Integrand size = 102, antiderivative size = 28 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\left (3+x^2\right ) \left (x-\frac {2 x}{\frac {x}{25}+\frac {1}{5} (-1+x) \log (x)}\right ) \] Output:
(x-2/(1/5*(-1+x)*ln(x)+1/25*x)*x)*(x^2+3)
Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x \left (3+x^2-\frac {50 \left (3+x^2\right )}{x+5 (-1+x) \log (x)}\right ) \] Input:
Integrate[(-750 + 750*x - 247*x^2 + 150*x^3 + 3*x^4 + (750 - 30*x + 780*x^ 2 - 530*x^3 + 30*x^4)*Log[x] + (75 - 150*x + 150*x^2 - 150*x^3 + 75*x^4)*L og[x]^2)/(x^2 + (-10*x + 10*x^2)*Log[x] + (25 - 50*x + 25*x^2)*Log[x]^2),x ]
Output:
x*(3 + x^2 - (50*(3 + x^2))/(x + 5*(-1 + x)*Log[x]))
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.
\(\displaystyle \int \frac {3 x^4+150 x^3-247 x^2+\left (75 x^4-150 x^3+150 x^2-150 x+75\right ) \log ^2(x)+\left (30 x^4-530 x^3+780 x^2-30 x+750\right ) \log (x)+750 x-750}{x^2+\left (25 x^2-50 x+25\right ) \log ^2(x)+\left (10 x^2-10 x\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {3 x^4+150 x^3-247 x^2+\left (75 x^4-150 x^3+150 x^2-150 x+75\right ) \log ^2(x)+\left (30 x^4-530 x^3+780 x^2-30 x+750\right ) \log (x)+750 x-750}{(x+5 x \log (x)-5 \log (x))^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (3 \left (x^2+1\right )-\frac {50 \left (2 x^3-3 x^2-3\right )}{(x-1) (x+5 x \log (x)-5 \log (x))}+\frac {50 \left (5 x^4-11 x^3+20 x^2-33 x+15\right )}{(x-1) (x+5 x \log (x)-5 \log (x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 250 \int \frac {x^3}{(5 \log (x) x+x-5 \log (x))^2}dx-300 \int \frac {x^2}{(5 \log (x) x+x-5 \log (x))^2}dx-100 \int \frac {x^2}{5 \log (x) x+x-5 \log (x)}dx-950 \int \frac {1}{(5 \log (x) x+x-5 \log (x))^2}dx-200 \int \frac {1}{(x-1) (5 \log (x) x+x-5 \log (x))^2}dx+700 \int \frac {x}{(5 \log (x) x+x-5 \log (x))^2}dx+50 \int \frac {1}{5 \log (x) x+x-5 \log (x)}dx+200 \int \frac {1}{(x-1) (5 \log (x) x+x-5 \log (x))}dx+50 \int \frac {x}{5 \log (x) x+x-5 \log (x)}dx+x^3+3 x\) |
Input:
Int[(-750 + 750*x - 247*x^2 + 150*x^3 + 3*x^4 + (750 - 30*x + 780*x^2 - 53 0*x^3 + 30*x^4)*Log[x] + (75 - 150*x + 150*x^2 - 150*x^3 + 75*x^4)*Log[x]^ 2)/(x^2 + (-10*x + 10*x^2)*Log[x] + (25 - 50*x + 25*x^2)*Log[x]^2),x]
Output:
$Aborted
Time = 0.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
risch | \(x^{3}+3 x -\frac {50 \left (x^{2}+3\right ) x}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(29\) |
default | \(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(57\) |
norman | \(\frac {x^{4}-147 x -15 \ln \left (x \right )+3 x^{2}-50 x^{3}+15 x^{2} \ln \left (x \right )-5 x^{3} \ln \left (x \right )+5 x^{4} \ln \left (x \right )}{5 x \ln \left (x \right )-5 \ln \left (x \right )+x}\) | \(57\) |
parallelrisch | \(\frac {25 x^{4} \ln \left (x \right )+5 x^{4}-25 x^{3} \ln \left (x \right )-250 x^{3}+75 x^{2} \ln \left (x \right )+15 x^{2}-735 x -75 \ln \left (x \right )}{25 x \ln \left (x \right )-25 \ln \left (x \right )+5 x}\) | \(60\) |
Input:
int(((75*x^4-150*x^3+150*x^2-150*x+75)*ln(x)^2+(30*x^4-530*x^3+780*x^2-30* x+750)*ln(x)+3*x^4+150*x^3-247*x^2+750*x-750)/((25*x^2-50*x+25)*ln(x)^2+(1 0*x^2-10*x)*ln(x)+x^2),x,method=_RETURNVERBOSE)
Output:
x^3+3*x-50*(x^2+3)*x/(5*x*ln(x)-5*ln(x)+x)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \] Input:
integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780* x^2-30*x+750)*log(x)+3*x^4+150*x^3-247*x^2+750*x-750)/((25*x^2-50*x+25)*lo g(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="fricas")
Output:
(x^4 - 50*x^3 + 3*x^2 + 5*(x^4 - x^3 + 3*x^2 - 3*x)*log(x) - 150*x)/(5*(x - 1)*log(x) + x)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 x + \frac {- 50 x^{3} - 150 x}{x + \left (5 x - 5\right ) \log {\left (x \right )}} \] Input:
integrate(((75*x**4-150*x**3+150*x**2-150*x+75)*ln(x)**2+(30*x**4-530*x**3 +780*x**2-30*x+750)*ln(x)+3*x**4+150*x**3-247*x**2+750*x-750)/((25*x**2-50 *x+25)*ln(x)**2+(10*x**2-10*x)*ln(x)+x**2),x)
Output:
x**3 + 3*x + (-50*x**3 - 150*x)/(x + (5*x - 5)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x^{4} - 50 \, x^{3} + 3 \, x^{2} + 5 \, {\left (x^{4} - x^{3} + 3 \, x^{2} - 3 \, x\right )} \log \left (x\right ) - 150 \, x}{5 \, {\left (x - 1\right )} \log \left (x\right ) + x} \] Input:
integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780* x^2-30*x+750)*log(x)+3*x^4+150*x^3-247*x^2+750*x-750)/((25*x^2-50*x+25)*lo g(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="maxima")
Output:
(x^4 - 50*x^3 + 3*x^2 + 5*(x^4 - x^3 + 3*x^2 - 3*x)*log(x) - 150*x)/(5*(x - 1)*log(x) + x)
Time = 0.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x^{3} + 3 \, x - \frac {50 \, {\left (x^{3} + 3 \, x\right )}}{5 \, x \log \left (x\right ) + x - 5 \, \log \left (x\right )} \] Input:
integrate(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780* x^2-30*x+750)*log(x)+3*x^4+150*x^3-247*x^2+750*x-750)/((25*x^2-50*x+25)*lo g(x)^2+(10*x^2-10*x)*log(x)+x^2),x, algorithm="giac")
Output:
x^3 + 3*x - 50*(x^3 + 3*x)/(5*x*log(x) + x - 5*log(x))
Time = 7.50 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=x\,\left (x^2+3\right )-\frac {x^2\,\left (x^2+3\right )-x\,\left (x^2+3\right )\,\left (x-50\right )}{x-5\,\ln \left (x\right )+5\,x\,\ln \left (x\right )} \] Input:
int((750*x + log(x)*(780*x^2 - 30*x - 530*x^3 + 30*x^4 + 750) + log(x)^2*( 150*x^2 - 150*x - 150*x^3 + 75*x^4 + 75) - 247*x^2 + 150*x^3 + 3*x^4 - 750 )/(log(x)^2*(25*x^2 - 50*x + 25) - log(x)*(10*x - 10*x^2) + x^2),x)
Output:
x*(x^2 + 3) - (x^2*(x^2 + 3) - x*(x^2 + 3)*(x - 50))/(x - 5*log(x) + 5*x*l og(x))
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {-750+750 x-247 x^2+150 x^3+3 x^4+\left (750-30 x+780 x^2-530 x^3+30 x^4\right ) \log (x)+\left (75-150 x+150 x^2-150 x^3+75 x^4\right ) \log ^2(x)}{x^2+\left (-10 x+10 x^2\right ) \log (x)+\left (25-50 x+25 x^2\right ) \log ^2(x)} \, dx=\frac {x \left (5 \,\mathrm {log}\left (x \right ) x^{3}-5 \,\mathrm {log}\left (x \right ) x^{2}+15 \,\mathrm {log}\left (x \right ) x -15 \,\mathrm {log}\left (x \right )+x^{3}-50 x^{2}+3 x -150\right )}{5 \,\mathrm {log}\left (x \right ) x -5 \,\mathrm {log}\left (x \right )+x} \] Input:
int(((75*x^4-150*x^3+150*x^2-150*x+75)*log(x)^2+(30*x^4-530*x^3+780*x^2-30 *x+750)*log(x)+3*x^4+150*x^3-247*x^2+750*x-750)/((25*x^2-50*x+25)*log(x)^2 +(10*x^2-10*x)*log(x)+x^2),x)
Output:
(x*(5*log(x)*x**3 - 5*log(x)*x**2 + 15*log(x)*x - 15*log(x) + x**3 - 50*x* *2 + 3*x - 150))/(5*log(x)*x - 5*log(x) + x)