Integrand size = 88, antiderivative size = 26 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=e^{-\frac {2 e}{5}+\frac {-2-\frac {5}{x^3}+x}{5+x}}+x+\log (x) \] Output:
ln(x)+exp((x-2-5/x^3)/(5+x)-2/5*exp(1))+x
Time = 5.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=e^{-\frac {25+10 (1+e) x^3+(-5+2 e) x^4}{5 x^3 (5+x)}}+x+\log (x) \] Input:
Integrate[(25*x^3 + 35*x^4 + 11*x^5 + x^6 + E^((-25 - 10*x^3 + 5*x^4 + E*( -10*x^3 - 2*x^4))/(25*x^3 + 5*x^4))*(75 + 20*x + 7*x^4))/(25*x^4 + 10*x^5 + x^6),x]
Output:
E^(-1/5*(25 + 10*(1 + E)*x^3 + (-5 + 2*E)*x^4)/(x^3*(5 + x))) + x + Log[x]
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.
\(\displaystyle \int \frac {\left (7 x^4+20 x+75\right ) \exp \left (\frac {5 x^4-10 x^3+e \left (-2 x^4-10 x^3\right )-25}{5 x^4+25 x^3}\right )+x^6+11 x^5+35 x^4+25 x^3}{x^6+10 x^5+25 x^4} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (7 x^4+20 x+75\right ) \exp \left (\frac {5 x^4-10 x^3+e \left (-2 x^4-10 x^3\right )-25}{5 x^4+25 x^3}\right )+x^6+11 x^5+35 x^4+25 x^3}{x^4 \left (x^2+10 x+25\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (7 x^4+20 x+75\right ) \exp \left (\frac {5 x^4-10 x^3+e \left (-2 x^4-10 x^3\right )-25}{5 x^4+25 x^3}\right )+x^6+11 x^5+35 x^4+25 x^3}{x^4 (x+5)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (7 x^4+20 x+75\right ) \exp \left (\frac {(5-2 e) x^4-10 (1+e) x^3-25}{5 x^3 (x+5)}\right )}{(x+5)^2 x^4}+\frac {x^2}{(x+5)^2}+\frac {11 x}{(x+5)^2}+\frac {35}{(x+5)^2}+\frac {25}{(x+5)^2 x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {\exp \left (\frac {(5-2 e) x^4-10 (1+e) x^3-25}{5 x^3 (x+5)}\right )}{x^4}dx-\frac {2}{5} \int \frac {\exp \left (\frac {(5-2 e) x^4-10 (1+e) x^3-25}{5 x^3 (x+5)}\right )}{x^3}dx+\frac {174}{25} \int \frac {\exp \left (\frac {(5-2 e) x^4-10 (1+e) x^3-25}{5 x^3 (x+5)}\right )}{(x+5)^2}dx+\frac {1}{25} \int \frac {\exp \left (\frac {(5-2 e) x^4-10 (1+e) x^3-25}{5 x^3 (x+5)}\right )}{x^2}dx+x+\log (x)\) |
Input:
Int[(25*x^3 + 35*x^4 + 11*x^5 + x^6 + E^((-25 - 10*x^3 + 5*x^4 + E*(-10*x^ 3 - 2*x^4))/(25*x^3 + 5*x^4))*(75 + 20*x + 7*x^4))/(25*x^4 + 10*x^5 + x^6) ,x]
Output:
$Aborted
Time = 1.60 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
risch | \(x +\ln \left (x \right )+{\mathrm e}^{-\frac {2 x^{4} {\mathrm e}+10 x^{3} {\mathrm e}-5 x^{4}+10 x^{3}+25}{5 x^{3} \left (5+x \right )}}\) | \(42\) |
parallelrisch | \(x +\ln \left (x \right )+{\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{3} \left (5+x \right )}}-\frac {15}{2}\) | \(43\) |
parts | \(x +\ln \left (x \right )+\frac {x^{4} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}+5 x^{3} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}}{x^{3} \left (5+x \right )}\) | \(106\) |
norman | \(\frac {x^{5}+x^{4} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}-25 x^{3}+5 x^{3} {\mathrm e}^{\frac {\left (-2 x^{4}-10 x^{3}\right ) {\mathrm e}+5 x^{4}-10 x^{3}-25}{5 x^{4}+25 x^{3}}}}{x^{3} \left (5+x \right )}+\ln \left (x \right )\) | \(113\) |
Input:
int(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+2 5*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x^6+10*x^5+25*x^4),x,method=_RETURNVERB OSE)
Output:
x+ln(x)+exp(-1/5*(2*x^4*exp(1)+10*x^3*exp(1)-5*x^4+10*x^3+25)/x^3/(5+x))
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=x + e^{\left (\frac {5 \, x^{4} - 10 \, x^{3} - 2 \, {\left (x^{4} + 5 \, x^{3}\right )} e - 25}{5 \, {\left (x^{4} + 5 \, x^{3}\right )}}\right )} + \log \left (x\right ) \] Input:
integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5 *x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x^6+10*x^5+25*x^4),x, algorithm=" fricas")
Output:
x + e^(1/5*(5*x^4 - 10*x^3 - 2*(x^4 + 5*x^3)*e - 25)/(x^4 + 5*x^3)) + log( x)
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=x + e^{\frac {5 x^{4} - 10 x^{3} + e \left (- 2 x^{4} - 10 x^{3}\right ) - 25}{5 x^{4} + 25 x^{3}}} + \log {\left (x \right )} \] Input:
integrate(((7*x**4+20*x+75)*exp(((-2*x**4-10*x**3)*exp(1)+5*x**4-10*x**3-2 5)/(5*x**4+25*x**3))+x**6+11*x**5+35*x**4+25*x**3)/(x**6+10*x**5+25*x**4), x)
Output:
x + exp((5*x**4 - 10*x**3 + E*(-2*x**4 - 10*x**3) - 25)/(5*x**4 + 25*x**3) ) + log(x)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=x + e^{\left (-\frac {174}{25 \, {\left (x + 5\right )}} - \frac {1}{25 \, x} + \frac {1}{5 \, x^{2}} - \frac {1}{x^{3}} - \frac {2}{5} \, e + 1\right )} + \log \left (x\right ) \] Input:
integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5 *x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x^6+10*x^5+25*x^4),x, algorithm=" maxima")
Output:
x + e^(-174/25/(x + 5) - 1/25/x + 1/5/x^2 - 1/x^3 - 2/5*e + 1) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.31 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=x + e^{\left (-\frac {2 \, x^{4} e}{5 \, {\left (x^{4} + 5 \, x^{3}\right )}} + \frac {x^{4}}{x^{4} + 5 \, x^{3}} - \frac {2 \, x^{3} e}{x^{4} + 5 \, x^{3}} - \frac {2 \, x^{3}}{x^{4} + 5 \, x^{3}} - \frac {5}{x^{4} + 5 \, x^{3}}\right )} + \log \left (x\right ) \] Input:
integrate(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5 *x^4+25*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x^6+10*x^5+25*x^4),x, algorithm=" giac")
Output:
x + e^(-2/5*x^4*e/(x^4 + 5*x^3) + x^4/(x^4 + 5*x^3) - 2*x^3*e/(x^4 + 5*x^3 ) - 2*x^3/(x^4 + 5*x^3) - 5/(x^4 + 5*x^3)) + log(x)
Time = 2.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.88 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=x+\ln \left (x\right )+{\mathrm {e}}^{-\frac {2\,x^4\,\mathrm {e}}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {10\,x^3\,\mathrm {e}}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{\frac {5\,x^4}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {10\,x^3}{5\,x^4+25\,x^3}}\,{\mathrm {e}}^{-\frac {25}{5\,x^4+25\,x^3}} \] Input:
int((exp(-(exp(1)*(10*x^3 + 2*x^4) + 10*x^3 - 5*x^4 + 25)/(25*x^3 + 5*x^4) )*(20*x + 7*x^4 + 75) + 25*x^3 + 35*x^4 + 11*x^5 + x^6)/(25*x^4 + 10*x^5 + x^6),x)
Output:
x + log(x) + exp(-(2*x^4*exp(1))/(25*x^3 + 5*x^4))*exp(-(10*x^3*exp(1))/(2 5*x^3 + 5*x^4))*exp((5*x^4)/(25*x^3 + 5*x^4))*exp(-(10*x^3)/(25*x^3 + 5*x^ 4))*exp(-25/(25*x^3 + 5*x^4))
Time = 0.20 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.42 \[ \int \frac {25 x^3+35 x^4+11 x^5+x^6+e^{\frac {-25-10 x^3+5 x^4+e \left (-10 x^3-2 x^4\right )}{25 x^3+5 x^4}} \left (75+20 x+7 x^4\right )}{25 x^4+10 x^5+x^6} \, dx=\frac {e^{\frac {2 e \,x^{4}+10 e \,x^{3}+35 x^{3}+25}{5 x^{4}+25 x^{3}}} \mathrm {log}\left (x \right )+e^{\frac {2 e \,x^{4}+10 e \,x^{3}+35 x^{3}+25}{5 x^{4}+25 x^{3}}} x +e}{e^{\frac {2 e \,x^{4}+10 e \,x^{3}+35 x^{3}+25}{5 x^{4}+25 x^{3}}}} \] Input:
int(((7*x^4+20*x+75)*exp(((-2*x^4-10*x^3)*exp(1)+5*x^4-10*x^3-25)/(5*x^4+2 5*x^3))+x^6+11*x^5+35*x^4+25*x^3)/(x^6+10*x^5+25*x^4),x)
Output:
(e**((2*e*x**4 + 10*e*x**3 + 35*x**3 + 25)/(5*x**4 + 25*x**3))*log(x) + e* *((2*e*x**4 + 10*e*x**3 + 35*x**3 + 25)/(5*x**4 + 25*x**3))*x + e)/e**((2* e*x**4 + 10*e*x**3 + 35*x**3 + 25)/(5*x**4 + 25*x**3))