Integrand size = 72, antiderivative size = 19 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 x^{(4+x) \left (e^2+(3+25 x)^2\right )} \]
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 x^{(4+x) \left (e^2+(3+25 x)^2\right )} \]
Integrate[x^(35 + 609*x + 2650*x^2 + 625*x^3 + E^2*(4 + x))*(72 + 1218*x + 5300*x^2 + 1250*x^3 + E^2*(8 + 2*x) + (1218*x + 2*E^2*x + 10600*x^2 + 375 0*x^3)*Log[x]),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.
\(\displaystyle \int x^{625 x^3+2650 x^2+609 x+e^2 (x+4)+35} \left (1250 x^3+5300 x^2+\left (3750 x^3+10600 x^2+2 e^2 x+1218 x\right ) \log (x)+1218 x+e^2 (2 x+8)+72\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+35} \left (1250 x^3+5300 x^2+\left (3750 x^3+10600 x^2+2 e^2 x+1218 x\right ) \log (x)+1218 x+e^2 (2 x+8)+72\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 e^2 (x+4) x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+35}+72 x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+35}+1218 x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+36}+5300 x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}+1250 x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+38}+2 \left (1875 x^2+5300 x+e^2+609\right ) x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+36} \log (x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (609+e^2\right ) \int \frac {\int x^{(x+4) \left (625 x^2+150 x+e^2+9\right )}dx}{x}dx+\frac {8 e^2 \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+35}dx}{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}+72 \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+35}dx+5300 \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}dx+1218 \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 \left (9+e^2\right )}dx+1250 \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+2 \left (19+2 e^2\right )}dx-10600 \int \frac {\int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}dx}{x}dx-3750 \int \frac {\int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+2 \left (19+2 e^2\right )}dx}{x}dx+10600 \log (x) \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}dx+2 \left (609+e^2\right ) \log (x) \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 \left (9+e^2\right )}dx+3750 \log (x) \int x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+2 \left (19+2 e^2\right )}dx+\frac {2 e^2 (x+4) x^{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 \left (9+e^2\right )}}{625 x^3+2650 x^2+\left (609+e^2\right ) x+4 e^2+37}\) |
Int[x^(35 + 609*x + 2650*x^2 + 625*x^3 + E^2*(4 + x))*(72 + 1218*x + 5300* x^2 + 1250*x^3 + E^2*(8 + 2*x) + (1218*x + 2*E^2*x + 10600*x^2 + 3750*x^3) *Log[x]),x]
3.18.89.3.1 Defintions of rubi rules used
Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11
method | result | size |
risch | \(2 x^{\left (625 x^{2}+{\mathrm e}^{2}+150 x +9\right ) \left (4+x \right )}\) | \(21\) |
norman | \(2 \,{\mathrm e}^{\left (\left (4+x \right ) {\mathrm e}^{2}+625 x^{3}+2650 x^{2}+609 x +36\right ) \ln \left (x \right )}\) | \(28\) |
parallelrisch | \(2 \,{\mathrm e}^{\left (\left (4+x \right ) {\mathrm e}^{2}+625 x^{3}+2650 x^{2}+609 x +36\right ) \ln \left (x \right )}\) | \(28\) |
int(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*ln(x)+(2*x+8)*exp(2)+1250*x^3+ 5300*x^2+1218*x+72)*exp(((4+x)*exp(2)+625*x^3+2650*x^2+609*x+36)*ln(x))/x, x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 \, x^{625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + 609 \, x + 36} \]
integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+12 50*x^3+5300*x^2+1218*x+72)*exp(((4+x)*exp(2)+625*x^3+2650*x^2+609*x+36)*lo g(x))/x,x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 e^{\left (625 x^{3} + 2650 x^{2} + 609 x + \left (x + 4\right ) e^{2} + 36\right ) \log {\left (x \right )}} \]
integrate(((2*exp(2)*x+3750*x**3+10600*x**2+1218*x)*ln(x)+(2*x+8)*exp(2)+1 250*x**3+5300*x**2+1218*x+72)*exp(((4+x)*exp(2)+625*x**3+2650*x**2+609*x+3 6)*ln(x))/x,x)
Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).
Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.00 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2 \, x^{36} e^{\left (625 \, x^{3} \log \left (x\right ) + 2650 \, x^{2} \log \left (x\right ) + x e^{2} \log \left (x\right ) + 609 \, x \log \left (x\right ) + 4 \, e^{2} \log \left (x\right )\right )} \]
integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+12 50*x^3+5300*x^2+1218*x+72)*exp(((4+x)*exp(2)+625*x^3+2650*x^2+609*x+36)*lo g(x))/x,x, algorithm=\
\[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=\int { \frac {2 \, {\left (625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + {\left (1875 \, x^{3} + 5300 \, x^{2} + x e^{2} + 609 \, x\right )} \log \left (x\right ) + 609 \, x + 36\right )} x^{625 \, x^{3} + 2650 \, x^{2} + {\left (x + 4\right )} e^{2} + 609 \, x + 36}}{x} \,d x } \]
integrate(((2*exp(2)*x+3750*x^3+10600*x^2+1218*x)*log(x)+(2*x+8)*exp(2)+12 50*x^3+5300*x^2+1218*x+72)*exp(((4+x)*exp(2)+625*x^3+2650*x^2+609*x+36)*lo g(x))/x,x, algorithm=\
integrate(2*(625*x^3 + 2650*x^2 + (x + 4)*e^2 + (1875*x^3 + 5300*x^2 + x*e ^2 + 609*x)*log(x) + 609*x + 36)*x^(625*x^3 + 2650*x^2 + (x + 4)*e^2 + 609 *x + 36)/x, x)
Time = 14.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int x^{35+609 x+2650 x^2+625 x^3+e^2 (4+x)} \left (72+1218 x+5300 x^2+1250 x^3+e^2 (8+2 x)+\left (1218 x+2 e^2 x+10600 x^2+3750 x^3\right ) \log (x)\right ) \, dx=2\,x^{625\,x^3}\,x^{2650\,x^2}\,x^{x\,{\mathrm {e}}^2}\,x^{4\,{\mathrm {e}}^2}\,x^{609\,x}\,x^{36} \]