Integrand size = 109, antiderivative size = 38 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {4}{-\frac {3}{x^2}+\frac {5-x}{4 \log (3)}}-\frac {-5+x-x^2}{\log (5)} \] Output:
4/(1/4*(5-x)/ln(3)-3/x^2)-(-x^2+x-5)/ln(5)
Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(38)=76\).
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.37 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {x \left (x^3 (750-486 \log (3))+12 (125-81 \log (3)) \log (3)+5 x^2 (-125+81 \log (3))+x^4 (-125+81 \log (3))-4 x \log (3) (375-500 \log (5)+81 \log (3) (-3+\log (625)))\right )}{\left (-5 x^2+x^3+12 \log (3)\right ) (-125+81 \log (3)) \log (5)} \] Input:
Integrate[(-25*x^4 + 60*x^5 - 21*x^6 + 2*x^7 + (120*x^2 - 264*x^3 + 48*x^4 )*Log[3] + (-144 + 288*x)*Log[3]^2 + (16*x^4*Log[3] - 384*x*Log[3]^2)*Log[ 5])/((25*x^4 - 10*x^5 + x^6 + (-120*x^2 + 24*x^3)*Log[3] + 144*Log[3]^2)*L og[5]),x]
Output:
(x*(x^3*(750 - 486*Log[3]) + 12*(125 - 81*Log[3])*Log[3] + 5*x^2*(-125 + 8 1*Log[3]) + x^4*(-125 + 81*Log[3]) - 4*x*Log[3]*(375 - 500*Log[5] + 81*Log [3]*(-3 + Log[625]))))/((-5*x^2 + x^3 + 12*Log[3])*(-125 + 81*Log[3])*Log[ 5])
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.
| \(\displaystyle \int \frac {2 x^7-21 x^6+60 x^5-25 x^4+\log (5) \left (16 x^4 \log (3)-384 x \log ^2(3)\right )+\left (48 x^4-264 x^3+120 x^2\right ) \log (3)+(288 x-144) \log ^2(3)}{\log (5) \left (x^6-10 x^5+25 x^4+\left (24 x^3-120 x^2\right ) \log (3)+144 \log ^2(3)\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {-2 x^7+21 x^6-60 x^5+25 x^4-16 \left (x^4 \log (3)-24 x \log ^2(3)\right ) \log (5)+144 (1-2 x) \log ^2(3)-24 \left (2 x^4-11 x^3+5 x^2\right ) \log (3)}{x^6-10 x^5+25 x^4+144 \log ^2(3)-24 \left (5 x^2-x^3\right ) \log (3)}dx}{\log (5)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {-2 x^7+21 x^6-60 x^5+25 x^4-16 \left (x^4 \log (3)-24 x \log ^2(3)\right ) \log (5)+144 (1-2 x) \log ^2(3)-24 \left (2 x^4-11 x^3+5 x^2\right ) \log (3)}{x^6-10 x^5+25 x^4+144 \log ^2(3)-24 \left (5 x^2-x^3\right ) \log (3)}dx}{\log (5)}\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle -\frac {\int \left (-2 x+\frac {16 \log (3) \left (-25 x^2+36 \log (3) x+60 \log (3)\right ) \log (5)}{\left (x^3-5 x^2+12 \log (3)\right )^2}-\frac {16 (x+5) \log (3) \log (5)}{x^3-5 x^2+12 \log (3)}+1\right )dx}{\log (5)}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {\int \left (-2 x+\frac {16 \log (3) \left (-25 x^2+36 \log (3) x+60 \log (3)\right ) \log (5)}{\left (x^3-5 x^2+12 \log (3)\right )^2}-\frac {16 (x+5) \log (3) \log (5)}{x^3-5 x^2+12 \log (3)}+1\right )dx}{\log (5)}\) |
Input:
Int[(-25*x^4 + 60*x^5 - 21*x^6 + 2*x^7 + (120*x^2 - 264*x^3 + 48*x^4)*Log[ 3] + (-144 + 288*x)*Log[3]^2 + (16*x^4*Log[3] - 384*x*Log[3]^2)*Log[5])/(( 25*x^4 - 10*x^5 + x^6 + (-120*x^2 + 24*x^3)*Log[3] + 144*Log[3]^2)*Log[5]) ,x]
Output:
$Aborted
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {-x +x^{2}-\frac {4 \ln \left (5\right ) \ln \left (3\right ) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \left (3\right )\right )}}{\ln \left (5\right )}\) | \(37\) |
| risch | \(-\frac {x}{\ln \left (5\right )}+\frac {x^{2}}{\ln \left (5\right )}-\frac {4 \ln \left (3\right ) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \left (3\right )\right )}\) | \(39\) |
| gosper | \(-\frac {-x^{5}+16 x^{2} \ln \left (3\right ) \ln \left (5\right )+6 x^{4}-12 x^{2} \ln \left (3\right )+12 x \ln \left (3\right )-25 x^{2}+60 \ln \left (3\right )}{\ln \left (5\right ) \left (x^{3}-5 x^{2}+12 \ln \left (3\right )\right )}\) | \(63\) |
| parallelrisch | \(-\frac {-x^{5}+16 x^{2} \ln \left (3\right ) \ln \left (5\right )+6 x^{4}-12 x^{2} \ln \left (3\right )+12 x \ln \left (3\right )-25 x^{2}+60 \ln \left (3\right )}{\ln \left (5\right ) \left (x^{3}-5 x^{2}+12 \ln \left (3\right )\right )}\) | \(63\) |
| norman | \(\frac {-\frac {\left (16 \ln \left (5\right ) \ln \left (3\right )-12 \ln \left (3\right )-25\right ) x^{2}}{\ln \left (5\right )}+\frac {x^{5}}{\ln \left (5\right )}-\frac {6 x^{4}}{\ln \left (5\right )}-\frac {12 \ln \left (3\right ) x}{\ln \left (5\right )}-\frac {60 \ln \left (3\right )}{\ln \left (5\right )}}{x^{3}-5 x^{2}+12 \ln \left (3\right )}\) | \(73\) |
Input:
int(((-384*x*ln(3)^2+16*x^4*ln(3))*ln(5)+(288*x-144)*ln(3)^2+(48*x^4-264*x ^3+120*x^2)*ln(3)+2*x^7-21*x^6+60*x^5-25*x^4)/(144*ln(3)^2+(24*x^3-120*x^2 )*ln(3)+x^6-10*x^5+25*x^4)/ln(5),x,method=_RETURNVERBOSE)
Output:
1/ln(5)*(-x+x^2-4/3*ln(5)*ln(3)*x^2/(1/12*x^3-5/12*x^2+ln(3)))
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {x^{5} - 6 \, x^{4} - 16 \, x^{2} \log \left (5\right ) \log \left (3\right ) + 5 \, x^{3} + 12 \, {\left (x^{2} - x\right )} \log \left (3\right )}{{\left (x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )\right )} \log \left (5\right )} \] Input:
integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48 *x^4-264*x^3+120*x^2)*log(3)+2*x^7-21*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24 *x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="fricas")
Output:
(x^5 - 6*x^4 - 16*x^2*log(5)*log(3) + 5*x^3 + 12*(x^2 - x)*log(3))/((x^3 - 5*x^2 + 12*log(3))*log(5))
Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.84 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {x^{2}}{\log {\left (5 \right )}} - \frac {16 x^{2} \log {\left (3 \right )}}{x^{3} - 5 x^{2} + 12 \log {\left (3 \right )}} - \frac {x}{\log {\left (5 \right )}} \] Input:
integrate(((-384*x*ln(3)**2+16*x**4*ln(3))*ln(5)+(288*x-144)*ln(3)**2+(48* x**4-264*x**3+120*x**2)*ln(3)+2*x**7-21*x**6+60*x**5-25*x**4)/(144*ln(3)** 2+(24*x**3-120*x**2)*ln(3)+x**6-10*x**5+25*x**4)/ln(5),x)
Output:
x**2/log(5) - 16*x**2*log(3)/(x**3 - 5*x**2 + 12*log(3)) - x/log(5)
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=-\frac {\frac {16 \, x^{2} \log \left (5\right ) \log \left (3\right )}{x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )} - x^{2} + x}{\log \left (5\right )} \] Input:
integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48 *x^4-264*x^3+120*x^2)*log(3)+2*x^7-21*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24 *x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="maxima")
Output:
-(16*x^2*log(5)*log(3)/(x^3 - 5*x^2 + 12*log(3)) - x^2 + x)/log(5)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=-\frac {\frac {16 \, x^{2} \log \left (5\right ) \log \left (3\right )}{x^{3} - 5 \, x^{2} + 12 \, \log \left (3\right )} - x^{2} + x}{\log \left (5\right )} \] Input:
integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48 *x^4-264*x^3+120*x^2)*log(3)+2*x^7-21*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24 *x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="giac")
Output:
-(16*x^2*log(5)*log(3)/(x^3 - 5*x^2 + 12*log(3)) - x^2 + x)/log(5)
Time = 0.12 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {x^2}{\ln \left (5\right )}-\frac {x}{\ln \left (5\right )}-\frac {16\,x^2\,\ln \left (3\right )}{x^3-5\,x^2+12\,\ln \left (3\right )} \] Input:
int((log(3)^2*(288*x - 144) - log(5)*(384*x*log(3)^2 - 16*x^4*log(3)) + lo g(3)*(120*x^2 - 264*x^3 + 48*x^4) - 25*x^4 + 60*x^5 - 21*x^6 + 2*x^7)/(log (5)*(144*log(3)^2 - log(3)*(120*x^2 - 24*x^3) + 25*x^4 - 10*x^5 + x^6)),x)
Output:
x^2/log(5) - x/log(5) - (16*x^2*log(3))/(12*log(3) - 5*x^2 + x^3)
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.89 \[ \int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{\left (25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)\right ) \log (5)} \, dx=\frac {-192 \,\mathrm {log}\left (5\right ) \mathrm {log}\left (3\right )^{2}-16 \,\mathrm {log}\left (5\right ) \mathrm {log}\left (3\right ) x^{3}+144 \mathrm {log}\left (3\right )^{2}+12 \,\mathrm {log}\left (3\right ) x^{3}-60 \,\mathrm {log}\left (3\right ) x +5 x^{5}-30 x^{4}+25 x^{3}}{5 \,\mathrm {log}\left (5\right ) \left (12 \,\mathrm {log}\left (3\right )+x^{3}-5 x^{2}\right )} \] Input:
int(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48*x^4-2 64*x^3+120*x^2)*log(3)+2*x^7-21*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24*x^3-1 20*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x)
Output:
( - 192*log(5)*log(3)**2 - 16*log(5)*log(3)*x**3 + 144*log(3)**2 + 12*log( 3)*x**3 - 60*log(3)*x + 5*x**5 - 30*x**4 + 25*x**3)/(5*log(5)*(12*log(3) + x**3 - 5*x**2))