Integrand size = 80, antiderivative size = 24 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=2+x-\frac {x}{12+e^{-5+2 x} x-x^2} \] Output:
x-x/(12+exp(2*x+ln(x)-5)-x^2)+2
Time = 3.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=x+\frac {e^5 x}{-e^{2 x} x+e^5 \left (-12+x^2\right )} \] Input:
Integrate[(132 - 25*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 + 2 *x - 2*x^2))/(144 - 24*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 - 2*x^2)),x]
Output:
x + (E^5*x)/(-(E^(2*x)*x) + E^5*(-12 + x^2))
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.
\(\displaystyle \int \frac {x^4+e^{4 x-10} x^2-25 x^2+e^{2 x-5} \left (-2 x^2+2 x+24\right ) x+132}{x^4+e^{4 x-10} x^2-24 x^2+e^{2 x-5} \left (24-2 x^2\right ) x+144} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{10} \left (x^4+e^{4 x-10} x^2-25 x^2+e^{2 x-5} \left (-2 x^2+2 x+24\right ) x+132\right )}{\left (-e^5 x^2+e^{2 x} x+12 e^5\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^{10} \int \frac {x^4+e^{4 x-10} x^2-25 x^2+2 e^{2 x-5} \left (-x^2+x+12\right ) x+132}{\left (-e^5 x^2+e^{2 x} x+12 e^5\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle e^{10} \int \left (-\frac {2 x}{e^5 \left (e^5 x^2-e^{2 x} x-12 e^5\right )}+\frac {2 x^3-x^2-24 x-12}{\left (e^5 x^2-e^{2 x} x-12 e^5\right )^2}+\frac {1}{e^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{10} \left (-12 \int \frac {1}{\left (e^5 x^2-e^{2 x} x-12 e^5\right )^2}dx-24 \int \frac {x}{\left (e^5 x^2-e^{2 x} x-12 e^5\right )^2}dx-\int \frac {x^2}{\left (e^5 x^2-e^{2 x} x-12 e^5\right )^2}dx-\frac {2 \int \frac {x}{e^5 x^2-e^{2 x} x-12 e^5}dx}{e^5}+2 \int \frac {x^3}{\left (e^5 x^2-e^{2 x} x-12 e^5\right )^2}dx+\frac {x}{e^{10}}\right )\) |
Input:
Int[(132 - 25*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 + 2*x - 2 *x^2))/(144 - 24*x^2 + E^(-10 + 4*x)*x^2 + x^4 + E^(-5 + 2*x)*x*(24 - 2*x^ 2)),x]
Output:
$Aborted
Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
risch | \(x +\frac {x}{x^{2}-x \,{\mathrm e}^{-5+2 x}-12}\) | \(21\) |
norman | \(\frac {x^{3}-11 x -x \,{\mathrm e}^{2 x +\ln \left (x \right )-5}}{x^{2}-{\mathrm e}^{2 x +\ln \left (x \right )-5}-12}\) | \(37\) |
parallelrisch | \(\frac {x^{3}-11 x -x \,{\mathrm e}^{2 x +\ln \left (x \right )-5}}{x^{2}-{\mathrm e}^{2 x +\ln \left (x \right )-5}-12}\) | \(37\) |
Input:
int((exp(2*x+ln(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+ln(x)-5)+x^4-25*x^2+132)/( exp(2*x+ln(x)-5)^2+(-2*x^2+24)*exp(2*x+ln(x)-5)+x^4-24*x^2+144),x,method=_ RETURNVERBOSE)
Output:
x+x/(x^2-x*exp(-5+2*x)-12)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=\frac {x^{3} - x e^{\left (2 \, x + \log \left (x\right ) - 5\right )} - 11 \, x}{x^{2} - e^{\left (2 \, x + \log \left (x\right ) - 5\right )} - 12} \] Input:
integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^ 2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+24)*exp(2*x+log(x)-5)+x^4-24*x^2+144), x, algorithm="fricas")
Output:
(x^3 - x*e^(2*x + log(x) - 5) - 11*x)/(x^2 - e^(2*x + log(x) - 5) - 12)
Time = 0.08 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=x - \frac {x}{- x^{2} + x e^{2 x - 5} + 12} \] Input:
integrate((exp(2*x+ln(x)-5)**2+(-2*x**2+2*x+24)*exp(2*x+ln(x)-5)+x**4-25*x **2+132)/(exp(2*x+ln(x)-5)**2+(-2*x**2+24)*exp(2*x+ln(x)-5)+x**4-24*x**2+1 44),x)
Output:
x - x/(-x**2 + x*exp(2*x - 5) + 12)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=\frac {x^{3} e^{5} - x^{2} e^{\left (2 \, x\right )} - 11 \, x e^{5}}{x^{2} e^{5} - x e^{\left (2 \, x\right )} - 12 \, e^{5}} \] Input:
integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^ 2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+24)*exp(2*x+log(x)-5)+x^4-24*x^2+144), x, algorithm="maxima")
Output:
(x^3*e^5 - x^2*e^(2*x) - 11*x*e^5)/(x^2*e^5 - x*e^(2*x) - 12*e^5)
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=\frac {x^{3} e^{5} - x^{2} e^{\left (2 \, x\right )} - 11 \, x e^{5}}{x^{2} e^{5} - x e^{\left (2 \, x\right )} - 12 \, e^{5}} \] Input:
integrate((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^ 2+132)/(exp(2*x+log(x)-5)^2+(-2*x^2+24)*exp(2*x+log(x)-5)+x^4-24*x^2+144), x, algorithm="giac")
Output:
(x^3*e^5 - x^2*e^(2*x) - 11*x*e^5)/(x^2*e^5 - x*e^(2*x) - 12*e^5)
Time = 1.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=x-\frac {x}{x\,{\mathrm {e}}^{2\,x-5}-x^2+12} \] Input:
int((exp(4*x + 2*log(x) - 10) + exp(2*x + log(x) - 5)*(2*x - 2*x^2 + 24) - 25*x^2 + x^4 + 132)/(exp(4*x + 2*log(x) - 10) - exp(2*x + log(x) - 5)*(2* x^2 - 24) - 24*x^2 + x^4 + 144),x)
Output:
x - x/(x*exp(2*x - 5) - x^2 + 12)
\[ \int \frac {132-25 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24+2 x-2 x^2\right )}{144-24 x^2+e^{-10+4 x} x^2+x^4+e^{-5+2 x} x \left (24-2 x^2\right )} \, dx=\left (\int \frac {x^{4}}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{10}-25 \left (\int \frac {x^{2}}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{10}+\int \frac {e^{4 x} x^{2}}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x -2 \left (\int \frac {e^{2 x} x^{3}}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{5}+2 \left (\int \frac {e^{2 x} x^{2}}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{5}+24 \left (\int \frac {e^{2 x} x}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{5}+132 \left (\int \frac {1}{e^{4 x} x^{2}-2 e^{2 x} e^{5} x^{3}+24 e^{2 x} e^{5} x +e^{10} x^{4}-24 e^{10} x^{2}+144 e^{10}}d x \right ) e^{10} \] Input:
int((exp(2*x+log(x)-5)^2+(-2*x^2+2*x+24)*exp(2*x+log(x)-5)+x^4-25*x^2+132) /(exp(2*x+log(x)-5)^2+(-2*x^2+24)*exp(2*x+log(x)-5)+x^4-24*x^2+144),x)
Output:
int(x**4/(e**(4*x)*x**2 - 2*e**(2*x)*e**5*x**3 + 24*e**(2*x)*e**5*x + e**1 0*x**4 - 24*e**10*x**2 + 144*e**10),x)*e**10 - 25*int(x**2/(e**(4*x)*x**2 - 2*e**(2*x)*e**5*x**3 + 24*e**(2*x)*e**5*x + e**10*x**4 - 24*e**10*x**2 + 144*e**10),x)*e**10 + int((e**(4*x)*x**2)/(e**(4*x)*x**2 - 2*e**(2*x)*e** 5*x**3 + 24*e**(2*x)*e**5*x + e**10*x**4 - 24*e**10*x**2 + 144*e**10),x) - 2*int((e**(2*x)*x**3)/(e**(4*x)*x**2 - 2*e**(2*x)*e**5*x**3 + 24*e**(2*x) *e**5*x + e**10*x**4 - 24*e**10*x**2 + 144*e**10),x)*e**5 + 2*int((e**(2*x )*x**2)/(e**(4*x)*x**2 - 2*e**(2*x)*e**5*x**3 + 24*e**(2*x)*e**5*x + e**10 *x**4 - 24*e**10*x**2 + 144*e**10),x)*e**5 + 24*int((e**(2*x)*x)/(e**(4*x) *x**2 - 2*e**(2*x)*e**5*x**3 + 24*e**(2*x)*e**5*x + e**10*x**4 - 24*e**10* x**2 + 144*e**10),x)*e**5 + 132*int(1/(e**(4*x)*x**2 - 2*e**(2*x)*e**5*x** 3 + 24*e**(2*x)*e**5*x + e**10*x**4 - 24*e**10*x**2 + 144*e**10),x)*e**10