\(\int \frac {275+128 x+15 x^2+e^{4 x^3} (5+300 x^2+60 x^3)+e^{2 x^3} (-90-20 x-1200 x^2-540 x^3-60 x^4)}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} (-200-90 x-10 x^2)} \, dx\) [2377]

Optimal result

Integrand size = 107, antiderivative size = 25 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log \left ((5+x) \left (-\frac {x}{5}+\left (4-e^{2 x^3}+x\right )^2\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 15.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log (5+x)+\log \left (80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2\right ) \] Input:

Integrate[(275 + 128*x + 15*x^2 + E^(4*x^3)*(5 + 300*x^2 + 60*x^3) + E^(2* 
x^3)*(-90 - 20*x - 1200*x^2 - 540*x^3 - 60*x^4))/(400 + 275*x + 64*x^2 + 5 
*x^3 + E^(4*x^3)*(25 + 5*x) + E^(2*x^3)*(-200 - 90*x - 10*x^2)),x]
 

Output:

Log[5 + x] + Log[80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 
 5*x^2]
 

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.

\(\displaystyle \int \frac {15 x^2+e^{4 x^3} \left (60 x^3+300 x^2+5\right )+e^{2 x^3} \left (-60 x^4-540 x^3-1200 x^2-20 x-90\right )+128 x+275}{5 x^3+e^{4 x^3} (5 x+25)+64 x^2+e^{2 x^3} \left (-10 x^2-90 x-200\right )+275 x+400} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {15 x^2+e^{4 x^3} \left (60 x^3+300 x^2+5\right )+e^{2 x^3} \left (-60 x^4-540 x^3-1200 x^2-20 x-90\right )+128 x+275}{(x+5) \left (-10 e^{2 x^3} x-40 e^{2 x^3}+5 e^{4 x^3}+5 x^2+39 x+80\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {12 x^3+60 x^2+1}{x+5}+\frac {-60 x^4+60 e^{2 x^3} x^3-468 x^3-10 e^{2 x^3}-960 x^2+240 e^{2 x^3} x^2+10 x+39}{-10 e^{2 x^3} x-40 e^{2 x^3}+5 e^{4 x^3}+5 x^2+39 x+80}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 39 \int \frac {1}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx-10 \int \frac {e^{2 x^3}}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx+10 \int \frac {x}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx-960 \int \frac {x^2}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx+240 \int \frac {e^{2 x^3} x^2}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx-468 \int \frac {x^3}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx+60 \int \frac {e^{2 x^3} x^3}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx-60 \int \frac {x^4}{5 x^2-10 e^{2 x^3} x+39 x-40 e^{2 x^3}+5 e^{4 x^3}+80}dx+4 x^3+\log (x+5)\)

Input:

Int[(275 + 128*x + 15*x^2 + E^(4*x^3)*(5 + 300*x^2 + 60*x^3) + E^(2*x^3)*( 
-90 - 20*x - 1200*x^2 - 540*x^3 - 60*x^4))/(400 + 275*x + 64*x^2 + 5*x^3 + 
 E^(4*x^3)*(25 + 5*x) + E^(2*x^3)*(-200 - 90*x - 10*x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32

method	result	size
risch	\(\ln \left (5+x \right )+\ln \left ({\mathrm e}^{4 x^{3}}+\left (-2 x -8\right ) {\mathrm e}^{2 x^{3}}+x^{2}+\frac {39 x}{5}+16\right )\)	\(33\)
parallelrisch	\(\ln \left (5+x \right )+\ln \left ({\mathrm e}^{4 x^{3}}-2 \,{\mathrm e}^{2 x^{3}} x -8 \,{\mathrm e}^{2 x^{3}}+x^{2}+\frac {39 x}{5}+16\right )\)	\(38\)
norman	\(\ln \left (5+x \right )+\ln \left (5 \,{\mathrm e}^{4 x^{3}}-10 \,{\mathrm e}^{2 x^{3}} x +5 x^{2}-40 \,{\mathrm e}^{2 x^{3}}+39 x +80\right )\)	\(42\)

Input:

int(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp( 
x^3)^2+15*x^2+128*x+275)/((25+5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^ 
2+5*x^3+64*x^2+275*x+400),x,method=_RETURNVERBOSE)
 

Output:

ln(5+x)+ln(exp(4*x^3)+(-2*x-8)*exp(2*x^3)+x^2+39/5*x+16)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log \left (5 \, x^{2} - 10 \, {\left (x + 4\right )} e^{\left (2 \, x^{3}\right )} + 39 \, x + 5 \, e^{\left (4 \, x^{3}\right )} + 80\right ) + \log \left (x + 5\right ) \] Input:

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90 
)*exp(x^3)^2+15*x^2+128*x+275)/((25+5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp 
(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="fricas")
 

Output:

log(5*x^2 - 10*(x + 4)*e^(2*x^3) + 39*x + 5*e^(4*x^3) + 80) + log(x + 5)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log {\left (x + 5 \right )} + \log {\left (x^{2} + \frac {39 x}{5} + \left (- 2 x - 8\right ) e^{2 x^{3}} + e^{4 x^{3}} + 16 \right )} \] Input:

integrate(((60*x**3+300*x**2+5)*exp(x**3)**4+(-60*x**4-540*x**3-1200*x**2- 
20*x-90)*exp(x**3)**2+15*x**2+128*x+275)/((25+5*x)*exp(x**3)**4+(-10*x**2- 
90*x-200)*exp(x**3)**2+5*x**3+64*x**2+275*x+400),x)
 

Output:

log(x + 5) + log(x**2 + 39*x/5 + (-2*x - 8)*exp(2*x**3) + exp(4*x**3) + 16 
)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log \left (x^{2} - 2 \, {\left (x + 4\right )} e^{\left (2 \, x^{3}\right )} + \frac {39}{5} \, x + e^{\left (4 \, x^{3}\right )} + 16\right ) + \log \left (x + 5\right ) \] Input:

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90 
)*exp(x^3)^2+15*x^2+128*x+275)/((25+5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp 
(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="maxima")
 

Output:

log(x^2 - 2*(x + 4)*e^(2*x^3) + 39/5*x + e^(4*x^3) + 16) + log(x + 5)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\log \left (5 \, x^{2} - 10 \, x e^{\left (2 \, x^{3}\right )} + 39 \, x + 5 \, e^{\left (4 \, x^{3}\right )} - 40 \, e^{\left (2 \, x^{3}\right )} + 80\right ) + \log \left (x + 5\right ) \] Input:

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90 
)*exp(x^3)^2+15*x^2+128*x+275)/((25+5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp 
(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="giac")
 

Output:

log(5*x^2 - 10*x*e^(2*x^3) + 39*x + 5*e^(4*x^3) - 40*e^(2*x^3) + 80) + log 
(x + 5)
 

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=\ln \left (x+5\right )+\ln \left (\frac {39\,x}{5}-8\,{\mathrm {e}}^{2\,x^3}+{\mathrm {e}}^{4\,x^3}-2\,x\,{\mathrm {e}}^{2\,x^3}+x^2+16\right ) \] Input:

int((128*x + exp(4*x^3)*(300*x^2 + 60*x^3 + 5) - exp(2*x^3)*(20*x + 1200*x 
^2 + 540*x^3 + 60*x^4 + 90) + 15*x^2 + 275)/(275*x + exp(4*x^3)*(5*x + 25) 
 - exp(2*x^3)*(90*x + 10*x^2 + 200) + 64*x^2 + 5*x^3 + 400),x)
 

Output:

log(x + 5) + log((39*x)/5 - 8*exp(2*x^3) + exp(4*x^3) - 2*x*exp(2*x^3) + x 
^2 + 16)
 

Reduce [F]

\[ \int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx=5 \left (\int \frac {e^{4 x^{3}}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )-90 \left (\int \frac {e^{2 x^{3}}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )+15 \left (\int \frac {x^{2}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )+60 \left (\int \frac {e^{4 x^{3}} x^{3}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )+300 \left (\int \frac {e^{4 x^{3}} x^{2}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )-60 \left (\int \frac {e^{2 x^{3}} x^{4}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )-540 \left (\int \frac {e^{2 x^{3}} x^{3}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )-1200 \left (\int \frac {e^{2 x^{3}} x^{2}}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )-20 \left (\int \frac {e^{2 x^{3}} x}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )+128 \left (\int \frac {x}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right )+275 \left (\int \frac {1}{5 e^{4 x^{3}} x +25 e^{4 x^{3}}-10 e^{2 x^{3}} x^{2}-90 e^{2 x^{3}} x -200 e^{2 x^{3}}+5 x^{3}+64 x^{2}+275 x +400}d x \right ) \] Input:

int(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp( 
x^3)^2+15*x^2+128*x+275)/((25+5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^ 
2+5*x^3+64*x^2+275*x+400),x)
 

Output:

5*int(e**(4*x**3)/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 
- 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 400),x) 
- 90*int(e**(2*x**3)/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x* 
*2 - 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 400), 
x) + 15*int(x**2/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 - 
 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 400),x) + 
 60*int((e**(4*x**3)*x**3)/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x* 
*3)*x**2 - 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 
 400),x) + 300*int((e**(4*x**3)*x**2)/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 
10*e**(2*x**3)*x**2 - 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x** 
2 + 275*x + 400),x) - 60*int((e**(2*x**3)*x**4)/(5*e**(4*x**3)*x + 25*e**( 
4*x**3) - 10*e**(2*x**3)*x**2 - 90*e**(2*x**3)*x - 200*e**(2*x**3) + 5*x** 
3 + 64*x**2 + 275*x + 400),x) - 540*int((e**(2*x**3)*x**3)/(5*e**(4*x**3)* 
x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 - 90*e**(2*x**3)*x - 200*e**(2*x* 
*3) + 5*x**3 + 64*x**2 + 275*x + 400),x) - 1200*int((e**(2*x**3)*x**2)/(5* 
e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 - 90*e**(2*x**3)*x - 
200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 400),x) - 20*int((e**(2*x**3) 
*x)/(5*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 - 90*e**(2*x** 
3)*x - 200*e**(2*x**3) + 5*x**3 + 64*x**2 + 275*x + 400),x) + 128*int(x/(5 
*e**(4*x**3)*x + 25*e**(4*x**3) - 10*e**(2*x**3)*x**2 - 90*e**(2*x**3)*...