3.22.75 \(\int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} (-625 x^2-50 x^3-x^4+(1875 x^3+150 x^4+3 x^5) \log (3))}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} (625 x^2+50 x^3+x^4)+3^{3 x} (250 x+6210 x^2+498 x^3+10 x^4)} \, dx\) [2175]

3.22.75.1 Optimal result

Integrand size = 128, antiderivative size = 25 \[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=\frac {x}{-5-3^{3 x}+\frac {-5+x}{x (25+x)}} \]

output

x/((-5+x)/x/(x+25)-5-exp(3*x*ln(3)))
 

3.22.75.2 Mathematica [F]

\[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=\int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx \]

input

Integrate[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^ 
3 - x^4 + (1875*x^3 + 150*x^4 + 3*x^5)*Log[3]))/(25 + 1240*x + 15426*x^2 + 
 1240*x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x + 6 
210*x^2 + 498*x^3 + 10*x^4)),x]
 

output

Integrate[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^ 
3 - x^4 + (1875*x^3 + 150*x^4 + 3*x^5)*Log[3]))/(25 + 1240*x + 15426*x^2 + 
 1240*x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x + 6 
210*x^2 + 498*x^3 + 10*x^4)), x]
 

3.22.75.3 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.

\(\displaystyle \int \frac {-5 x^4-248 x^3-3115 x^2+3^{3 x} \left (-x^4-50 x^3-625 x^2+\left (3 x^5+150 x^4+1875 x^3\right ) \log (3)\right )-250 x}{25 x^4+1240 x^3+15426 x^2+3^{6 x} \left (x^4+50 x^3+625 x^2\right )+3^{3 x} \left (10 x^4+498 x^3+6210 x^2+250 x\right )+1240 x+25} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-5 x^4-248 x^3-3115 x^2+3^{3 x} \left (-x^4-50 x^3-625 x^2+\left (3 x^5+150 x^4+1875 x^3\right ) \log (3)\right )-250 x}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x (x+25) (x \log (27)-1)}{27^x x^2+5 x^2+25\ 27^x x+124 x+5}+\frac {x \left (-15 x^4 \log (3)-747 x^3 \log (3)+x^2 (1-9315 \log (3))-5 x (2+75 \log (3))-125\right )}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -125 \int \frac {x}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx-25 \int \frac {x}{27^x x^2+5 x^2+25\ 27^x x+124 x+5}dx-5 (2+75 \log (3)) \int \frac {x^2}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx-(1-25 \log (27)) \int \frac {x^2}{27^x x^2+5 x^2+25\ 27^x x+124 x+5}dx-15 \log (3) \int \frac {x^5}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx-747 \log (3) \int \frac {x^4}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx+(1-9315 \log (3)) \int \frac {x^3}{\left (27^x x^2+5 x^2+25\ 27^x x+124 x+5\right )^2}dx+\log (27) \int \frac {x^3}{27^x x^2+5 x^2+25\ 27^x x+124 x+5}dx\)

input

Int[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^3 - x^ 
4 + (1875*x^3 + 150*x^4 + 3*x^5)*Log[3]))/(25 + 1240*x + 15426*x^2 + 1240* 
x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x + 6210*x^ 
2 + 498*x^3 + 10*x^4)),x]
 

output

$Aborted
 

3.22.75.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.22.75.4 Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36

method	result	size
risch	\(-\frac {x^{2} \left (x +25\right )}{27^{x} x^{2}+25 \,27^{x} x +5 x^{2}+124 x +5}\)	\(34\)
parallelrisch	\(-\frac {x^{3}+25 x^{2}}{{\mathrm e}^{3 x \ln \left (3\right )} x^{2}+25 \,{\mathrm e}^{3 x \ln \left (3\right )} x +5 x^{2}+124 x +5}\)	\(43\)
norman	\(\frac {-x^{3}-25 x^{2}}{{\mathrm e}^{3 x \ln \left (3\right )} x^{2}+25 \,{\mathrm e}^{3 x \ln \left (3\right )} x +5 x^{2}+124 x +5}\)	\(44\)

input

int((((3*x^5+150*x^4+1875*x^3)*ln(3)-x^4-50*x^3-625*x^2)*exp(3*x*ln(3))-5* 
x^4-248*x^3-3115*x^2-250*x)/((x^4+50*x^3+625*x^2)*exp(3*x*ln(3))^2+(10*x^4 
+498*x^3+6210*x^2+250*x)*exp(3*x*ln(3))+25*x^4+1240*x^3+15426*x^2+1240*x+2 
5),x,method=_RETURNVERBOSE)
 

output

-x^2*(x+25)/(27^x*x^2+25*27^x*x+5*x^2+124*x+5)
 

3.22.75.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=-\frac {x^{3} + 25 \, x^{2}}{{\left (x^{2} + 25 \, x\right )} 3^{3 \, x} + 5 \, x^{2} + 124 \, x + 5} \]

input

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*lo 
g(3))-5*x^4-248*x^3-3115*x^2-250*x)/((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^ 
2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*x^ 
2+1240*x+25),x, algorithm=\
 

output

-(x^3 + 25*x^2)/((x^2 + 25*x)*3^(3*x) + 5*x^2 + 124*x + 5)
 

3.22.75.6 Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=\frac {- x^{3} - 25 x^{2}}{5 x^{2} + 124 x + \left (x^{2} + 25 x\right ) e^{3 x \log {\left (3 \right )}} + 5} \]

input

integrate((((3*x**5+150*x**4+1875*x**3)*ln(3)-x**4-50*x**3-625*x**2)*exp(3 
*x*ln(3))-5*x**4-248*x**3-3115*x**2-250*x)/((x**4+50*x**3+625*x**2)*exp(3* 
x*ln(3))**2+(10*x**4+498*x**3+6210*x**2+250*x)*exp(3*x*ln(3))+25*x**4+1240 
*x**3+15426*x**2+1240*x+25),x)
 

output

(-x**3 - 25*x**2)/(5*x**2 + 124*x + (x**2 + 25*x)*exp(3*x*log(3)) + 5)
 

3.22.75.7 Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=-\frac {x^{3} + 25 \, x^{2}}{{\left (x^{2} + 25 \, x\right )} 3^{3 \, x} + 5 \, x^{2} + 124 \, x + 5} \]

input

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*lo 
g(3))-5*x^4-248*x^3-3115*x^2-250*x)/((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^ 
2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*x^ 
2+1240*x+25),x, algorithm=\
 

output

-(x^3 + 25*x^2)/((x^2 + 25*x)*3^(3*x) + 5*x^2 + 124*x + 5)
 

3.22.75.8 Giac [F]

\[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=\int { -\frac {5 \, x^{4} + 248 \, x^{3} + {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2} - 3 \, {\left (x^{5} + 50 \, x^{4} + 625 \, x^{3}\right )} \log \left (3\right )\right )} 3^{3 \, x} + 3115 \, x^{2} + 250 \, x}{25 \, x^{4} + 1240 \, x^{3} + {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2}\right )} 3^{6 \, x} + 2 \, {\left (5 \, x^{4} + 249 \, x^{3} + 3105 \, x^{2} + 125 \, x\right )} 3^{3 \, x} + 15426 \, x^{2} + 1240 \, x + 25} \,d x } \]

input

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*lo 
g(3))-5*x^4-248*x^3-3115*x^2-250*x)/((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^ 
2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*x^ 
2+1240*x+25),x, algorithm=\
 

output

integrate(-(5*x^4 + 248*x^3 + (x^4 + 50*x^3 + 625*x^2 - 3*(x^5 + 50*x^4 + 
625*x^3)*log(3))*3^(3*x) + 3115*x^2 + 250*x)/(25*x^4 + 1240*x^3 + (x^4 + 5 
0*x^3 + 625*x^2)*3^(6*x) + 2*(5*x^4 + 249*x^3 + 3105*x^2 + 125*x)*3^(3*x) 
+ 15426*x^2 + 1240*x + 25), x)
 

3.22.75.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx=\int -\frac {250\,x+{\mathrm {e}}^{3\,x\,\ln \left (3\right )}\,\left (625\,x^2-\ln \left (3\right )\,\left (3\,x^5+150\,x^4+1875\,x^3\right )+50\,x^3+x^4\right )+3115\,x^2+248\,x^3+5\,x^4}{1240\,x+{\mathrm {e}}^{6\,x\,\ln \left (3\right )}\,\left (x^4+50\,x^3+625\,x^2\right )+15426\,x^2+1240\,x^3+25\,x^4+{\mathrm {e}}^{3\,x\,\ln \left (3\right )}\,\left (10\,x^4+498\,x^3+6210\,x^2+250\,x\right )+25} \,d x \]

input

int(-(250*x + exp(3*x*log(3))*(625*x^2 - log(3)*(1875*x^3 + 150*x^4 + 3*x^ 
5) + 50*x^3 + x^4) + 3115*x^2 + 248*x^3 + 5*x^4)/(1240*x + exp(6*x*log(3)) 
*(625*x^2 + 50*x^3 + x^4) + 15426*x^2 + 1240*x^3 + 25*x^4 + exp(3*x*log(3) 
)*(250*x + 6210*x^2 + 498*x^3 + 10*x^4) + 25),x)
 

output

int(-(250*x + exp(3*x*log(3))*(625*x^2 - log(3)*(1875*x^3 + 150*x^4 + 3*x^ 
5) + 50*x^3 + x^4) + 3115*x^2 + 248*x^3 + 5*x^4)/(1240*x + exp(6*x*log(3)) 
*(625*x^2 + 50*x^3 + x^4) + 15426*x^2 + 1240*x^3 + 25*x^4 + exp(3*x*log(3) 
)*(250*x + 6210*x^2 + 498*x^3 + 10*x^4) + 25), x)