{3}+3 x \ln \left (2\right )+x -3}\) | \(28\) |
| risch | \(-\frac {x \ln \left (3+x \right )}{2 \left (x^{3}+3 x \ln \left (2\right )+x -3\right )}+\frac {x^{2}}{10 x^{3}+30 x \ln \left (2\right )+10 x -30}\) | \(40\) |
| parts | \(\frac {\frac {\left (\ln \left (2\right )+\frac {11}{3}\right ) x^{2}}{\frac {x^{3}}{3}+x \ln \left (2\right )+\frac {x}{3}-1}+5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-3+\textit {\_Z}^{3}+\left (3 \ln \left (2\right )+1\right ) \textit {\_Z} \right )}{\sum }\frac {\left (\textit {\_R}^{2}-3 \textit {\_R} -1\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )+1}\right )}{30 \ln \left (2\right )+110}-\frac {\ln \left (3+x \right )}{2 \left (3 \ln \left (2\right )+11\right )}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9 \textit {\_Z}^{2}+\left (3 \ln \left (2\right )+28\right ) \textit {\_Z} -9 \ln \left (2\right )-33\right )}{\sum }\frac {\left (\textit {\_R}^{2}-9 \textit {\_R} +17\right ) \ln \left (3+x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )-18 \textit {\_R} +28}}{2 \left (3 \ln \left (2\right )+11\right )}+\frac {\ln \left (3+x \right ) \left (3+x \right ) \left (\left (3+x \right )^{2}-10-9 x \right )}{2 \left (\left (3+x \right )^{3}+3 \left (3+x \right ) \ln \left (2\right )-9 \left (3+x \right )^{2}-9 \ln \left (2\right )+51+28 x \right ) \left (3 \ln \left (2\right )+11\right )}\) | \(223\) |
| derivativedivides | \(\frac {\frac {\left (\ln \left (2\right )+\frac {11}{3}\right ) \left (3+x \right )^{2}+\left (-6 \ln \left (2\right )-22\right ) \left (3+x \right )+9 \ln \left (2\right )+33}{\frac {\left (3+x \right )^{3}}{3}+\left (3+x \right ) \ln \left (2\right )-3 \left (3+x \right )^{2}-3 \ln \left (2\right )+17+\frac {28 x}{3}}+5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9 \textit {\_Z}^{2}+\left (3 \ln \left (2\right )+28\right ) \textit {\_Z} -9 \ln \left (2\right )-33\right )}{\sum }\frac {\left (\textit {\_R}^{2}-9 \textit {\_R} +17\right ) \ln \left (3+x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )-18 \textit {\_R} +28}\right )}{30 \ln \left (2\right )+110}-\frac {\ln \left (3+x \right )}{2 \left (3 \ln \left (2\right )+11\right )}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9 \textit {\_Z}^{2}+\left (3 \ln \left (2\right )+28\right ) \textit {\_Z} -9 \ln \left (2\right )-33\right )}{\sum }\frac {\left (\textit {\_R}^{2}-9 \textit {\_R} +17\right ) \ln \left (3+x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )-18 \textit {\_R} +28}}{2 \left (3 \ln \left (2\right )+11\right )}+\frac {\ln \left (3+x \right ) \left (3+x \right ) \left (\left (3+x \right )^{2}-10-9 x \right )}{2 \left (\left (3+x \right )^{3}+3 \left (3+x \right ) \ln \left (2\right )-9 \left (3+x \right )^{2}-9 \ln \left (2\right )+51+28 x \right ) \left (3 \ln \left (2\right )+11\right )}\) | \(270\) |
| default | \(\frac {\frac {\left (\ln \left (2\right )+\frac {11}{3}\right ) \left (3+x \right )^{2}+\left (-6 \ln \left (2\right )-22\right ) \left (3+x \right )+9 \ln \left (2\right )+33}{\frac {\left (3+x \right )^{3}}{3}+\left (3+x \right ) \ln \left (2\right )-3 \left (3+x \right )^{2}-3 \ln \left (2\right )+17+\frac {28 x}{3}}+5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9 \textit {\_Z}^{2}+\left (3 \ln \left (2\right )+28\right ) \textit {\_Z} -9 \ln \left (2\right )-33\right )}{\sum }\frac {\left (\textit {\_R}^{2}-9 \textit {\_R} +17\right ) \ln \left (3+x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )-18 \textit {\_R} +28}\right )}{30 \ln \left (2\right )+110}-\frac {\ln \left (3+x \right )}{2 \left (3 \ln \left (2\right )+11\right )}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}-9 \textit {\_Z}^{2}+\left (3 \ln \left (2\right )+28\right ) \textit {\_Z} -9 \ln \left (2\right )-33\right )}{\sum }\frac {\left (\textit {\_R}^{2}-9 \textit {\_R} +17\right ) \ln \left (3+x -\textit {\_R} \right )}{3 \textit {\_R}^{2}+3 \ln \left (2\right )-18 \textit {\_R} +28}}{2 \left (3 \ln \left (2\right )+11\right )}+\frac {\ln \left (3+x \right ) \left (3+x \right ) \left (\left (3+x \right )^{2}-10-9 x \right )}{2 \left (\left (3+x \right )^{3}+3 \left (3+x \right ) \ln \left (2\right )-9 \left (3+x \right )^{2}-9 \ln \left (2\right )+51+28 x \right ) \left (3 \ln \left (2\right )+11\right )}\) | \(270\) |
int(((10*x^4+30*x^3+15*x+45)*ln(3+x)+(3*x^3-6*x^2)*ln(2)-x^5-8*x^4+x^3-8*x ^2-3*x)/((90*x^3+270*x^2)*ln(2)^2+(60*x^5+180*x^4+60*x^3-540*x)*ln(2)+10*x ^7+30*x^6+20*x^5-170*x^3-30*x^2-90*x+270),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x^{2} - 5 \, x \log \left (x + 3\right )}{10 \, {\left (x^{3} + 3 \, x \log \left (2\right ) + x - 3\right )}} \end {dmath*}
integrate(((10*x^4+30*x^3+15*x+45)*log(3+x)+(3*x^3-6*x^2)*log(2)-x^5-8*x^4 +x^3-8*x^2-3*x)/((90*x^3+270*x^2)*log(2)^2+(60*x^5+180*x^4+60*x^3-540*x)*l og(2)+10*x^7+30*x^6+20*x^5-170*x^3-30*x^2-90*x+270),x, algorithm=\
Time = 0.65 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x^{2}}{10 x^{3} + x \left (10 + 30 \log {\left (2 \right )}\right ) - 30} - \frac {x \log {\left (x + 3 \right )}}{2 x^{3} + 2 x + 6 x \log {\left (2 \right )} - 6} \end {dmath*}
integrate(((10*x**4+30*x**3+15*x+45)*ln(3+x)+(3*x**3-6*x**2)*ln(2)-x**5-8* x**4+x**3-8*x**2-3*x)/((90*x**3+270*x**2)*ln(2)**2+(60*x**5+180*x**4+60*x* *3-540*x)*ln(2)+10*x**7+30*x**6+20*x**5-170*x**3-30*x**2-90*x+270),x)
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x^{2} - 5 \, x \log \left (x + 3\right )}{10 \, {\left (x^{3} + x {\left (3 \, \log \left (2\right ) + 1\right )} - 3\right )}} \end {dmath*}
integrate(((10*x^4+30*x^3+15*x+45)*log(3+x)+(3*x^3-6*x^2)*log(2)-x^5-8*x^4 +x^3-8*x^2-3*x)/((90*x^3+270*x^2)*log(2)^2+(60*x^5+180*x^4+60*x^3-540*x)*l og(2)+10*x^7+30*x^6+20*x^5-170*x^3-30*x^2-90*x+270),x, algorithm=\
Time = 0.33 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x^{2}}{10 \, {\left (x^{3} + 3 \, x \log \left (2\right ) + x - 3\right )}} - \frac {x \log \left (x + 3\right )}{2 \, {\left (x^{3} + 3 \, x \log \left (2\right ) + x - 3\right )}} \end {dmath*}
integrate(((10*x^4+30*x^3+15*x+45)*log(3+x)+(3*x^3-6*x^2)*log(2)-x^5-8*x^4 +x^3-8*x^2-3*x)/((90*x^3+270*x^2)*log(2)^2+(60*x^5+180*x^4+60*x^3-540*x)*l og(2)+10*x^7+30*x^6+20*x^5-170*x^3-30*x^2-90*x+270),x, algorithm=\
Time = 13.80 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \begin {dmath*} \int \frac {-3 x-8 x^2+x^3-8 x^4-x^5+\left (-6 x^2+3 x^3\right ) \log (2)+\left (45+15 x+30 x^3+10 x^4\right ) \log (3+x)}{270-90 x-30 x^2-170 x^3+20 x^5+30 x^6+10 x^7+\left (-540 x+60 x^3+180 x^4+60 x^5\right ) \log (2)+\left (270 x^2+90 x^3\right ) \log ^2(2)} \, dx=\frac {x\,\left (x-5\,\ln \left (x+3\right )\right )}{10\,\left (x+3\,x\,\ln \left (2\right )+x^3-3\right )} \end {dmath*}