Integrand size = 111, antiderivative size = 27 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=\frac {5}{-6+\frac {1}{5} \log (3) (-\log (3)+\log (5+(1+x) (5+x)))} \] Output:
5/(ln(3)*(1/5*ln((5+x)*(1+x)+5)-1/5*ln(3))-6)
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=-\frac {50 \log (3)}{\log (9) \left (30+\log ^2(3)-\log (3) \log \left (10+6 x+x^2\right )\right )} \] Input:
Integrate[((-150 - 50*x)*Log[3])/(9000 + 5400*x + 900*x^2 + (600 + 360*x + 60*x^2)*Log[3]^2 + (10 + 6*x + x^2)*Log[3]^4 + ((-600 - 360*x - 60*x^2)*L og[3] + (-20 - 12*x - 2*x^2)*Log[3]^3)*Log[10 + 6*x + x^2] + (10 + 6*x + x ^2)*Log[3]^2*Log[10 + 6*x + x^2]^2),x]
Output:
(-50*Log[3])/(Log[9]*(30 + Log[3]^2 - Log[3]*Log[10 + 6*x + x^2]))
Time = 0.51 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {27, 6, 27, 7239, 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.
\(\displaystyle \int \frac {(-50 x-150) \log (3)}{900 x^2+\left (x^2+6 x+10\right ) \log ^4(3)+\left (\left (-2 x^2-12 x-20\right ) \log ^3(3)+\left (-60 x^2-360 x-600\right ) \log (3)\right ) \log \left (x^2+6 x+10\right )+\left (x^2+6 x+10\right ) \log ^2(3) \log ^2\left (x^2+6 x+10\right )+\left (60 x^2+360 x+600\right ) \log ^2(3)+5400 x+9000} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log (3) \int -\frac {50 (x+3)}{900 x^2+5400 x+\left (x^2+6 x+10\right ) \log ^2(3) \log ^2\left (x^2+6 x+10\right )-2 \left (\log ^3(3) \left (x^2+6 x+10\right )+30 \log (3) \left (x^2+6 x+10\right )\right ) \log \left (x^2+6 x+10\right )+\left (x^2+6 x+10\right ) \log ^4(3)+60 \left (x^2+6 x+10\right ) \log ^2(3)+9000}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \log (3) \int -\frac {50 (x+3)}{900 x^2+5400 x+\left (x^2+6 x+10\right ) \log ^2(3) \log ^2\left (x^2+6 x+10\right )-2 \left (\log ^3(3) \left (x^2+6 x+10\right )+30 \log (3) \left (x^2+6 x+10\right )\right ) \log \left (x^2+6 x+10\right )+\left (x^2+6 x+10\right ) \left (60 \log ^2(3)+\log ^4(3)\right )+9000}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -50 \log (3) \int \frac {x+3}{900 x^2+5400 x+\left (x^2+6 x+10\right ) \log ^2(3) \log ^2\left (x^2+6 x+10\right )-2 \left (\log ^3(3) \left (x^2+6 x+10\right )+30 \log (3) \left (x^2+6 x+10\right )\right ) \log \left (x^2+6 x+10\right )+\left (x^2+6 x+10\right ) \log ^2(3) \left (60+\log ^2(3)\right )+9000}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -50 \log (3) \int \frac {x+3}{\left (x^2+6 x+10\right ) \left (30 \left (1+\frac {\log ^2(3)}{30}\right )-\log (3) \log \left (x^2+6 x+10\right )\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle -\frac {50 \log (3)}{\log (9) \left (-\log (3) \log \left (x^2+6 x+10\right )+30+\log ^2(3)\right )}\) |
Input:
Int[((-150 - 50*x)*Log[3])/(9000 + 5400*x + 900*x^2 + (600 + 360*x + 60*x^ 2)*Log[3]^2 + (10 + 6*x + x^2)*Log[3]^4 + ((-600 - 360*x - 60*x^2)*Log[3] + (-20 - 12*x - 2*x^2)*Log[3]^3)*Log[10 + 6*x + x^2] + (10 + 6*x + x^2)*Lo g[3]^2*Log[10 + 6*x + x^2]^2),x]
Output:
(-50*Log[3])/(Log[9]*(30 + Log[3]^2 - Log[3]*Log[10 + 6*x + x^2]))
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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 = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
norman | \(-\frac {25}{\ln \left (3\right )^{2}-\ln \left (3\right ) \ln \left (x^{2}+6 x +10\right )+30}\) | \(24\) |
risch | \(-\frac {25}{\ln \left (3\right )^{2}-\ln \left (3\right ) \ln \left (x^{2}+6 x +10\right )+30}\) | \(24\) |
parallelrisch | \(-\frac {25}{\ln \left (3\right )^{2}-\ln \left (3\right ) \ln \left (x^{2}+6 x +10\right )+30}\) | \(24\) |
default | \(\frac {25}{-\ln \left (3\right )^{2}+\ln \left (3\right ) \ln \left (x^{2}+6 x +10\right )-30}\) | \(25\) |
Input:
int((-50*x-150)*ln(3)/((x^2+6*x+10)*ln(3)^2*ln(x^2+6*x+10)^2+((-2*x^2-12*x -20)*ln(3)^3+(-60*x^2-360*x-600)*ln(3))*ln(x^2+6*x+10)+(x^2+6*x+10)*ln(3)^ 4+(60*x^2+360*x+600)*ln(3)^2+900*x^2+5400*x+9000),x,method=_RETURNVERBOSE)
Output:
-25/(ln(3)^2-ln(3)*ln(x^2+6*x+10)+30)
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=-\frac {25}{\log \left (3\right )^{2} - \log \left (3\right ) \log \left (x^{2} + 6 \, x + 10\right ) + 30} \] Input:
integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2 *x^2-12*x-20)*log(3)^3+(-60*x^2-360*x-600)*log(3))*log(x^2+6*x+10)+(x^2+6* x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algorit hm="fricas")
Output:
-25/(log(3)^2 - log(3)*log(x^2 + 6*x + 10) + 30)
Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=\frac {25}{\log {\left (3 \right )} \log {\left (x^{2} + 6 x + 10 \right )} - 30 - \log {\left (3 \right )}^{2}} \] Input:
integrate((-50*x-150)*ln(3)/((x**2+6*x+10)*ln(3)**2*ln(x**2+6*x+10)**2+((- 2*x**2-12*x-20)*ln(3)**3+(-60*x**2-360*x-600)*ln(3))*ln(x**2+6*x+10)+(x**2 +6*x+10)*ln(3)**4+(60*x**2+360*x+600)*ln(3)**2+900*x**2+5400*x+9000),x)
Output:
25/(log(3)*log(x**2 + 6*x + 10) - 30 - log(3)**2)
Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=-\frac {25 \, \log \left (3\right )}{\log \left (3\right )^{3} - \log \left (3\right )^{2} \log \left (x^{2} + 6 \, x + 10\right ) + 30 \, \log \left (3\right )} \] Input:
integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2 *x^2-12*x-20)*log(3)^3+(-60*x^2-360*x-600)*log(3))*log(x^2+6*x+10)+(x^2+6* x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algorit hm="maxima")
Output:
-25*log(3)/(log(3)^3 - log(3)^2*log(x^2 + 6*x + 10) + 30*log(3))
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=-\frac {25}{\log \left (3\right )^{2} - \log \left (3\right ) \log \left (x^{2} + 6 \, x + 10\right ) + 30} \] Input:
integrate((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2 *x^2-12*x-20)*log(3)^3+(-60*x^2-360*x-600)*log(3))*log(x^2+6*x+10)+(x^2+6* x+10)*log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x, algorit hm="giac")
Output:
-25/(log(3)^2 - log(3)*log(x^2 + 6*x + 10) + 30)
Time = 3.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=\frac {25}{\ln \left (3\right )\,\left (\ln \left (x^2+6\,x+10\right )-\frac {{\ln \left (3\right )}^2+30}{\ln \left (3\right )}\right )} \] Input:
int(-(log(3)*(50*x + 150))/(5400*x - log(6*x + x^2 + 10)*(log(3)*(360*x + 60*x^2 + 600) + log(3)^3*(12*x + 2*x^2 + 20)) + log(3)^4*(6*x + x^2 + 10) + log(3)^2*(360*x + 60*x^2 + 600) + 900*x^2 + log(3)^2*log(6*x + x^2 + 10) ^2*(6*x + x^2 + 10) + 9000),x)
Output:
25/(log(3)*(log(6*x + x^2 + 10) - (log(3)^2 + 30)/log(3)))
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {(-150-50 x) \log (3)}{9000+5400 x+900 x^2+\left (600+360 x+60 x^2\right ) \log ^2(3)+\left (10+6 x+x^2\right ) \log ^4(3)+\left (\left (-600-360 x-60 x^2\right ) \log (3)+\left (-20-12 x-2 x^2\right ) \log ^3(3)\right ) \log \left (10+6 x+x^2\right )+\left (10+6 x+x^2\right ) \log ^2(3) \log ^2\left (10+6 x+x^2\right )} \, dx=\frac {25 \,\mathrm {log}\left (x^{2}+6 x +10\right ) \mathrm {log}\left (3\right )}{\mathrm {log}\left (x^{2}+6 x +10\right ) \mathrm {log}\left (3\right )^{3}+30 \,\mathrm {log}\left (x^{2}+6 x +10\right ) \mathrm {log}\left (3\right )-\mathrm {log}\left (3\right )^{4}-60 \mathrm {log}\left (3\right )^{2}-900} \] Input:
int((-50*x-150)*log(3)/((x^2+6*x+10)*log(3)^2*log(x^2+6*x+10)^2+((-2*x^2-1 2*x-20)*log(3)^3+(-60*x^2-360*x-600)*log(3))*log(x^2+6*x+10)+(x^2+6*x+10)* log(3)^4+(60*x^2+360*x+600)*log(3)^2+900*x^2+5400*x+9000),x)
Output:
(25*log(x**2 + 6*x + 10)*log(3))/(log(x**2 + 6*x + 10)*log(3)**3 + 30*log( x**2 + 6*x + 10)*log(3) - log(3)**4 - 60*log(3)**2 - 900)