Integrand size = 165, antiderivative size = 26 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {4}{\frac {25}{9}+2 x^2+x \log \left (\frac {3}{(-5+x)^4 x^2}\right )} \] Output:
4/(2*x^2+x*ln(3/x^2/(-5+x)^4)+25/9)
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{25+18 x^2+9 x \log \left (\frac {3}{(-5+x)^4 x^2}\right )} \] Input:
Integrate[(-3240 + 8424*x - 1296*x^2 + (1620 - 324*x)*Log[3/(625*x^2 - 500 *x^3 + 150*x^4 - 20*x^5 + x^6)])/(-3125 + 625*x - 4500*x^2 + 900*x^3 - 162 0*x^4 + 324*x^5 + (-2250*x + 450*x^2 - 1620*x^3 + 324*x^4)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)] + (-405*x^2 + 81*x^3)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)]^2),x]
Output:
36/(25 + 18*x^2 + 9*x*Log[3/((-5 + x)^4*x^2)])
Time = 0.55 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {-1296 x^2+(1620-324 x) \log \left (\frac {3}{x^6-20 x^5+150 x^4-500 x^3+625 x^2}\right )+8424 x-3240}{324 x^5-1620 x^4+900 x^3-4500 x^2+\left (81 x^3-405 x^2\right ) \log ^2\left (\frac {3}{x^6-20 x^5+150 x^4-500 x^3+625 x^2}\right )+\left (324 x^4-1620 x^3+450 x^2-2250 x\right ) \log \left (\frac {3}{x^6-20 x^5+150 x^4-500 x^3+625 x^2}\right )+625 x-3125} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {324 \left (4 x^2+(x-5) \log \left (\frac {3}{(x-5)^4 x^2}\right )-26 x+10\right )}{(5-x) \left (18 x^2+9 x \log \left (\frac {3}{(x-5)^4 x^2}\right )+25\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 324 \int \frac {4 x^2-26 x-(5-x) \log \left (\frac {3}{(5-x)^4 x^2}\right )+10}{(5-x) \left (18 x^2+9 \log \left (\frac {3}{(5-x)^4 x^2}\right ) x+25\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {36}{18 x^2+9 x \log \left (\frac {3}{(5-x)^4 x^2}\right )+25}\) |
Input:
Int[(-3240 + 8424*x - 1296*x^2 + (1620 - 324*x)*Log[3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)])/(-3125 + 625*x - 4500*x^2 + 900*x^3 - 1620*x^4 + 324*x^5 + (-2250*x + 450*x^2 - 1620*x^3 + 324*x^4)*Log[3/(625*x^2 - 500* x^3 + 150*x^4 - 20*x^5 + x^6)] + (-405*x^2 + 81*x^3)*Log[3/(625*x^2 - 500* x^3 + 150*x^4 - 20*x^5 + x^6)]^2),x]
Output:
36/(25 + 18*x^2 + 9*x*Log[3/((5 - x)^4*x^2)])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.74 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58
method | result | size |
parallelrisch | \(\frac {36}{9 \ln \left (\frac {3}{x^{2} \left (x^{4}-20 x^{3}+150 x^{2}-500 x +625\right )}\right ) x +18 x^{2}+25}\) | \(41\) |
norman | \(\frac {36}{18 x^{2}+9 \ln \left (\frac {3}{x^{6}-20 x^{5}+150 x^{4}-500 x^{3}+625 x^{2}}\right ) x +25}\) | \(44\) |
risch | \(\frac {36}{18 x^{2}+9 \ln \left (\frac {3}{x^{6}-20 x^{5}+150 x^{4}-500 x^{3}+625 x^{2}}\right ) x +25}\) | \(44\) |
Input:
int(((-324*x+1620)*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+842 4*x-3240)/((81*x^3-405*x^2)*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))^2+( 324*x^4-1620*x^3+450*x^2-2250*x)*ln(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2) )+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x,method=_RETURNVERBOSE)
Output:
36/(9*ln(3/x^2/(x^4-20*x^3+150*x^2-500*x+625))*x+18*x^2+25)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{18 \, x^{2} + 9 \, x \log \left (\frac {3}{x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}}\right ) + 25} \] Input:
integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296* x^2+8424*x-3240)/((81*x^3-405*x^2)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x ^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x^3 +625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="fri cas")
Output:
36/(18*x^2 + 9*x*log(3/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2)) + 25)
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{18 x^{2} + 9 x \log {\left (\frac {3}{x^{6} - 20 x^{5} + 150 x^{4} - 500 x^{3} + 625 x^{2}} \right )} + 25} \] Input:
integrate(((-324*x+1620)*ln(3/(x**6-20*x**5+150*x**4-500*x**3+625*x**2))-1 296*x**2+8424*x-3240)/((81*x**3-405*x**2)*ln(3/(x**6-20*x**5+150*x**4-500* x**3+625*x**2))**2+(324*x**4-1620*x**3+450*x**2-2250*x)*ln(3/(x**6-20*x**5 +150*x**4-500*x**3+625*x**2))+324*x**5-1620*x**4+900*x**3-4500*x**2+625*x- 3125),x)
Output:
36/(18*x**2 + 9*x*log(3/(x**6 - 20*x**5 + 150*x**4 - 500*x**3 + 625*x**2)) + 25)
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{18 \, x^{2} + 9 \, x \log \left (3\right ) - 36 \, x \log \left (x - 5\right ) - 18 \, x \log \left (x\right ) + 25} \] Input:
integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296* x^2+8424*x-3240)/((81*x^3-405*x^2)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x ^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x^3 +625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="max ima")
Output:
36/(18*x^2 + 9*x*log(3) - 36*x*log(x - 5) - 18*x*log(x) + 25)
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{18 \, x^{2} + 9 \, x \log \left (\frac {3}{x^{6} - 20 \, x^{5} + 150 \, x^{4} - 500 \, x^{3} + 625 \, x^{2}}\right ) + 25} \] Input:
integrate(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296* x^2+8424*x-3240)/((81*x^3-405*x^2)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x ^2))^2+(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x^3 +625*x^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x, algorithm="gia c")
Output:
36/(18*x^2 + 9*x*log(3/(x^6 - 20*x^5 + 150*x^4 - 500*x^3 + 625*x^2)) + 25)
Time = 3.51 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{9\,x\,\ln \left (\frac {3}{x^6-20\,x^5+150\,x^4-500\,x^3+625\,x^2}\right )+18\,x^2+25} \] Input:
int((log(3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6))*(324*x - 1620) - 8424*x + 1296*x^2 + 3240)/(log(3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x ^6))*(2250*x - 450*x^2 + 1620*x^3 - 324*x^4) - 625*x + log(3/(625*x^2 - 50 0*x^3 + 150*x^4 - 20*x^5 + x^6))^2*(405*x^2 - 81*x^3) + 4500*x^2 - 900*x^3 + 1620*x^4 - 324*x^5 + 3125),x)
Output:
36/(9*x*log(3/(625*x^2 - 500*x^3 + 150*x^4 - 20*x^5 + x^6)) + 18*x^2 + 25)
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-3240+8424 x-1296 x^2+(1620-324 x) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )}{-3125+625 x-4500 x^2+900 x^3-1620 x^4+324 x^5+\left (-2250 x+450 x^2-1620 x^3+324 x^4\right ) \log \left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )+\left (-405 x^2+81 x^3\right ) \log ^2\left (\frac {3}{625 x^2-500 x^3+150 x^4-20 x^5+x^6}\right )} \, dx=\frac {36}{9 \,\mathrm {log}\left (\frac {3}{x^{6}-20 x^{5}+150 x^{4}-500 x^{3}+625 x^{2}}\right ) x +18 x^{2}+25} \] Input:
int(((-324*x+1620)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))-1296*x^2+84 24*x-3240)/((81*x^3-405*x^2)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x^2))^2 +(324*x^4-1620*x^3+450*x^2-2250*x)*log(3/(x^6-20*x^5+150*x^4-500*x^3+625*x ^2))+324*x^5-1620*x^4+900*x^3-4500*x^2+625*x-3125),x)
Output:
36/(9*log(3/(x**6 - 20*x**5 + 150*x**4 - 500*x**3 + 625*x**2))*x + 18*x**2 + 25)