Integrand size = 50, antiderivative size = 30 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=4 \left (e^x+x^2 \left (1+e^{4+x (3+x)}+2 x-e^x x\right )\right ) \] Output:
4*(exp(4+(3+x)*x)+2*x-exp(x)*x+1)*x^2+4*exp(x)
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=4 x^2+8 x^3+\frac {4 e^{4+3 x+x^2} x \left (3 x+2 x^2\right )}{3+2 x}-4 e^x \left (-1+x^3\right ) \] Input:
Integrate[8*x + 24*x^2 + E^x*(4 - 12*x^2 - 4*x^3) + E^(4 + 3*x + x^2)*(8*x + 12*x^2 + 8*x^3),x]
Output:
4*x^2 + 8*x^3 + (4*E^(4 + 3*x + x^2)*x*(3*x + 2*x^2))/(3 + 2*x) - 4*E^x*(- 1 + x^3)
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2009}
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.
\(\displaystyle \int \left (24 x^2+e^x \left (-4 x^3-12 x^2+4\right )+e^{x^2+3 x+4} \left (8 x^3+12 x^2+8 x\right )+8 x\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 e^x x^3+8 x^3+4 x^2+\frac {4 e^{x^2+3 x+4} \left (2 x^2+3 x\right ) x}{2 x+3}+4 e^x\) |
Input:
Int[8*x + 24*x^2 + E^x*(4 - 12*x^2 - 4*x^3) + E^(4 + 3*x + x^2)*(8*x + 12* x^2 + 8*x^3),x]
Output:
4*E^x + 4*x^2 + 8*x^3 - 4*E^x*x^3 + (4*E^(4 + 3*x + x^2)*x*(3*x + 2*x^2))/ (3 + 2*x)
Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23
method | result | size |
norman | \(4 x^{2}+8 x^{3}-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 \,{\mathrm e}^{x}\) | \(37\) |
risch | \(4 x^{2}+8 x^{3}-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 \,{\mathrm e}^{x}\) | \(37\) |
parallelrisch | \(4 x^{2}+8 x^{3}-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x^{2}+3 x +4} x^{2}+4 \,{\mathrm e}^{x}\) | \(37\) |
default | \(-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x}+8 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}+3 x}}{2}+\frac {3 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{4}\right )+12 \,{\mathrm e}^{4} \left (\frac {x \,{\mathrm e}^{x^{2}+3 x}}{2}-\frac {3 \,{\mathrm e}^{x^{2}+3 x}}{4}-\frac {7 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{8}\right )+8 \,{\mathrm e}^{4} \left (\frac {x^{2} {\mathrm e}^{x^{2}+3 x}}{2}-\frac {3 x \,{\mathrm e}^{x^{2}+3 x}}{4}+\frac {5 \,{\mathrm e}^{x^{2}+3 x}}{8}+\frac {9 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{16}\right )+4 x^{2}+8 x^{3}\) | \(151\) |
parts | \(-4 \,{\mathrm e}^{x} x^{3}+4 \,{\mathrm e}^{x}+8 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{x^{2}+3 x}}{2}+\frac {3 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{4}\right )+12 \,{\mathrm e}^{4} \left (\frac {x \,{\mathrm e}^{x^{2}+3 x}}{2}-\frac {3 \,{\mathrm e}^{x^{2}+3 x}}{4}-\frac {7 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{8}\right )+8 \,{\mathrm e}^{4} \left (\frac {x^{2} {\mathrm e}^{x^{2}+3 x}}{2}-\frac {3 x \,{\mathrm e}^{x^{2}+3 x}}{4}+\frac {5 \,{\mathrm e}^{x^{2}+3 x}}{8}+\frac {9 i \sqrt {\pi }\, {\mathrm e}^{-\frac {9}{4}} \operatorname {erf}\left (i x +\frac {3}{2} i\right )}{16}\right )+4 x^{2}+8 x^{3}\) | \(151\) |
orering | \(\frac {\left (32 x^{11}+336 x^{10}+1192 x^{9}+2388 x^{8}+2664 x^{7}+1716 x^{6}+1566 x^{5}+2937 x^{4}+3246 x^{3}+2002 x^{2}+390 x -4\right ) \left (\left (8 x^{3}+12 x^{2}+8 x \right ) {\mathrm e}^{x^{2}+3 x +4}+\left (-4 x^{3}-12 x^{2}+4\right ) {\mathrm e}^{x}+24 x^{2}+8 x \right )}{96 x^{10}+800 x^{9}+2520 x^{8}+4368 x^{7}+4196 x^{6}+2148 x^{5}+1464 x^{4}+2096 x^{3}+1440 x^{2}+432 x +24}-\frac {\left (16 x^{11}+80 x^{10}+220 x^{9}+304 x^{8}+216 x^{7}+196 x^{6}+529 x^{5}+774 x^{4}+604 x^{3}+165 x^{2}-32 x -9\right ) \left (\left (24 x^{2}+24 x +8\right ) {\mathrm e}^{x^{2}+3 x +4}+\left (8 x^{3}+12 x^{2}+8 x \right ) \left (3+2 x \right ) {\mathrm e}^{x^{2}+3 x +4}+\left (-12 x^{2}-24 x \right ) {\mathrm e}^{x}+\left (-4 x^{3}-12 x^{2}+4\right ) {\mathrm e}^{x}+48 x +8\right )}{2 \left (24 x^{10}+200 x^{9}+630 x^{8}+1092 x^{7}+1049 x^{6}+537 x^{5}+366 x^{4}+524 x^{3}+360 x^{2}+108 x +6\right )}+\frac {\left (16 x^{10}+48 x^{9}+44 x^{8}+40 x^{6}+116 x^{5}+177 x^{4}+130 x^{3}-10 x^{2}-10 x -2\right ) \left (\left (48 x +24\right ) {\mathrm e}^{x^{2}+3 x +4}+2 \left (24 x^{2}+24 x +8\right ) \left (3+2 x \right ) {\mathrm e}^{x^{2}+3 x +4}+2 \left (8 x^{3}+12 x^{2}+8 x \right ) {\mathrm e}^{x^{2}+3 x +4}+\left (8 x^{3}+12 x^{2}+8 x \right ) \left (3+2 x \right )^{2} {\mathrm e}^{x^{2}+3 x +4}+\left (-24 x -24\right ) {\mathrm e}^{x}+2 \left (-12 x^{2}-24 x \right ) {\mathrm e}^{x}+\left (-4 x^{3}-12 x^{2}+4\right ) {\mathrm e}^{x}+48\right )}{96 x^{10}+800 x^{9}+2520 x^{8}+4368 x^{7}+4196 x^{6}+2148 x^{5}+1464 x^{4}+2096 x^{3}+1440 x^{2}+432 x +24}\) | \(583\) |
Input:
int((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x, x,method=_RETURNVERBOSE)
Output:
4*x^2+8*x^3-4*exp(x)*x^3+4*exp(x^2+3*x+4)*x^2+4*exp(x)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=8 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 3 \, x + 4\right )} + 4 \, x^{2} - 4 \, {\left (x^{3} - 1\right )} e^{x} \] Input:
integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^ 2+8*x,x, algorithm="fricas")
Output:
8*x^3 + 4*x^2*e^(x^2 + 3*x + 4) + 4*x^2 - 4*(x^3 - 1)*e^x
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=8 x^{3} + 4 x^{2} e^{x^{2} + 3 x + 4} + 4 x^{2} + \left (4 - 4 x^{3}\right ) e^{x} \] Input:
integrate((8*x**3+12*x**2+8*x)*exp(x**2+3*x+4)+(-4*x**3-12*x**2+4)*exp(x)+ 24*x**2+8*x,x)
Output:
8*x**3 + 4*x**2*exp(x**2 + 3*x + 4) + 4*x**2 + (4 - 4*x**3)*exp(x)
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=8 \, x^{3} + 4 \, x^{2} e^{\left (x^{2} + 3 \, x + 4\right )} + 4 \, x^{2} - 4 \, {\left (x^{3} - 1\right )} e^{x} \] Input:
integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^ 2+8*x,x, algorithm="maxima")
Output:
8*x^3 + 4*x^2*e^(x^2 + 3*x + 4) + 4*x^2 - 4*(x^3 - 1)*e^x
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=8 \, x^{3} + 4 \, x^{2} + {\left ({\left (2 \, x + 3\right )}^{2} - 12 \, x - 9\right )} e^{\left (x^{2} + 3 \, x + 4\right )} - 4 \, {\left (x^{3} - 1\right )} e^{x} \] Input:
integrate((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^ 2+8*x,x, algorithm="giac")
Output:
8*x^3 + 4*x^2 + ((2*x + 3)^2 - 12*x - 9)*e^(x^2 + 3*x + 4) - 4*(x^3 - 1)*e ^x
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=4\,{\mathrm {e}}^x-4\,x^3\,{\mathrm {e}}^x+4\,x^2\,{\mathrm {e}}^{x^2+3\,x+4}+4\,x^2+8\,x^3 \] Input:
int(8*x - exp(x)*(12*x^2 + 4*x^3 - 4) + exp(3*x + x^2 + 4)*(8*x + 12*x^2 + 8*x^3) + 24*x^2,x)
Output:
4*exp(x) - 4*x^3*exp(x) + 4*x^2*exp(3*x + x^2 + 4) + 4*x^2 + 8*x^3
Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \left (8 x+24 x^2+e^x \left (4-12 x^2-4 x^3\right )+e^{4+3 x+x^2} \left (8 x+12 x^2+8 x^3\right )\right ) \, dx=4 e^{x^{2}+3 x} e^{4} x^{2}-4 e^{x} x^{3}+4 e^{x}+8 x^{3}+4 x^{2} \] Input:
int((8*x^3+12*x^2+8*x)*exp(x^2+3*x+4)+(-4*x^3-12*x^2+4)*exp(x)+24*x^2+8*x, x)
Output:
4*(e**(x**2 + 3*x)*e**4*x**2 - e**x*x**3 + e**x + 2*x**3 + x**2)