\(\int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} (75-375 x+575 x^2)}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} (9-78 x+241 x^2-312 x^3+144 x^4)+e^{\frac {x}{-1+3 x}} (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6)} \, dx\) [945]

Optimal result

Integrand size = 156, antiderivative size = 32 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\frac {5}{\left (5-\frac {e^{\frac {x}{-1+3 x}}}{5 x}+x\right ) (-3+4 x)} \] Output:

5/(-3+4*x)/(5+x-1/5*exp(x/(-1+3*x))/x)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\frac {25 x}{(-3+4 x) \left (-e^{\frac {x}{-1+3 x}}+5 x (5+x)\right )} \] Input:

Integrate[(-2125*x^2 + 11750*x^3 - 13125*x^4 - 9000*x^5 + E^(x/(-1 + 3*x)) 
*(75 - 375*x + 575*x^2))/(5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 
 18025*x^6 + 28200*x^7 + 3600*x^8 + E^((2*x)/(-1 + 3*x))*(9 - 78*x + 241*x 
^2 - 312*x^3 + 144*x^4) + E^(x/(-1 + 3*x))*(-450*x + 3810*x^2 - 11270*x^3 
+ 13190*x^4 - 4080*x^5 - 1440*x^6)),x]
 

Output:

(25*x)/((-3 + 4*x)*(-E^(x/(-1 + 3*x)) + 5*x*(5 + x)))
 

Rubi [F]

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.

\(\displaystyle \int \frac {-9000 x^5-13125 x^4+11750 x^3-2125 x^2+e^{\frac {x}{3 x-1}} \left (575 x^2-375 x+75\right )}{3600 x^8+28200 x^7+18025 x^6-136700 x^5+131350 x^4-46500 x^3+5625 x^2+e^{\frac {2 x}{3 x-1}} \left (144 x^4-312 x^3+241 x^2-78 x+9\right )+e^{\frac {x}{3 x-1}} \left (-1440 x^6-4080 x^5+13190 x^4-11270 x^3+3810 x^2-450 x\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {25 \left (e^{\frac {x}{3 x-1}} \left (23 x^2-15 x+3\right )-5 (1-3 x)^2 x^2 (8 x+17)\right )}{\left (12 x^2-13 x+3\right )^2 \left (e^{\frac {x}{3 x-1}}-5 x (x+5)\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 25 \int -\frac {5 (1-3 x)^2 x^2 (8 x+17)-e^{-\frac {x}{1-3 x}} \left (23 x^2-15 x+3\right )}{\left (12 x^2-13 x+3\right )^2 \left (e^{-\frac {x}{1-3 x}}-5 x (x+5)\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -25 \int \frac {5 (1-3 x)^2 x^2 (8 x+17)-e^{-\frac {x}{1-3 x}} \left (23 x^2-15 x+3\right )}{\left (12 x^2-13 x+3\right )^2 \left (e^{-\frac {x}{1-3 x}}-5 x (x+5)\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -25 \int \left (\frac {8 x+17}{5 (x+5)^2 (4 x-3)^2}+\frac {18 x^3+34 x^2-23 x+5}{5 x (x+5)^2 (3 x-1)^2 (4 x-3) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}+\frac {144 x^4+187 x^3-288 x^2+106 x-15}{5 x (x+5)^2 (3 x-1)^2 (4 x-3)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -25 \left (-\frac {1}{75} \int \frac {1}{x \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx-\frac {1}{115} \int \frac {1}{(x+5)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx+\frac {653 \int \frac {1}{(x+5) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx}{3385600}-\frac {9}{400} \int \frac {1}{(3 x-1)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx-\frac {531 \int \frac {1}{(3 x-1) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx}{32000}+\frac {14816 \int \frac {1}{(4 x-3) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )^2}dx}{198375}-\frac {1}{75} \int \frac {1}{x \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx-\frac {2}{115} \int \frac {1}{(x+5)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx+\frac {653 \int \frac {1}{(x+5) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx}{3385600}-\frac {9}{400} \int \frac {1}{(3 x-1)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx-\frac {531 \int \frac {1}{(3 x-1) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx}{32000}+\frac {16}{115} \int \frac {1}{(4 x-3)^2 \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx+\frac {14816 \int \frac {1}{(4 x-3) \left (5 e^{\frac {x}{1-3 x}} x^2+25 e^{\frac {x}{1-3 x}} x-1\right )}dx}{198375}+\frac {1}{5 (3-4 x) (x+5)}\right )\)

Input:

Int[(-2125*x^2 + 11750*x^3 - 13125*x^4 - 9000*x^5 + E^(x/(-1 + 3*x))*(75 - 
 375*x + 575*x^2))/(5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 18025 
*x^6 + 28200*x^7 + 3600*x^8 + E^((2*x)/(-1 + 3*x))*(9 - 78*x + 241*x^2 - 3 
12*x^3 + 144*x^4) + E^(x/(-1 + 3*x))*(-450*x + 3810*x^2 - 11270*x^3 + 1319 
0*x^4 - 4080*x^5 - 1440*x^6)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {25 x}{\left (-3+4 x \right ) \left (5 x^{2}+25 x -{\mathrm e}^{\frac {x}{-1+3 x}}\right )}\)	\(34\)
parallelrisch	\(\frac {25 x}{20 x^{3}+85 x^{2}-4 x \,{\mathrm e}^{\frac {x}{-1+3 x}}-75 x +3 \,{\mathrm e}^{\frac {x}{-1+3 x}}}\)	\(45\)
norman	\(\frac {75 x^{2}-25 x}{60 x^{4}+235 x^{3}-12 \,{\mathrm e}^{\frac {x}{-1+3 x}} x^{2}-310 x^{2}+13 x \,{\mathrm e}^{\frac {x}{-1+3 x}}+75 x -3 \,{\mathrm e}^{\frac {x}{-1+3 x}}}\)	\(72\)

Input:

int(((575*x^2-375*x+75)*exp(x/(-1+3*x))-9000*x^5-13125*x^4+11750*x^3-2125* 
x^2)/((144*x^4-312*x^3+241*x^2-78*x+9)*exp(x/(-1+3*x))^2+(-1440*x^6-4080*x 
^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(-1+3*x))+3600*x^8+28200*x^7+ 
18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x,method=_RETURNVERBOS 
E)
 

Output:

25*x/(-3+4*x)/(5*x^2+25*x-exp(x/(-1+3*x)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\frac {25 \, x}{20 \, x^{3} + 85 \, x^{2} - {\left (4 \, x - 3\right )} e^{\left (\frac {x}{3 \, x - 1}\right )} - 75 \, x} \] Input:

integrate(((575*x^2-375*x+75)*exp(x/(-1+3*x))-9000*x^5-13125*x^4+11750*x^3 
-2125*x^2)/((144*x^4-312*x^3+241*x^2-78*x+9)*exp(x/(-1+3*x))^2+(-1440*x^6- 
4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(-1+3*x))+3600*x^8+2820 
0*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="fr 
icas")
 

Output:

25*x/(20*x^3 + 85*x^2 - (4*x - 3)*e^(x/(3*x - 1)) - 75*x)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=- \frac {25 x}{- 20 x^{3} - 85 x^{2} + 75 x + \left (4 x - 3\right ) e^{\frac {x}{3 x - 1}}} \] Input:

integrate(((575*x**2-375*x+75)*exp(x/(-1+3*x))-9000*x**5-13125*x**4+11750* 
x**3-2125*x**2)/((144*x**4-312*x**3+241*x**2-78*x+9)*exp(x/(-1+3*x))**2+(- 
1440*x**6-4080*x**5+13190*x**4-11270*x**3+3810*x**2-450*x)*exp(x/(-1+3*x)) 
+3600*x**8+28200*x**7+18025*x**6-136700*x**5+131350*x**4-46500*x**3+5625*x 
**2),x)
 

Output:

-25*x/(-20*x**3 - 85*x**2 + 75*x + (4*x - 3)*exp(x/(3*x - 1)))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(((575*x^2-375*x+75)*exp(x/(-1+3*x))-9000*x^5-13125*x^4+11750*x^3 
-2125*x^2)/((144*x^4-312*x^3+241*x^2-78*x+9)*exp(x/(-1+3*x))^2+(-1440*x^6- 
4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(-1+3*x))+3600*x^8+2820 
0*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="ma 
xima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (31) = 62\).

Time = 0.47 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.66 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\frac {25 \, {\left (\frac {x}{3 \, x - 1} - \frac {6 \, x^{2}}{{\left (3 \, x - 1\right )}^{2}} + \frac {9 \, x^{3}}{{\left (3 \, x - 1\right )}^{3}}\right )}}{\frac {23 \, x e^{\left (\frac {x}{3 \, x - 1}\right )}}{3 \, x - 1} - \frac {57 \, x^{2} e^{\left (\frac {x}{3 \, x - 1}\right )}}{{\left (3 \, x - 1\right )}^{2}} + \frac {45 \, x^{3} e^{\left (\frac {x}{3 \, x - 1}\right )}}{{\left (3 \, x - 1\right )}^{3}} - \frac {75 \, x}{3 \, x - 1} + \frac {365 \, x^{2}}{{\left (3 \, x - 1\right )}^{2}} - \frac {400 \, x^{3}}{{\left (3 \, x - 1\right )}^{3}} - 3 \, e^{\left (\frac {x}{3 \, x - 1}\right )}} \] Input:

integrate(((575*x^2-375*x+75)*exp(x/(-1+3*x))-9000*x^5-13125*x^4+11750*x^3 
-2125*x^2)/((144*x^4-312*x^3+241*x^2-78*x+9)*exp(x/(-1+3*x))^2+(-1440*x^6- 
4080*x^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(-1+3*x))+3600*x^8+2820 
0*x^7+18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x, algorithm="gi 
ac")
 

Output:

25*(x/(3*x - 1) - 6*x^2/(3*x - 1)^2 + 9*x^3/(3*x - 1)^3)/(23*x*e^(x/(3*x - 
 1))/(3*x - 1) - 57*x^2*e^(x/(3*x - 1))/(3*x - 1)^2 + 45*x^3*e^(x/(3*x - 1 
))/(3*x - 1)^3 - 75*x/(3*x - 1) + 365*x^2/(3*x - 1)^2 - 400*x^3/(3*x - 1)^ 
3 - 3*e^(x/(3*x - 1)))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\int -\frac {2125\,x^2-{\mathrm {e}}^{\frac {x}{3\,x-1}}\,\left (575\,x^2-375\,x+75\right )-11750\,x^3+13125\,x^4+9000\,x^5}{{\mathrm {e}}^{\frac {2\,x}{3\,x-1}}\,\left (144\,x^4-312\,x^3+241\,x^2-78\,x+9\right )-{\mathrm {e}}^{\frac {x}{3\,x-1}}\,\left (1440\,x^6+4080\,x^5-13190\,x^4+11270\,x^3-3810\,x^2+450\,x\right )+5625\,x^2-46500\,x^3+131350\,x^4-136700\,x^5+18025\,x^6+28200\,x^7+3600\,x^8} \,d x \] Input:

int(-(2125*x^2 - exp(x/(3*x - 1))*(575*x^2 - 375*x + 75) - 11750*x^3 + 131 
25*x^4 + 9000*x^5)/(exp((2*x)/(3*x - 1))*(241*x^2 - 78*x - 312*x^3 + 144*x 
^4 + 9) - exp(x/(3*x - 1))*(450*x - 3810*x^2 + 11270*x^3 - 13190*x^4 + 408 
0*x^5 + 1440*x^6) + 5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 18025 
*x^6 + 28200*x^7 + 3600*x^8),x)
 

Output:

int(-(2125*x^2 - exp(x/(3*x - 1))*(575*x^2 - 375*x + 75) - 11750*x^3 + 131 
25*x^4 + 9000*x^5)/(exp((2*x)/(3*x - 1))*(241*x^2 - 78*x - 312*x^3 + 144*x 
^4 + 9) - exp(x/(3*x - 1))*(450*x - 3810*x^2 + 11270*x^3 - 13190*x^4 + 408 
0*x^5 + 1440*x^6) + 5625*x^2 - 46500*x^3 + 131350*x^4 - 136700*x^5 + 18025 
*x^6 + 28200*x^7 + 3600*x^8), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {-2125 x^2+11750 x^3-13125 x^4-9000 x^5+e^{\frac {x}{-1+3 x}} \left (75-375 x+575 x^2\right )}{5625 x^2-46500 x^3+131350 x^4-136700 x^5+18025 x^6+28200 x^7+3600 x^8+e^{\frac {2 x}{-1+3 x}} \left (9-78 x+241 x^2-312 x^3+144 x^4\right )+e^{\frac {x}{-1+3 x}} \left (-450 x+3810 x^2-11270 x^3+13190 x^4-4080 x^5-1440 x^6\right )} \, dx=\frac {4 e^{\frac {x}{3 x -1}} x -3 e^{\frac {x}{3 x -1}}-20 x^{3}-85 x^{2}}{12 e^{\frac {x}{3 x -1}} x -9 e^{\frac {x}{3 x -1}}-60 x^{3}-255 x^{2}+225 x} \] Input:

int(((575*x^2-375*x+75)*exp(x/(-1+3*x))-9000*x^5-13125*x^4+11750*x^3-2125* 
x^2)/((144*x^4-312*x^3+241*x^2-78*x+9)*exp(x/(-1+3*x))^2+(-1440*x^6-4080*x 
^5+13190*x^4-11270*x^3+3810*x^2-450*x)*exp(x/(-1+3*x))+3600*x^8+28200*x^7+ 
18025*x^6-136700*x^5+131350*x^4-46500*x^3+5625*x^2),x)
 

Output:

(4*e**(x/(3*x - 1))*x - 3*e**(x/(3*x - 1)) - 20*x**3 - 85*x**2)/(3*(4*e**( 
x/(3*x - 1))*x - 3*e**(x/(3*x - 1)) - 20*x**3 - 85*x**2 + 75*x))