Integrand size = 57, antiderivative size = 23 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2 x+\frac {4-13 x}{\log \left (\frac {3 e^x x^2}{2}\right )} \] Output:
(-13*x+4)/ln(3/2*exp(x)*x^2)+2*x
Time = 6.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2 x+\frac {4-13 x}{\log \left (\frac {3 e^x x^2}{2}\right )} \] Input:
Integrate[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x ^2)/2]^2)/(x*Log[(3*E^x*x^2)/2]^2),x]
Output:
2*x + (4 - 13*x)/Log[(3*E^x*x^2)/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.
\(\displaystyle \int \frac {13 x^2+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )-13 x \log \left (\frac {3 e^x x^2}{2}\right )+22 x-8}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {13 x^2+22 x-8}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )}-\frac {13}{\log \left (\frac {3 e^x x^2}{2}\right )}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 22 \int \frac {1}{\log ^2\left (\frac {3 e^x x^2}{2}\right )}dx-8 \int \frac {1}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )}dx+13 \int \frac {x}{\log ^2\left (\frac {3 e^x x^2}{2}\right )}dx-13 \int \frac {1}{\log \left (\frac {3 e^x x^2}{2}\right )}dx+2 x\) |
Input:
Int[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x^2)/2] ^2)/(x*Log[(3*E^x*x^2)/2]^2),x]
Output:
$Aborted
Time = 0.93 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22
method | result | size |
norman | \(\frac {4+2 x \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-13 x}{\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\) | \(28\) |
parallelrisch | \(\frac {8+4 x \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-26 x}{2 \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\) | \(29\) |
default | \(2 x +\frac {13 \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-13 x +4}{\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\) | \(31\) |
parts | \(2 x +\frac {13 \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-13 x +4}{\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\) | \(31\) |
risch | \(2 x +\frac {26 x -8}{-2 \ln \left (3\right )+2 \ln \left (2\right )-4 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x}\right )+i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-2 i \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) \pi +i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{x}\right )-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{3}}\) | \(165\) |
Input:
int((2*x*ln(3/2*exp(x)*x^2)^2-13*x*ln(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/ln( 3/2*exp(x)*x^2)^2,x,method=_RETURNVERBOSE)
Output:
(4+2*x*ln(3/2*exp(x)*x^2)-13*x)/ln(3/2*exp(x)*x^2)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=\frac {2 \, x \log \left (\frac {3}{2} \, x^{2} e^{x}\right ) - 13 \, x + 4}{\log \left (\frac {3}{2} \, x^{2} e^{x}\right )} \] Input:
integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x- 8)/x/log(3/2*exp(x)*x^2)^2,x, algorithm="fricas")
Output:
(2*x*log(3/2*x^2*e^x) - 13*x + 4)/log(3/2*x^2*e^x)
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2 x + \frac {4 - 13 x}{\log {\left (\frac {3 x^{2} e^{x}}{2} \right )}} \] Input:
integrate((2*x*ln(3/2*exp(x)*x**2)**2-13*x*ln(3/2*exp(x)*x**2)+13*x**2+22* x-8)/x/ln(3/2*exp(x)*x**2)**2,x)
Output:
2*x + (4 - 13*x)/log(3*x**2*exp(x)/2)
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2 \, x - \frac {13 \, x - 4}{x + \log \left (3\right ) - \log \left (2\right ) + 2 \, \log \left (x\right )} \] Input:
integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x- 8)/x/log(3/2*exp(x)*x^2)^2,x, algorithm="maxima")
Output:
2*x - (13*x - 4)/(x + log(3) - log(2) + 2*log(x))
Time = 0.16 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2 \, x - \frac {13 \, x - 4}{x + \log \left (\frac {3}{2} \, x^{2}\right )} \] Input:
integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x- 8)/x/log(3/2*exp(x)*x^2)^2,x, algorithm="giac")
Output:
2*x - (13*x - 4)/(x + log(3/2*x^2))
Time = 3.53 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=2\,x-\frac {13\,x-4}{\ln \left (\frac {3\,x^2\,{\mathrm {e}}^x}{2}\right )} \] Input:
int((22*x - 13*x*log((3*x^2*exp(x))/2) + 2*x*log((3*x^2*exp(x))/2)^2 + 13* x^2 - 8)/(x*log((3*x^2*exp(x))/2)^2),x)
Output:
2*x - (13*x - 4)/log((3*x^2*exp(x))/2)
Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx=\frac {2 \,\mathrm {log}\left (\frac {3 e^{x} x^{2}}{2}\right ) x -13 x +4}{\mathrm {log}\left (\frac {3 e^{x} x^{2}}{2}\right )} \] Input:
int((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/l og(3/2*exp(x)*x^2)^2,x)
Output:
(2*log((3*e**x*x**2)/2)*x - 13*x + 4)/log((3*e**x*x**2)/2)