Integrand size = 94, antiderivative size = 35 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=e^x+\frac {1}{4} \left (1-\frac {2 \left (-3-\frac {4}{x}+2 x\right )}{x \left (-3-x+x^3\right )}\right ) \] Output:
1/4-1/2/(x^3-x-3)/x*(2*x-4/x-3)+exp(x)
Time = 1.85 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=\frac {1}{2} \left (2 e^x+\frac {4+3 x-2 x^2}{x^2 \left (-3-x+x^3\right )}\right ) \] Input:
Integrate[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12* x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9))/(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]
Output:
(2*E^x + (4 + 3*x - 2*x^2)/(x^2*(-3 - x + x^3)))/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.
\(\displaystyle \int \frac {6 x^5-12 x^4-22 x^3+6 x^2+e^x \left (2 x^9-4 x^7-12 x^6+2 x^5+12 x^4+18 x^3\right )+21 x+24}{2 x^9-4 x^7-12 x^6+2 x^5+12 x^4+18 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {6 x^5-12 x^4-22 x^3+6 x^2+e^x \left (2 x^9-4 x^7-12 x^6+2 x^5+12 x^4+18 x^3\right )+21 x+24}{x^3 \left (2 x^6-4 x^4-12 x^3+2 x^2+12 x+18\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {6 x^5-12 x^4-22 x^3+6 x^2+e^x \left (2 x^9-4 x^7-12 x^6+2 x^5+12 x^4+18 x^3\right )+21 x+24}{2 x^3 \left (x^3-x-3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {6 x^5-12 x^4-22 x^3+6 x^2+21 x+2 e^x \left (x^9-2 x^7-6 x^6+x^5+6 x^4+9 x^3\right )+24}{x^3 \left (-x^3+x+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {6 x^2}{\left (x^3-x-3\right )^2}-\frac {12 x}{\left (x^3-x-3\right )^2}+2 e^x-\frac {22}{\left (x^3-x-3\right )^2}+\frac {6}{\left (x^3-x-3\right )^2 x}+\frac {21}{\left (x^3-x-3\right )^2 x^2}+\frac {24}{\left (x^3-x-3\right )^2 x^3}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {1}{2} \int \left (\frac {6 x^2}{\left (x^3-x-3\right )^2}-\frac {12 x}{\left (x^3-x-3\right )^2}+2 e^x-\frac {22}{\left (x^3-x-3\right )^2}+\frac {6}{\left (x^3-x-3\right )^2 x}+\frac {21}{\left (x^3-x-3\right )^2 x^2}+\frac {24}{\left (x^3-x-3\right )^2 x^3}\right )dx\) |
Input:
Int[(24 + 21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + E^x*(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9))/(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4*x^7 + 2*x^9),x]
Output:
$Aborted
Time = 0.77 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {-x^{2}+\frac {3}{2} x +2}{x^{2} \left (x^{3}-x -3\right )}+{\mathrm e}^{x}\) | \(28\) |
parts | \(-\frac {2}{3 x^{2}}-\frac {5}{18 x}-\frac {-5 x^{2}-12 x +23}{18 \left (x^{3}-x -3\right )}+{\mathrm e}^{x}\) | \(36\) |
norman | \(\frac {2+x^{5} {\mathrm e}^{x}-x^{2}+\frac {3 x}{2}-3 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x^{3}}{x^{2} \left (x^{3}-x -3\right )}\) | \(45\) |
parallelrisch | \(\frac {2 x^{5} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{3}-6 \,{\mathrm e}^{x} x^{2}-2 x^{2}+3 x +4}{2 x^{2} \left (x^{3}-x -3\right )}\) | \(47\) |
orering | \(-\frac {\left (2 x^{10}-3 x^{9}-32 x^{8}+54 x^{7}+160 x^{6}-57 x^{5}-180 x^{4}-249 x^{3}+66 x^{2}+246 x +216\right ) \left (\left (2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}\right ) {\mathrm e}^{x}+6 x^{5}-12 x^{4}-22 x^{3}+6 x^{2}+21 x +24\right )}{\left (6 x^{9}+12 x^{8}-88 x^{7}-132 x^{6}+169 x^{5}+260 x^{4}+159 x^{3}-189 x^{2}-318 x -216\right ) \left (2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}\right )}+\frac {\left (2 x^{6}+3 x^{5}-18 x^{4}-25 x^{3}+19 x^{2}+33 x +24\right ) x \left (x^{3}-x -3\right ) \left (\frac {\left (18 x^{8}-28 x^{6}-72 x^{5}+10 x^{4}+48 x^{3}+54 x^{2}\right ) {\mathrm e}^{x}+\left (2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}\right ) {\mathrm e}^{x}+30 x^{4}-48 x^{3}-66 x^{2}+12 x +21}{2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}}-\frac {\left (\left (2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}\right ) {\mathrm e}^{x}+6 x^{5}-12 x^{4}-22 x^{3}+6 x^{2}+21 x +24\right ) \left (18 x^{8}-28 x^{6}-72 x^{5}+10 x^{4}+48 x^{3}+54 x^{2}\right )}{\left (2 x^{9}-4 x^{7}-12 x^{6}+2 x^{5}+12 x^{4}+18 x^{3}\right )^{2}}\right )}{6 x^{9}+12 x^{8}-88 x^{7}-132 x^{6}+169 x^{5}+260 x^{4}+159 x^{3}-189 x^{2}-318 x -216}\) | \(528\) |
default | \(\text {Expression too large to display}\) | \(736\) |
Input:
int(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6 *x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3),x,method=_RETURNVER BOSE)
Output:
(-x^2+3/2*x+2)/x^2/(x^3-x-3)+exp(x)
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=-\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \] Input:
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22 *x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3),x, algorithm= "fricas")
Output:
-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=\frac {- 2 x^{2} + 3 x + 4}{2 x^{5} - 2 x^{3} - 6 x^{2}} + e^{x} \] Input:
integrate(((2*x**9-4*x**7-12*x**6+2*x**5+12*x**4+18*x**3)*exp(x)+6*x**5-12 *x**4-22*x**3+6*x**2+21*x+24)/(2*x**9-4*x**7-12*x**6+2*x**5+12*x**4+18*x** 3),x)
Output:
(-2*x**2 + 3*x + 4)/(2*x**5 - 2*x**3 - 6*x**2) + exp(x)
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=-\frac {2 \, x^{2} - 2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )} e^{x} - 3 \, x - 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \] Input:
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22 *x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3),x, algorithm= "maxima")
Output:
-1/2*(2*x^2 - 2*(x^5 - x^3 - 3*x^2)*e^x - 3*x - 4)/(x^5 - x^3 - 3*x^2)
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=\frac {2 \, x^{5} e^{x} - 2 \, x^{3} e^{x} - 6 \, x^{2} e^{x} - 2 \, x^{2} + 3 \, x + 4}{2 \, {\left (x^{5} - x^{3} - 3 \, x^{2}\right )}} \] Input:
integrate(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22 *x^3+6*x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3),x, algorithm= "giac")
Output:
1/2*(2*x^5*e^x - 2*x^3*e^x - 6*x^2*e^x - 2*x^2 + 3*x + 4)/(x^5 - x^3 - 3*x ^2)
Time = 2.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx={\mathrm {e}}^x-\frac {-x^2+\frac {3\,x}{2}+2}{x^2\,\left (-x^3+x+3\right )} \] Input:
int((21*x + 6*x^2 - 22*x^3 - 12*x^4 + 6*x^5 + exp(x)*(18*x^3 + 12*x^4 + 2* x^5 - 12*x^6 - 4*x^7 + 2*x^9) + 24)/(18*x^3 + 12*x^4 + 2*x^5 - 12*x^6 - 4* x^7 + 2*x^9),x)
Output:
exp(x) - ((3*x)/2 - x^2 + 2)/(x^2*(x - x^3 + 3))
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {24+21 x+6 x^2-22 x^3-12 x^4+6 x^5+e^x \left (18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9\right )}{18 x^3+12 x^4+2 x^5-12 x^6-4 x^7+2 x^9} \, dx=\frac {6 e^{x} x^{5}-6 e^{x} x^{3}-18 e^{x} x^{2}-2 x^{5}+2 x^{3}+9 x +12}{6 x^{2} \left (x^{3}-x -3\right )} \] Input:
int(((2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3)*exp(x)+6*x^5-12*x^4-22*x^3+6 *x^2+21*x+24)/(2*x^9-4*x^7-12*x^6+2*x^5+12*x^4+18*x^3),x)
Output:
(6*e**x*x**5 - 6*e**x*x**3 - 18*e**x*x**2 - 2*x**5 + 2*x**3 + 9*x + 12)/(6 *x**2*(x**3 - x - 3))