Integrand size = 85, antiderivative size = 31 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {e^{3+\frac {1}{2} (-3-x)+x}}{\left (\frac {e^x}{2 x^2}+2 x\right )^2} \] Output:
exp(3/2-1/2*x)*exp(x)/(2*x+1/2*exp(x)/x^2)^2
Time = 1.60 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {4 e^{\frac {3}{2}+\frac {x}{2}} x^4}{\left (e^x+4 x^3\right )^2} \] Input:
Integrate[(E^((3 - x)/2 + 2*x)*(16*x^3 - 6*x^4) + E^((3 - x)/2 + x)*(-32*x ^6 + 8*x^7))/(E^(3*x) + 12*E^(2*x)*x^3 + 48*E^x*x^6 + 64*x^9),x]
Output:
(4*E^(3/2 + x/2)*x^4)/(E^x + 4*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.
\(\displaystyle \int \frac {e^{\frac {3-x}{2}+x} \left (8 x^7-32 x^6\right )+e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )}{64 x^9+48 e^x x^6+12 e^{2 x} x^3+e^{3 x}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 e^{\frac {x}{2}+\frac {3}{2}} x^3 \left (4 (x-4) x^3-e^x (3 x-8)\right )}{\left (4 x^3+e^x\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {e^{\frac {x}{2}+\frac {3}{2}} x^3 \left (e^x (8-3 x)-4 (4-x) x^3\right )}{\left (4 x^3+e^x\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {16 e^{\frac {x}{2}+\frac {3}{2}} (x-3) x^6}{\left (4 x^3+e^x\right )^3}-\frac {e^{\frac {x}{2}+\frac {3}{2}} x^3 (3 x-8)}{\left (4 x^3+e^x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (8 \int \frac {e^{\frac {x}{2}+\frac {3}{2}} x^3}{\left (4 x^3+e^x\right )^2}dx+16 \int \frac {e^{\frac {x}{2}+\frac {3}{2}} x^7}{\left (4 x^3+e^x\right )^3}dx-48 \int \frac {e^{\frac {x}{2}+\frac {3}{2}} x^6}{\left (4 x^3+e^x\right )^3}dx-3 \int \frac {e^{\frac {x}{2}+\frac {3}{2}} x^4}{\left (4 x^3+e^x\right )^2}dx\right )\) |
Input:
Int[(E^((3 - x)/2 + 2*x)*(16*x^3 - 6*x^4) + E^((3 - x)/2 + x)*(-32*x^6 + 8 *x^7))/(E^(3*x) + 12*E^(2*x)*x^3 + 48*E^x*x^6 + 64*x^9),x]
Output:
$Aborted
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71
method | result | size |
risch | \(\frac {4 x^{4} {\mathrm e}^{\frac {3}{2}+\frac {x}{2}}}{\left (4 x^{3}+{\mathrm e}^{x}\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{\frac {3}{2}-\frac {x}{2}} x^{4} {\mathrm e}^{x}}{16 x^{6}+8 \,{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x}}\) | \(33\) |
Input:
int(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x) *exp(x))/(exp(x)^3+12*exp(x)^2*x^3+48*x^6*exp(x)+64*x^9),x,method=_RETURNV ERBOSE)
Output:
4/(4*x^3+exp(x))^2*x^4*exp(3/2+1/2*x)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {4 \, x^{4} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )}}{16 \, x^{6} e^{6} + 8 \, x^{3} e^{\left (x + 6\right )} + e^{\left (2 \, x + 6\right )}} \] Input:
integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2- 1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2*x^3+48*x^6*exp(x)+64*x^9),x, algorith m="fricas")
Output:
4*x^4*e^(1/2*x + 15/2)/(16*x^6*e^6 + 8*x^3*e^(x + 6) + e^(2*x + 6))
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {4 x^{4} e^{\frac {3}{2}} \sqrt {e^{x}}}{16 x^{6} + 8 x^{3} e^{x} + e^{2 x}} \] Input:
integrate(((-6*x**4+16*x**3)*exp(3/2-1/2*x)*exp(x)**2+(8*x**7-32*x**6)*exp (3/2-1/2*x)*exp(x))/(exp(x)**3+12*exp(x)**2*x**3+48*x**6*exp(x)+64*x**9),x )
Output:
4*x**4*exp(3/2)*sqrt(exp(x))/(16*x**6 + 8*x**3*exp(x) + exp(2*x))
Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {4 \, x^{4} e^{\left (\frac {1}{2} \, x + \frac {3}{2}\right )}}{16 \, x^{6} + 8 \, x^{3} e^{x} + e^{\left (2 \, x\right )}} \] Input:
integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2- 1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2*x^3+48*x^6*exp(x)+64*x^9),x, algorith m="maxima")
Output:
4*x^4*e^(1/2*x + 3/2)/(16*x^6 + 8*x^3*e^x + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.23 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {8 \, {\left ({\left (x + 3\right )}^{4} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )} - 12 \, {\left (x + 3\right )}^{3} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )} + 54 \, {\left (x + 3\right )}^{2} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )} - 108 \, {\left (x + 3\right )} e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )} + 81 \, e^{\left (\frac {1}{2} \, x + \frac {15}{2}\right )}\right )}}{16 \, {\left (x + 3\right )}^{6} e^{6} - 288 \, {\left (x + 3\right )}^{5} e^{6} + 2160 \, {\left (x + 3\right )}^{4} e^{6} - 8640 \, {\left (x + 3\right )}^{3} e^{6} + 8 \, {\left (x + 3\right )}^{3} e^{\left (x + 6\right )} + 19440 \, {\left (x + 3\right )}^{2} e^{6} - 72 \, {\left (x + 3\right )}^{2} e^{\left (x + 6\right )} - 23328 \, {\left (x + 3\right )} e^{6} + 216 \, {\left (x + 3\right )} e^{\left (x + 6\right )} + 11664 \, e^{6} + e^{\left (2 \, x + 6\right )} - 216 \, e^{\left (x + 6\right )}} \] Input:
integrate(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2- 1/2*x)*exp(x))/(exp(x)^3+12*exp(x)^2*x^3+48*x^6*exp(x)+64*x^9),x, algorith m="giac")
Output:
8*((x + 3)^4*e^(1/2*x + 15/2) - 12*(x + 3)^3*e^(1/2*x + 15/2) + 54*(x + 3) ^2*e^(1/2*x + 15/2) - 108*(x + 3)*e^(1/2*x + 15/2) + 81*e^(1/2*x + 15/2))/ (16*(x + 3)^6*e^6 - 288*(x + 3)^5*e^6 + 2160*(x + 3)^4*e^6 - 8640*(x + 3)^ 3*e^6 + 8*(x + 3)^3*e^(x + 6) + 19440*(x + 3)^2*e^6 - 72*(x + 3)^2*e^(x + 6) - 23328*(x + 3)*e^6 + 216*(x + 3)*e^(x + 6) + 11664*e^6 + e^(2*x + 6) - 216*e^(x + 6))
Time = 3.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=\frac {4\,x^4\,{\mathrm {e}}^{x/2}\,{\mathrm {e}}^{3/2}}{{\mathrm {e}}^{2\,x}+8\,x^3\,{\mathrm {e}}^x+16\,x^6} \] Input:
int((exp(2*x)*exp(3/2 - x/2)*(16*x^3 - 6*x^4) - exp(3/2 - x/2)*exp(x)*(32* x^6 - 8*x^7))/(exp(3*x) + 48*x^6*exp(x) + 12*x^3*exp(2*x) + 64*x^9),x)
Output:
(4*x^4*exp(x/2)*exp(3/2))/(exp(2*x) + 8*x^3*exp(x) + 16*x^6)
\[ \int \frac {e^{\frac {3-x}{2}+2 x} \left (16 x^3-6 x^4\right )+e^{\frac {3-x}{2}+x} \left (-32 x^6+8 x^7\right )}{e^{3 x}+12 e^{2 x} x^3+48 e^x x^6+64 x^9} \, dx=2 \sqrt {e}\, e \left (-3 \left (\int \frac {e^{2 x} x^{4}}{e^{\frac {7 x}{2}}+12 e^{\frac {5 x}{2}} x^{3}+48 e^{\frac {3 x}{2}} x^{6}+64 e^{\frac {x}{2}} x^{9}}d x \right )+8 \left (\int \frac {e^{2 x} x^{3}}{e^{\frac {7 x}{2}}+12 e^{\frac {5 x}{2}} x^{3}+48 e^{\frac {3 x}{2}} x^{6}+64 e^{\frac {x}{2}} x^{9}}d x \right )+4 \left (\int \frac {e^{x} x^{7}}{e^{\frac {7 x}{2}}+12 e^{\frac {5 x}{2}} x^{3}+48 e^{\frac {3 x}{2}} x^{6}+64 e^{\frac {x}{2}} x^{9}}d x \right )-16 \left (\int \frac {e^{x} x^{6}}{e^{\frac {7 x}{2}}+12 e^{\frac {5 x}{2}} x^{3}+48 e^{\frac {3 x}{2}} x^{6}+64 e^{\frac {x}{2}} x^{9}}d x \right )\right ) \] Input:
int(((-6*x^4+16*x^3)*exp(3/2-1/2*x)*exp(x)^2+(8*x^7-32*x^6)*exp(3/2-1/2*x) *exp(x))/(exp(x)^3+12*exp(x)^2*x^3+48*x^6*exp(x)+64*x^9),x)
Output:
2*sqrt(e)*e*( - 3*int((e**(2*x)*x**4)/(e**((7*x)/2) + 12*e**((5*x)/2)*x**3 + 48*e**((3*x)/2)*x**6 + 64*e**(x/2)*x**9),x) + 8*int((e**(2*x)*x**3)/(e* *((7*x)/2) + 12*e**((5*x)/2)*x**3 + 48*e**((3*x)/2)*x**6 + 64*e**(x/2)*x** 9),x) + 4*int((e**x*x**7)/(e**((7*x)/2) + 12*e**((5*x)/2)*x**3 + 48*e**((3 *x)/2)*x**6 + 64*e**(x/2)*x**9),x) - 16*int((e**x*x**6)/(e**((7*x)/2) + 12 *e**((5*x)/2)*x**3 + 48*e**((3*x)/2)*x**6 + 64*e**(x/2)*x**9),x))