Integrand size = 133, antiderivative size = 31 \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=e^{3+\frac {3-x}{\log \left (-\frac {3}{2}+\frac {1}{3} \left (2-\frac {10}{x}-x\right )\right )}} \]
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=e^{3+\frac {3-x}{\log \left (\frac {1}{6} \left (-5-\frac {20}{x}-2 x\right )\right )}} \]
Integrate[(E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2)/(6*x)])*(60 - 20*x - 6*x^2 + 2*x^3 + (-20*x - 5*x^2 - 2*x^3)*Log[(- 20 - 5*x - 2*x^2)/(6*x)]))/((20*x + 5*x^2 + 2*x^3)*Log[(-20 - 5*x - 2*x^2) /(6*x)]^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.
| \(\displaystyle \int \frac {\left (2 x^3-6 x^2+\left (-2 x^3-5 x^2-20 x\right ) \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-20 x+60\right ) \exp \left (\frac {3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-x+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\left (2 x^3+5 x^2+20 x\right ) \log ^2\left (\frac {-2 x^2-5 x-20}{6 x}\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (2 x^3-6 x^2+\left (-2 x^3-5 x^2-20 x\right ) \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-20 x+60\right ) \exp \left (\frac {3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-x+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{x \left (2 x^2+5 x+20\right ) \log ^2\left (\frac {-2 x^2-5 x-20}{6 x}\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 \left (x^3-3 x^2-10 x+30\right ) \exp \left (\frac {3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-x+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{x \left (2 x^2+5 x+20\right ) \log ^2\left (\frac {1}{6} \left (-2 x-\frac {20}{x}-5\right )\right )}-\frac {\exp \left (\frac {3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )-x+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-2 x-\frac {20}{x}-5\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx-\frac {44}{3} i \sqrt {\frac {5}{3}} \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\left (-4 x+3 i \sqrt {15}-5\right ) \log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx+3 \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{x \log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx-\frac {17}{9} \left (9+i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\left (4 x-3 i \sqrt {15}+5\right ) \log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx-\frac {17}{9} \left (9-i \sqrt {15}\right ) \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\left (4 x+3 i \sqrt {15}+5\right ) \log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx-\frac {44}{3} i \sqrt {\frac {5}{3}} \int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\left (4 x+3 i \sqrt {15}+5\right ) \log ^2\left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx-\int \frac {\exp \left (\frac {-x+3 \log \left (\frac {-2 x^2-5 x-20}{6 x}\right )+3}{\log \left (\frac {-2 x^2-5 x-20}{6 x}\right )}\right )}{\log \left (\frac {1}{6} \left (-2 x-5-\frac {20}{x}\right )\right )}dx\) |
Int[(E^((3 - x + 3*Log[(-20 - 5*x - 2*x^2)/(6*x)])/Log[(-20 - 5*x - 2*x^2) /(6*x)])*(60 - 20*x - 6*x^2 + 2*x^3 + (-20*x - 5*x^2 - 2*x^3)*Log[(-20 - 5 *x - 2*x^2)/(6*x)]))/((20*x + 5*x^2 + 2*x^3)*Log[(-20 - 5*x - 2*x^2)/(6*x) ]^2),x]
3.13.76.3.1 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_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.58 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39
| method | result | size |
| risch | \({\mathrm e}^{-\frac {-3 \ln \left (\frac {-2 x^{2}-5 x -20}{6 x}\right )-3+x}{\ln \left (\frac {-2 x^{2}-5 x -20}{6 x}\right )}}\) | \(43\) |
| parallelrisch | \({\mathrm e}^{\frac {3 \ln \left (-\frac {2 x^{2}+5 x +20}{6 x}\right )+3-x}{\ln \left (-\frac {2 x^{2}+5 x +20}{6 x}\right )}}\) | \(44\) |
int(((-2*x^3-5*x^2-20*x)*ln(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x+60)*ex p((3*ln(1/6*(-2*x^2-5*x-20)/x)+3-x)/ln(1/6*(-2*x^2-5*x-20)/x))/(2*x^3+5*x^ 2+20*x)/ln(1/6*(-2*x^2-5*x-20)/x)^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=e^{\left (-\frac {x - 3 \, \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 3}{\log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )}\right )} \]
integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x +60)*exp((3*log(1/6*(-2*x^2-5*x-20)/x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2 *x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm=\
Exception generated. \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=\text {Exception raised: TypeError} \]
integrate(((-2*x**3-5*x**2-20*x)*ln(1/6*(-2*x**2-5*x-20)/x)+2*x**3-6*x**2- 20*x+60)*exp((3*ln(1/6*(-2*x**2-5*x-20)/x)+3-x)/ln(1/6*(-2*x**2-5*x-20)/x) )/(2*x**3+5*x**2+20*x)/ln(1/6*(-2*x**2-5*x-20)/x)**2,x)
\[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - 6 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2} + 20 \, x\right )} \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 20 \, x + 60\right )} e^{\left (-\frac {x - 3 \, \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right ) - 3}{\log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )}\right )}}{{\left (2 \, x^{3} + 5 \, x^{2} + 20 \, x\right )} \log \left (-\frac {2 \, x^{2} + 5 \, x + 20}{6 \, x}\right )^{2}} \,d x } \]
integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x +60)*exp((3*log(1/6*(-2*x^2-5*x-20)/x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2 *x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm=\
integrate((2*x^3 - 6*x^2 - (2*x^3 + 5*x^2 + 20*x)*log(-1/6*(2*x^2 + 5*x + 20)/x) - 20*x + 60)*e^(-(x - 3*log(-1/6*(2*x^2 + 5*x + 20)/x) - 3)/log(-1/ 6*(2*x^2 + 5*x + 20)/x))/((2*x^3 + 5*x^2 + 20*x)*log(-1/6*(2*x^2 + 5*x + 2 0)/x)^2), x)
Time = 1.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx=e^{\left (-\frac {x}{\log \left (-\frac {1}{3} \, x - \frac {10}{3 \, x} - \frac {5}{6}\right )} + \frac {3}{\log \left (-\frac {1}{3} \, x - \frac {10}{3 \, x} - \frac {5}{6}\right )} + 3\right )} \]
integrate(((-2*x^3-5*x^2-20*x)*log(1/6*(-2*x^2-5*x-20)/x)+2*x^3-6*x^2-20*x +60)*exp((3*log(1/6*(-2*x^2-5*x-20)/x)+3-x)/log(1/6*(-2*x^2-5*x-20)/x))/(2 *x^3+5*x^2+20*x)/log(1/6*(-2*x^2-5*x-20)/x)^2,x, algorithm=\
Time = 9.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.26 \[ \int \frac {e^{\frac {3-x+3 \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}{\log \left (\frac {-20-5 x-2 x^2}{6 x}\right )}} \left (60-20 x-6 x^2+2 x^3+\left (-20 x-5 x^2-2 x^3\right ) \log \left (\frac {-20-5 x-2 x^2}{6 x}\right )\right )}{\left (20 x+5 x^2+2 x^3\right ) \log ^2\left (\frac {-20-5 x-2 x^2}{6 x}\right )} \, dx={\mathrm {e}}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}-\frac {x}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}}\,{\left (-\frac {1}{6\,x}\right )}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}}\,{\left (2\,x^2+5\,x+20\right )}^{\frac {3}{\ln \left (-\frac {2\,x^2+5\,x+20}{6\,x}\right )}} \]
int(-(exp((3*log(-((5*x)/6 + x^2/3 + 10/3)/x) - x + 3)/log(-((5*x)/6 + x^2 /3 + 10/3)/x))*(20*x + log(-((5*x)/6 + x^2/3 + 10/3)/x)*(20*x + 5*x^2 + 2* x^3) + 6*x^2 - 2*x^3 - 60))/(log(-((5*x)/6 + x^2/3 + 10/3)/x)^2*(20*x + 5* x^2 + 2*x^3)),x)