Integrand size = 214, antiderivative size = 29 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-e^{3+x}+\frac {x}{5+\frac {1}{2} x (5+x)^2}\right )}{x^3} \]
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-e^{3+x}+\frac {2 x}{10+25 x+10 x^2+x^3}\right )}{x^3} \]
Integrate[(-20*x + 20*x^3 + 4*x^4 + E^(3 + x)*(100*x + 500*x^2 + 825*x^3 + 520*x^4 + 150*x^5 + 20*x^6 + x^7) + (60*x + 150*x^2 + 60*x^3 + 6*x^4 + E^ (3 + x)*(-300 - 1500*x - 2475*x^2 - 1560*x^3 - 450*x^4 - 60*x^5 - 3*x^6))* Log[(2*x + E^(3 + x)*(-10 - 25*x - 10*x^2 - x^3))/(10 + 25*x + 10*x^2 + x^ 3)])/(-20*x^5 - 50*x^6 - 20*x^7 - 2*x^8 + E^(3 + x)*(100*x^4 + 500*x^5 + 8 25*x^6 + 520*x^7 + 150*x^8 + 20*x^9 + x^10)),x]
Time = 7.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7292, 7293, 2009}
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.
\(\displaystyle \int \frac {4 x^4+20 x^3+\left (6 x^4+60 x^3+150 x^2+e^{x+3} \left (-3 x^6-60 x^5-450 x^4-1560 x^3-2475 x^2-1500 x-300\right )+60 x\right ) \log \left (\frac {e^{x+3} \left (-x^3-10 x^2-25 x-10\right )+2 x}{x^3+10 x^2+25 x+10}\right )+e^{x+3} \left (x^7+20 x^6+150 x^5+520 x^4+825 x^3+500 x^2+100 x\right )-20 x}{-2 x^8-20 x^7-50 x^6-20 x^5+e^{x+3} \left (x^{10}+20 x^9+150 x^8+520 x^7+825 x^6+500 x^5+100 x^4\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^4+20 x^3+\left (6 x^4+60 x^3+150 x^2+e^{x+3} \left (-3 x^6-60 x^5-450 x^4-1560 x^3-2475 x^2-1500 x-300\right )+60 x\right ) \log \left (\frac {e^{x+3} \left (-x^3-10 x^2-25 x-10\right )+2 x}{x^3+10 x^2+25 x+10}\right )+e^{x+3} \left (x^7+20 x^6+150 x^5+520 x^4+825 x^3+500 x^2+100 x\right )-20 x}{x^4 \left (x^3+10 x^2+25 x+10\right ) \left (e^{x+3} x^3+10 e^{x+3} x^2+25 e^{x+3} x-2 x+10 e^{x+3}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 \left (x^4+12 x^3+35 x^2+10 x-10\right )}{x^3 \left (x^3+10 x^2+25 x+10\right ) \left (e^{x+3} x^3+10 e^{x+3} x^2+25 e^{x+3} x-2 x+10 e^{x+3}\right )}+\frac {x-3 \log \left (\frac {2 x}{x^3+10 x^2+25 x+10}-e^{x+3}\right )}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\log \left (\frac {2 x}{x^3+10 x^2+25 x+10}-e^{x+3}\right )}{x^3}\) |
Int[(-20*x + 20*x^3 + 4*x^4 + E^(3 + x)*(100*x + 500*x^2 + 825*x^3 + 520*x ^4 + 150*x^5 + 20*x^6 + x^7) + (60*x + 150*x^2 + 60*x^3 + 6*x^4 + E^(3 + x )*(-300 - 1500*x - 2475*x^2 - 1560*x^3 - 450*x^4 - 60*x^5 - 3*x^6))*Log[(2 *x + E^(3 + x)*(-10 - 25*x - 10*x^2 - x^3))/(10 + 25*x + 10*x^2 + x^3)])/( -20*x^5 - 50*x^6 - 20*x^7 - 2*x^8 + E^(3 + x)*(100*x^4 + 500*x^5 + 825*x^6 + 520*x^7 + 150*x^8 + 20*x^9 + x^10)),x]
3.19.72.3.1 Defintions of rubi rules used
Time = 35.57 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59
method | result | size |
parallelrisch | \(\frac {\ln \left (\frac {\left (-x^{3}-10 x^{2}-25 x -10\right ) {\mathrm e}^{3+x}+2 x}{x^{3}+10 x^{2}+25 x +10}\right )}{x^{3}}\) | \(46\) |
risch | \(\frac {\ln \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}}-\frac {2 i \pi {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) \operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )-i \pi \,\operatorname {csgn}\left (i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{x^{3}+10 x^{2}+25 x +10}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{2}-i \pi {\operatorname {csgn}\left (\frac {i \left (x^{3} {\mathrm e}^{3+x}+10 x^{2} {\mathrm e}^{3+x}+\left (25 \,{\mathrm e}^{3+x}-2\right ) x +10 \,{\mathrm e}^{3+x}\right )}{x^{3}+10 x^{2}+25 x +10}\right )}^{3}-2 i \pi +2 \ln \left (x^{3}+10 x^{2}+25 x +10\right )}{2 x^{3}}\) | \(474\) |
int((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+6*x^4+ 60*x^3+150*x^2+60*x)*ln(((-x^3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+10*x^2+2 5*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3+x)+4*x^4 +20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4)*exp(3 +x)-2*x^8-20*x^7-50*x^6-20*x^5),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-\frac {{\left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right )} e^{\left (x + 3\right )} - 2 \, x}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \]
integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+ 6*x^4+60*x^3+150*x^2+60*x)*log(((-x^3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+1 0*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3+x )+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4 )*exp(3+x)-2*x^8-20*x^7-50*x^6-20*x^5),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log {\left (\frac {2 x + \left (- x^{3} - 10 x^{2} - 25 x - 10\right ) e^{x + 3}}{x^{3} + 10 x^{2} + 25 x + 10} \right )}}{x^{3}} \]
integrate((((-3*x**6-60*x**5-450*x**4-1560*x**3-2475*x**2-1500*x-300)*exp( 3+x)+6*x**4+60*x**3+150*x**2+60*x)*ln(((-x**3-10*x**2-25*x-10)*exp(3+x)+2* x)/(x**3+10*x**2+25*x+10))+(x**7+20*x**6+150*x**5+520*x**4+825*x**3+500*x* *2+100*x)*exp(3+x)+4*x**4+20*x**3-20*x)/((x**10+20*x**9+150*x**8+520*x**7+ 825*x**6+500*x**5+100*x**4)*exp(3+x)-2*x**8-20*x**7-50*x**6-20*x**5),x)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=-\frac {\log \left (x^{3} + 10 \, x^{2} + 25 \, x + 10\right ) - \log \left (-{\left (x^{3} e^{3} + 10 \, x^{2} e^{3} + 25 \, x e^{3} + 10 \, e^{3}\right )} e^{x} + 2 \, x\right )}{x^{3}} \]
integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+ 6*x^4+60*x^3+150*x^2+60*x)*log(((-x^3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+1 0*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3+x )+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4 )*exp(3+x)-2*x^8-20*x^7-50*x^6-20*x^5),x, algorithm=\
-(log(x^3 + 10*x^2 + 25*x + 10) - log(-(x^3*e^3 + 10*x^2*e^3 + 25*x*e^3 + 10*e^3)*e^x + 2*x))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\log \left (-\frac {x^{3} e^{\left (x + 3\right )} + 10 \, x^{2} e^{\left (x + 3\right )} + 25 \, x e^{\left (x + 3\right )} - 2 \, x + 10 \, e^{\left (x + 3\right )}}{x^{3} + 10 \, x^{2} + 25 \, x + 10}\right )}{x^{3}} \]
integrate((((-3*x^6-60*x^5-450*x^4-1560*x^3-2475*x^2-1500*x-300)*exp(3+x)+ 6*x^4+60*x^3+150*x^2+60*x)*log(((-x^3-10*x^2-25*x-10)*exp(3+x)+2*x)/(x^3+1 0*x^2+25*x+10))+(x^7+20*x^6+150*x^5+520*x^4+825*x^3+500*x^2+100*x)*exp(3+x )+4*x^4+20*x^3-20*x)/((x^10+20*x^9+150*x^8+520*x^7+825*x^6+500*x^5+100*x^4 )*exp(3+x)-2*x^8-20*x^7-50*x^6-20*x^5),x, algorithm=\
log(-(x^3*e^(x + 3) + 10*x^2*e^(x + 3) + 25*x*e^(x + 3) - 2*x + 10*e^(x + 3))/(x^3 + 10*x^2 + 25*x + 10))/x^3
Time = 12.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-20 x+20 x^3+4 x^4+e^{3+x} \left (100 x+500 x^2+825 x^3+520 x^4+150 x^5+20 x^6+x^7\right )+\left (60 x+150 x^2+60 x^3+6 x^4+e^{3+x} \left (-300-1500 x-2475 x^2-1560 x^3-450 x^4-60 x^5-3 x^6\right )\right ) \log \left (\frac {2 x+e^{3+x} \left (-10-25 x-10 x^2-x^3\right )}{10+25 x+10 x^2+x^3}\right )}{-20 x^5-50 x^6-20 x^7-2 x^8+e^{3+x} \left (100 x^4+500 x^5+825 x^6+520 x^7+150 x^8+20 x^9+x^{10}\right )} \, dx=\frac {\ln \left (\frac {2\,x-{\mathrm {e}}^3\,{\mathrm {e}}^x\,\left (x^3+10\,x^2+25\,x+10\right )}{x^3+10\,x^2+25\,x+10}\right )}{x^3} \]
int(-(log((2*x - exp(x + 3)*(25*x + 10*x^2 + x^3 + 10))/(25*x + 10*x^2 + x ^3 + 10))*(60*x + 150*x^2 + 60*x^3 + 6*x^4 - exp(x + 3)*(1500*x + 2475*x^2 + 1560*x^3 + 450*x^4 + 60*x^5 + 3*x^6 + 300)) - 20*x + exp(x + 3)*(100*x + 500*x^2 + 825*x^3 + 520*x^4 + 150*x^5 + 20*x^6 + x^7) + 20*x^3 + 4*x^4)/ (20*x^5 - exp(x + 3)*(100*x^4 + 500*x^5 + 825*x^6 + 520*x^7 + 150*x^8 + 20 *x^9 + x^10) + 50*x^6 + 20*x^7 + 2*x^8),x)