Integrand size = 94, antiderivative size = 33 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {1}{3} \left (x^2+\frac {3 x^2 (4+x)}{16-e^x-x-5 x^2}\right ) \] Output:
1/(16-exp(x)-5*x^2-x)*x^2*(4+x)+1/3*x^2
Time = 2.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {1}{3} x^2 \left (1-\frac {3 (4+x)}{-16+e^x+x+5 x^2}\right ) \] Input:
Integrate[(896*x + 2*E^(2*x)*x + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + E^x*( -88*x + 7*x^2 + 23*x^3))/(768 + 3*E^(2*x) - 96*x - 477*x^2 + 30*x^3 + 75*x ^4 + E^x*(-96 + 6*x + 30*x^2)),x]
Output:
(x^2*(1 - (3*(4 + x))/(-16 + E^x + x + 5*x^2)))/3
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.
\(\displaystyle \int \frac {50 x^5+5 x^4-324 x^3+68 x^2+e^x \left (23 x^3+7 x^2-88 x\right )+2 e^{2 x} x+896 x}{75 x^4+30 x^3-477 x^2+e^x \left (30 x^2+6 x-96\right )-96 x+3 e^{2 x}+768} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {50 x^5+5 x^4-324 x^3+68 x^2+e^x \left (23 x^3+7 x^2-88 x\right )+2 e^{2 x} x+896 x}{3 \left (-5 x^2-x-e^x+16\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {50 x^5+5 x^4-324 x^3+68 x^2+2 e^{2 x} x+896 x-e^x \left (-23 x^3-7 x^2+88 x\right )}{\left (-5 x^2-x-e^x+16\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{3} \int \left (-\frac {3 \left (5 x^3+11 x^2-53 x-68\right ) x^2}{\left (5 x^2+x+e^x-16\right )^2}+\frac {3 \left (x^2+x-8\right ) x}{5 x^2+x+e^x-16}+2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (204 \int \frac {x^2}{\left (5 x^2+x+e^x-16\right )^2}dx-24 \int \frac {x}{5 x^2+x+e^x-16}dx+3 \int \frac {x^2}{5 x^2+x+e^x-16}dx-15 \int \frac {x^5}{\left (5 x^2+x+e^x-16\right )^2}dx-33 \int \frac {x^4}{\left (5 x^2+x+e^x-16\right )^2}dx+159 \int \frac {x^3}{\left (5 x^2+x+e^x-16\right )^2}dx+3 \int \frac {x^3}{5 x^2+x+e^x-16}dx+x^2\right )\) |
Input:
Int[(896*x + 2*E^(2*x)*x + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + E^x*(-88*x + 7*x^2 + 23*x^3))/(768 + 3*E^(2*x) - 96*x - 477*x^2 + 30*x^3 + 75*x^4 + E ^x*(-96 + 6*x + 30*x^2)),x]
Output:
$Aborted
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {x^{2}}{3}-\frac {x^{2} \left (4+x \right )}{5 x^{2}+{\mathrm e}^{x}+x -16}\) | \(27\) |
parallelrisch | \(\frac {5 x^{4}-2 x^{3}+{\mathrm e}^{x} x^{2}-28 x^{2}}{15 x^{2}+3 \,{\mathrm e}^{x}+3 x -48}\) | \(37\) |
norman | \(\frac {\frac {28 x}{15}+\frac {28 \,{\mathrm e}^{x}}{15}-\frac {2 x^{3}}{3}+\frac {5 x^{4}}{3}+\frac {{\mathrm e}^{x} x^{2}}{3}-\frac {448}{15}}{5 x^{2}+{\mathrm e}^{x}+x -16}\) | \(40\) |
Input:
int((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+8 96*x)/(3*exp(x)^2+(30*x^2+6*x-96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+768),x ,method=_RETURNVERBOSE)
Output:
1/3*x^2-x^2*(4+x)/(5*x^2+exp(x)+x-16)
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \] Input:
integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68 *x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+ 768),x, algorithm="fricas")
Output:
1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {x^{2}}{3} + \frac {- x^{3} - 4 x^{2}}{5 x^{2} + x + e^{x} - 16} \] Input:
integrate((2*x*exp(x)**2+(23*x**3+7*x**2-88*x)*exp(x)+50*x**5+5*x**4-324*x **3+68*x**2+896*x)/(3*exp(x)**2+(30*x**2+6*x-96)*exp(x)+75*x**4+30*x**3-47 7*x**2-96*x+768),x)
Output:
x**2/3 + (-x**3 - 4*x**2)/(5*x**2 + x + exp(x) - 16)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \] Input:
integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68 *x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+ 768),x, algorithm="maxima")
Output:
1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {5 \, x^{4} - 2 \, x^{3} + x^{2} e^{x} - 28 \, x^{2}}{3 \, {\left (5 \, x^{2} + x + e^{x} - 16\right )}} \] Input:
integrate((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68 *x^2+896*x)/(3*exp(x)^2+(30*x^2+6*x-96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+ 768),x, algorithm="giac")
Output:
1/3*(5*x^4 - 2*x^3 + x^2*e^x - 28*x^2)/(5*x^2 + x + e^x - 16)
Time = 0.68 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=-\frac {x^2\,\left (2\,x-{\mathrm {e}}^x-5\,x^2+28\right )}{3\,\left (x+{\mathrm {e}}^x+5\,x^2-16\right )} \] Input:
int((896*x + 2*x*exp(2*x) + 68*x^2 - 324*x^3 + 5*x^4 + 50*x^5 + exp(x)*(7* x^2 - 88*x + 23*x^3))/(3*exp(2*x) - 96*x + exp(x)*(6*x + 30*x^2 - 96) - 47 7*x^2 + 30*x^3 + 75*x^4 + 768),x)
Output:
-(x^2*(2*x - exp(x) - 5*x^2 + 28))/(3*(x + exp(x) + 5*x^2 - 16))
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {896 x+2 e^{2 x} x+68 x^2-324 x^3+5 x^4+50 x^5+e^x \left (-88 x+7 x^2+23 x^3\right )}{768+3 e^{2 x}-96 x-477 x^2+30 x^3+75 x^4+e^x \left (-96+6 x+30 x^2\right )} \, dx=\frac {5 e^{x} x^{2}+28 e^{x}+25 x^{4}-10 x^{3}+28 x -448}{15 e^{x}+75 x^{2}+15 x -240} \] Input:
int((2*x*exp(x)^2+(23*x^3+7*x^2-88*x)*exp(x)+50*x^5+5*x^4-324*x^3+68*x^2+8 96*x)/(3*exp(x)^2+(30*x^2+6*x-96)*exp(x)+75*x^4+30*x^3-477*x^2-96*x+768),x )
Output:
(5*e**x*x**2 + 28*e**x + 25*x**4 - 10*x**3 + 28*x - 448)/(15*(e**x + 5*x** 2 + x - 16))