Integrand size = 38, antiderivative size = 16 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=\left (x^2\right )^{25 \left (\frac {5 x}{4}+x \log (x)\right )} \] Output:
exp(25*ln(x^2)*(x*ln(x)+5/4*x))
Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=x^{25 x \log \left (x^2\right )} \left (x^2\right )^{125 x/4} \] Input:
Integrate[((x^2)^((125*x + 100*x*Log[x])/4)*(250 + 200*Log[x] + (225 + 100 *Log[x])*Log[x^2]))/4,x]
Output:
x^(25*x*Log[x^2])*(x^2)^((125*x)/4)
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.
\(\displaystyle \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left ((100 \log (x)+225) \log \left (x^2\right )+200 \log (x)+250\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int 25 \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)} \left (8 \log (x)+(4 \log (x)+9) \log \left (x^2\right )+10\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {25}{4} \int \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)} \left (8 \log (x)+(4 \log (x)+9) \log \left (x^2\right )+10\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {25}{4} \int \left (8 \log (x) \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)}+(4 \log (x)+9) \log \left (x^2\right ) \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)}+10 \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {25}{4} \left (10 \int \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)}dx+8 \int \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)} \log (x)dx+9 \int \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)} \log \left (x^2\right )dx+4 \int \left (x^2\right )^{\frac {25}{4} (4 \log (x) x+5 x)} \log (x) \log \left (x^2\right )dx\right )\) |
Input:
Int[((x^2)^((125*x + 100*x*Log[x])/4)*(250 + 200*Log[x] + (225 + 100*Log[x ])*Log[x^2]))/4,x]
Output:
$Aborted
Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {25 x \left (4 \ln \left (x \right )+5\right ) \ln \left (x^{2}\right )}{4}}\) | \(15\) |
default | \({\mathrm e}^{\frac {\left (100 x \ln \left (x \right )+125 x \right ) \ln \left (x^{2}\right )}{4}}\) | \(17\) |
norman | \({\mathrm e}^{\frac {\left (100 x \ln \left (x \right )+125 x \right ) \ln \left (x^{2}\right )}{4}}\) | \(17\) |
risch | \({\mathrm e}^{\frac {25 x \left (4 \ln \left (x \right )+5\right ) \left (-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+4 \ln \left (x \right )\right )}{8}}\) | \(65\) |
Input:
int(1/4*((100*ln(x)+225)*ln(x^2)+200*ln(x)+250)*exp(1/4*(100*x*ln(x)+125*x )*ln(x^2)),x,method=_RETURNVERBOSE)
Output:
exp(25/4*x*(4*ln(x)+5)*ln(x^2))
Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=e^{\left (50 \, x \log \left (x\right )^{2} + \frac {125}{2} \, x \log \left (x\right )\right )} \] Input:
integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*lo g(x)+125*x)*log(x^2)),x, algorithm="fricas")
Output:
e^(50*x*log(x)^2 + 125/2*x*log(x))
Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=e^{2 \cdot \left (25 x \log {\left (x \right )} + \frac {125 x}{4}\right ) \log {\left (x \right )}} \] Input:
integrate(1/4*((100*ln(x)+225)*ln(x**2)+200*ln(x)+250)*exp(1/4*(100*x*ln(x )+125*x)*ln(x**2)),x)
Output:
exp(2*(25*x*log(x) + 125*x/4)*log(x))
Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=e^{\left (50 \, x \log \left (x\right )^{2} + \frac {125}{2} \, x \log \left (x\right )\right )} \] Input:
integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*lo g(x)+125*x)*log(x^2)),x, algorithm="maxima")
Output:
e^(50*x*log(x)^2 + 125/2*x*log(x))
\[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=\int { \frac {25}{4} \, {\left ({\left (4 \, \log \left (x\right ) + 9\right )} \log \left (x^{2}\right ) + 8 \, \log \left (x\right ) + 10\right )} {\left (x^{2}\right )}^{25 \, x \log \left (x\right ) + \frac {125}{4} \, x} \,d x } \] Input:
integrate(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*lo g(x)+125*x)*log(x^2)),x, algorithm="giac")
Output:
integrate(25/4*((4*log(x) + 9)*log(x^2) + 8*log(x) + 10)*(x^2)^(25*x*log(x ) + 125/4*x), x)
Time = 2.90 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=x^{25\,x\,\ln \left (x^2\right )}\,{\left (x^2\right )}^{\frac {125\,x}{4}} \] Input:
int((exp((log(x^2)*(125*x + 100*x*log(x)))/4)*(200*log(x) + log(x^2)*(100* log(x) + 225) + 250))/4,x)
Output:
x^(25*x*log(x^2))*(x^2)^((125*x)/4)
\[ \int \frac {1}{4} \left (x^2\right )^{\frac {1}{4} (125 x+100 x \log (x))} \left (250+200 \log (x)+(225+100 \log (x)) \log \left (x^2\right )\right ) \, dx=\frac {125 \left (\int x^{50 \,\mathrm {log}\left (x \right ) x +\frac {125 x}{2}}d x \right )}{2}+25 \left (\int x^{50 \,\mathrm {log}\left (x \right ) x +\frac {125 x}{2}} \mathrm {log}\left (x^{2}\right ) \mathrm {log}\left (x \right )d x \right )+\frac {225 \left (\int x^{50 \,\mathrm {log}\left (x \right ) x +\frac {125 x}{2}} \mathrm {log}\left (x^{2}\right )d x \right )}{4}+50 \left (\int x^{50 \,\mathrm {log}\left (x \right ) x +\frac {125 x}{2}} \mathrm {log}\left (x \right )d x \right ) \] Input:
int(1/4*((100*log(x)+225)*log(x^2)+200*log(x)+250)*exp(1/4*(100*x*log(x)+1 25*x)*log(x^2)),x)
Output:
(25*(10*int(x**((100*log(x)*x + 125*x)/2),x) + 4*int(x**((100*log(x)*x + 1 25*x)/2)*log(x**2)*log(x),x) + 9*int(x**((100*log(x)*x + 125*x)/2)*log(x** 2),x) + 8*int(x**((100*log(x)*x + 125*x)/2)*log(x),x)))/4