Integrand size = 116, antiderivative size = 27 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=2+\frac {e^x}{\left (-1+x-\frac {-1+\frac {1}{4} (-2+x)+\log (x)}{x}\right )^2} \]
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\frac {16 e^x x^2}{\left (-6+5 x-4 x^2+4 \log (x)\right )^2} \]
Integrate[(E^x*(-320*x - 96*x^2 + 208*x^3 - 64*x^4) + E^x*(128*x + 64*x^2) *Log[x])/(-216 + 540*x - 882*x^2 + 845*x^3 - 588*x^4 + 240*x^5 - 64*x^6 + (432 - 720*x + 876*x^2 - 480*x^3 + 192*x^4)*Log[x] + (-288 + 240*x - 192*x ^2)*Log[x]^2 + 64*Log[x]^3),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.
\(\displaystyle \int \frac {e^x \left (64 x^2+128 x\right ) \log (x)+e^x \left (-64 x^4+208 x^3-96 x^2-320 x\right )}{-64 x^6+240 x^5-588 x^4+845 x^3-882 x^2+\left (-192 x^2+240 x-288\right ) \log ^2(x)+\left (192 x^4-480 x^3+876 x^2-720 x+432\right ) \log (x)+540 x+64 \log ^3(x)-216} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {16 e^x x \left (4 x^3-13 x^2+6 x-4 (x+2) \log (x)+20\right )}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 16 \int \frac {e^x x \left (4 x^3-13 x^2+6 x-4 (x+2) \log (x)+20\right )}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 16 \int \left (\frac {e^x x (x+2)}{\left (4 x^2-5 x-4 \log (x)+6\right )^2}-\frac {2 e^x x \left (8 x^2-5 x-4\right )}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 16 \left (8 \int \frac {e^x x}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}dx+10 \int \frac {e^x x^2}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}dx+2 \int \frac {e^x x}{\left (4 x^2-5 x-4 \log (x)+6\right )^2}dx+\int \frac {e^x x^2}{\left (4 x^2-5 x-4 \log (x)+6\right )^2}dx-16 \int \frac {e^x x^3}{\left (4 x^2-5 x-4 \log (x)+6\right )^3}dx\right )\) |
Int[(E^x*(-320*x - 96*x^2 + 208*x^3 - 64*x^4) + E^x*(128*x + 64*x^2)*Log[x ])/(-216 + 540*x - 882*x^2 + 845*x^3 - 588*x^4 + 240*x^5 - 64*x^6 + (432 - 720*x + 876*x^2 - 480*x^3 + 192*x^4)*Log[x] + (-288 + 240*x - 192*x^2)*Lo g[x]^2 + 64*Log[x]^3),x]
3.30.82.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {16 x^{2} {\mathrm e}^{x}}{\left (4 x^{2}-5 x -4 \ln \left (x \right )+6\right )^{2}}\) | \(24\) |
parallelrisch | \(\frac {16 \,{\mathrm e}^{x} x^{2}}{16 x^{4}-40 x^{3}-32 x^{2} \ln \left (x \right )+73 x^{2}+40 x \ln \left (x \right )+16 \ln \left (x \right )^{2}-60 x -48 \ln \left (x \right )+36}\) | \(52\) |
int(((64*x^2+128*x)*exp(x)*ln(x)+(-64*x^4+208*x^3-96*x^2-320*x)*exp(x))/(6 4*ln(x)^3+(-192*x^2+240*x-288)*ln(x)^2+(192*x^4-480*x^3+876*x^2-720*x+432) *ln(x)-64*x^6+240*x^5-588*x^4+845*x^3-882*x^2+540*x-216),x,method=_RETURNV ERBOSE)
Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\frac {16 \, x^{2} e^{x}}{16 \, x^{4} - 40 \, x^{3} + 73 \, x^{2} - 8 \, {\left (4 \, x^{2} - 5 \, x + 6\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 60 \, x + 36} \]
integrate(((64*x^2+128*x)*exp(x)*log(x)+(-64*x^4+208*x^3-96*x^2-320*x)*exp (x))/(64*log(x)^3+(-192*x^2+240*x-288)*log(x)^2+(192*x^4-480*x^3+876*x^2-7 20*x+432)*log(x)-64*x^6+240*x^5-588*x^4+845*x^3-882*x^2+540*x-216),x, algo rithm=\
16*x^2*e^x/(16*x^4 - 40*x^3 + 73*x^2 - 8*(4*x^2 - 5*x + 6)*log(x) + 16*log (x)^2 - 60*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\frac {16 x^{2} e^{x}}{16 x^{4} - 40 x^{3} - 32 x^{2} \log {\left (x \right )} + 73 x^{2} + 40 x \log {\left (x \right )} - 60 x + 16 \log {\left (x \right )}^{2} - 48 \log {\left (x \right )} + 36} \]
integrate(((64*x**2+128*x)*exp(x)*ln(x)+(-64*x**4+208*x**3-96*x**2-320*x)* exp(x))/(64*ln(x)**3+(-192*x**2+240*x-288)*ln(x)**2+(192*x**4-480*x**3+876 *x**2-720*x+432)*ln(x)-64*x**6+240*x**5-588*x**4+845*x**3-882*x**2+540*x-2 16),x)
16*x**2*exp(x)/(16*x**4 - 40*x**3 - 32*x**2*log(x) + 73*x**2 + 40*x*log(x) - 60*x + 16*log(x)**2 - 48*log(x) + 36)
Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\frac {16 \, x^{2} e^{x}}{16 \, x^{4} - 40 \, x^{3} + 73 \, x^{2} - 8 \, {\left (4 \, x^{2} - 5 \, x + 6\right )} \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 60 \, x + 36} \]
integrate(((64*x^2+128*x)*exp(x)*log(x)+(-64*x^4+208*x^3-96*x^2-320*x)*exp (x))/(64*log(x)^3+(-192*x^2+240*x-288)*log(x)^2+(192*x^4-480*x^3+876*x^2-7 20*x+432)*log(x)-64*x^6+240*x^5-588*x^4+845*x^3-882*x^2+540*x-216),x, algo rithm=\
16*x^2*e^x/(16*x^4 - 40*x^3 + 73*x^2 - 8*(4*x^2 - 5*x + 6)*log(x) + 16*log (x)^2 - 60*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\frac {16 \, x^{2} e^{x}}{16 \, x^{4} - 40 \, x^{3} - 32 \, x^{2} \log \left (x\right ) + 73 \, x^{2} + 40 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2} - 60 \, x - 48 \, \log \left (x\right ) + 36} \]
integrate(((64*x^2+128*x)*exp(x)*log(x)+(-64*x^4+208*x^3-96*x^2-320*x)*exp (x))/(64*log(x)^3+(-192*x^2+240*x-288)*log(x)^2+(192*x^4-480*x^3+876*x^2-7 20*x+432)*log(x)-64*x^6+240*x^5-588*x^4+845*x^3-882*x^2+540*x-216),x, algo rithm=\
16*x^2*e^x/(16*x^4 - 40*x^3 - 32*x^2*log(x) + 73*x^2 + 40*x*log(x) + 16*lo g(x)^2 - 60*x - 48*log(x) + 36)
Timed out. \[ \int \frac {e^x \left (-320 x-96 x^2+208 x^3-64 x^4\right )+e^x \left (128 x+64 x^2\right ) \log (x)}{-216+540 x-882 x^2+845 x^3-588 x^4+240 x^5-64 x^6+\left (432-720 x+876 x^2-480 x^3+192 x^4\right ) \log (x)+\left (-288+240 x-192 x^2\right ) \log ^2(x)+64 \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (64\,x^4-208\,x^3+96\,x^2+320\,x\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (64\,x^2+128\,x\right )}{540\,x-{\ln \left (x\right )}^2\,\left (192\,x^2-240\,x+288\right )+\ln \left (x\right )\,\left (192\,x^4-480\,x^3+876\,x^2-720\,x+432\right )+64\,{\ln \left (x\right )}^3-882\,x^2+845\,x^3-588\,x^4+240\,x^5-64\,x^6-216} \,d x \]
int(-(exp(x)*(320*x + 96*x^2 - 208*x^3 + 64*x^4) - exp(x)*log(x)*(128*x + 64*x^2))/(540*x - log(x)^2*(192*x^2 - 240*x + 288) + log(x)*(876*x^2 - 720 *x - 480*x^3 + 192*x^4 + 432) + 64*log(x)^3 - 882*x^2 + 845*x^3 - 588*x^4 + 240*x^5 - 64*x^6 - 216),x)