3.29.25 \(\int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} (832 x-40 x^2)}{x^2}} \, dx\) [2825]

3.29.25.1 Optimal result

Integrand size = 69, antiderivative size = 23 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=\frac {4}{-26-\frac {e^{3+2 x}}{x^2}+\frac {5 x}{4}} \]

output

4/(5/4*x-exp(-2*ln(x)+2*x+3)-26)
 

3.29.25.2 Mathematica [A] (verified)

Time = 2.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=\frac {16 x^2}{-4 e^{3+2 x}+x^2 (-104+5 x)} \]

input

Integrate[(-80*x + (E^(3 + 2*x)*(-128 + 128*x))/x^2)/((16*E^(6 + 4*x))/x^3 
 + 10816*x - 1040*x^2 + 25*x^3 + (E^(3 + 2*x)*(832*x - 40*x^2))/x^2),x]
 

output

(16*x^2)/(-4*E^(3 + 2*x) + x^2*(-104 + 5*x))
 

3.29.25.3 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.

\(\displaystyle \int \frac {\frac {e^{2 x+3} (128 x-128)}{x^2}-80 x}{25 x^3+\frac {16 e^{4 x+6}}{x^3}-1040 x^2+\frac {e^{2 x+3} \left (832 x-40 x^2\right )}{x^2}+10816 x} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {128 e^{2 x+3} (x-1) x-80 x^4}{\left ((104-5 x) x^2+4 e^{2 x+3}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {16 x^3 \left (10 x^2-223 x+208\right )}{\left (5 x^3-104 x^2-4 e^{2 x+3}\right )^2}-\frac {32 (x-1) x}{5 x^3-104 x^2-4 e^{2 x+3}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3328 \int \frac {x^3}{\left (5 x^3-104 x^2-4 e^{2 x+3}\right )^2}dx+32 \int \frac {x}{5 x^3-104 x^2-4 e^{2 x+3}}dx-32 \int \frac {x^2}{5 x^3-104 x^2-4 e^{2 x+3}}dx+160 \int \frac {x^5}{\left (5 x^3-104 x^2-4 e^{2 x+3}\right )^2}dx-3568 \int \frac {x^4}{\left (5 x^3-104 x^2-4 e^{2 x+3}\right )^2}dx\)

input

Int[(-80*x + (E^(3 + 2*x)*(-128 + 128*x))/x^2)/((16*E^(6 + 4*x))/x^3 + 108 
16*x - 1040*x^2 + 25*x^3 + (E^(3 + 2*x)*(832*x - 40*x^2))/x^2),x]
 

output

$Aborted
 

3.29.25.3.1 Defintions of rubi rules used

rule 2009

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

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

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

3.29.25.4 Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

method	result	size
risch	\(\frac {16}{5 x -\frac {4 \,{\mathrm e}^{3+2 x}}{x^{2}}-104}\)	\(21\)
norman	\(\frac {16}{5 x -4 \,{\mathrm e}^{-2 \ln \left (x \right )+2 x +3}-104}\)	\(22\)
parallelrisch	\(\frac {16}{5 x -4 \,{\mathrm e}^{-2 \ln \left (x \right )+2 x +3}-104}\)	\(22\)

input

int(((128*x-128)*exp(-2*ln(x)+2*x+3)-80*x)/(16*x*exp(-2*ln(x)+2*x+3)^2+(-4 
0*x^2+832*x)*exp(-2*ln(x)+2*x+3)+25*x^3-1040*x^2+10816*x),x,method=_RETURN 
VERBOSE)
 

output

16/(5*x-4/x^2*exp(3+2*x)-104)
 

3.29.25.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=\frac {16}{5 \, x - 4 \, e^{\left (2 \, x - 2 \, \log \left (x\right ) + 3\right )} - 104} \]

input

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+ 
3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2*x+3)+25*x^3-1040*x^2+10816*x),x, algo 
rithm=\
 

output

16/(5*x - 4*e^(2*x - 2*log(x) + 3) - 104)
 

3.29.25.6 Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=- \frac {4 x^{2}}{- \frac {5 x^{3}}{4} + 26 x^{2} + e^{2 x + 3}} \]

input

integrate(((128*x-128)*exp(-2*ln(x)+2*x+3)-80*x)/(16*x*exp(-2*ln(x)+2*x+3) 
**2+(-40*x**2+832*x)*exp(-2*ln(x)+2*x+3)+25*x**3-1040*x**2+10816*x),x)
 

output

-4*x**2/(-5*x**3/4 + 26*x**2 + exp(2*x + 3))
 

3.29.25.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=\frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \]

input

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+ 
3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2*x+3)+25*x^3-1040*x^2+10816*x),x, algo 
rithm=\
 

output

16*x^2/(5*x^3 - 104*x^2 - 4*e^(2*x + 3))
 

3.29.25.8 Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=\frac {16 \, x^{2}}{5 \, x^{3} - 104 \, x^{2} - 4 \, e^{\left (2 \, x + 3\right )}} \]

input

integrate(((128*x-128)*exp(-2*log(x)+2*x+3)-80*x)/(16*x*exp(-2*log(x)+2*x+ 
3)^2+(-40*x^2+832*x)*exp(-2*log(x)+2*x+3)+25*x^3-1040*x^2+10816*x),x, algo 
rithm=\
 

output

16*x^2/(5*x^3 - 104*x^2 - 4*e^(2*x + 3))
 

3.29.25.9 Mupad [B] (verification not implemented)

Time = 9.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-80 x+\frac {e^{3+2 x} (-128+128 x)}{x^2}}{\frac {16 e^{6+4 x}}{x^3}+10816 x-1040 x^2+25 x^3+\frac {e^{3+2 x} \left (832 x-40 x^2\right )}{x^2}} \, dx=-\frac {16\,x^2}{4\,{\mathrm {e}}^{2\,x+3}+104\,x^2-5\,x^3} \]

input

int(-(80*x - exp(2*x - 2*log(x) + 3)*(128*x - 128))/(10816*x + exp(2*x - 2 
*log(x) + 3)*(832*x - 40*x^2) + 16*x*exp(4*x - 4*log(x) + 6) - 1040*x^2 + 
25*x^3),x)
 

output

-(16*x^2)/(4*exp(2*x + 3) + 104*x^2 - 5*x^3)