Integrand size = 156, antiderivative size = 24 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\left (-5+\frac {5 x^2}{\log \left (2+\frac {x}{15+\frac {1}{x}+x}\right )}\right )^2 \] Output:
(5*x^2/ln(2+x/(15+x+1/x))-5)^2
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(24)=48\).
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=50 \left (\frac {x^4}{2 \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}-\frac {x^2}{\log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}\right ) \] Input:
Integrate[(-100*x^5 - 750*x^6 + (300*x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300*x^7)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)] + (-200*x - 6000*x^2 - 45500*x^3 - 7500*x^4 - 300*x^5)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)] ^2)/((2 + 60*x + 455*x^2 + 75*x^3 + 3*x^4)*Log[(2 + 30*x + 3*x^2)/(1 + 15* x + x^2)]^3),x]
Output:
50*(x^4/(2*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2) - x^2/Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)])
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.
| \(\displaystyle \int \frac {-750 x^6-100 x^5+\left (-300 x^5-7500 x^4-45500 x^3-6000 x^2-200 x\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )+\left (300 x^7+7500 x^6+45500 x^5+6750 x^4+300 x^3\right ) \log \left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}{\left (3 x^4+75 x^3+455 x^2+60 x+2\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {(15 x+224) \left (-750 x^6-100 x^5+\left (-300 x^5-7500 x^4-45500 x^3-6000 x^2-200 x\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )+\left (300 x^7+7500 x^6+45500 x^5+6750 x^4+300 x^3\right ) \log \left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )\right )}{\left (x^2+15 x+1\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}-\frac {3 (15 x+149) \left (-750 x^6-100 x^5+\left (-300 x^5-7500 x^4-45500 x^3-6000 x^2-200 x\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )+\left (300 x^7+7500 x^6+45500 x^5+6750 x^4+300 x^3\right ) \log \left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )\right )}{\left (3 x^2+30 x+2\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -117500 \int \frac {1}{\log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+98000 \sqrt {\frac {3}{73}} \int \frac {1}{\left (-6 x+2 \sqrt {219}-30\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-\frac {333000 \int \frac {1}{\left (-2 x+\sqrt {221}-15\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx}{\sqrt {221}}+\frac {18650}{3} \int \frac {x}{\log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-250 \int \frac {x^2}{\log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+\frac {2486350}{221} \left (221-15 \sqrt {221}\right ) \int \frac {1}{\left (2 x-\sqrt {221}+15\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+\frac {2486350}{221} \left (221+15 \sqrt {221}\right ) \int \frac {1}{\left (2 x+\sqrt {221}+15\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-\frac {333000 \int \frac {1}{\left (2 x+\sqrt {221}+15\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx}{\sqrt {221}}-\frac {4380400}{219} \left (73-5 \sqrt {219}\right ) \int \frac {1}{\left (6 x-2 \sqrt {219}+30\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-\frac {4380400}{219} \left (73+5 \sqrt {219}\right ) \int \frac {1}{\left (6 x+2 \sqrt {219}+30\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+98000 \sqrt {\frac {3}{73}} \int \frac {1}{\left (6 x+2 \sqrt {219}+30\right ) \log ^3\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+250 \int \frac {1}{\log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-1000 \sqrt {\frac {3}{73}} \int \frac {1}{\left (-6 x+2 \sqrt {219}-30\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+\frac {1500 \int \frac {1}{\left (-2 x+\sqrt {221}-15\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx}{\sqrt {221}}+100 \int \frac {x^3}{\log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-\frac {11150}{221} \left (221-15 \sqrt {221}\right ) \int \frac {1}{\left (2 x-\sqrt {221}+15\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-\frac {11150}{221} \left (221+15 \sqrt {221}\right ) \int \frac {1}{\left (2 x+\sqrt {221}+15\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+\frac {1500 \int \frac {1}{\left (2 x+\sqrt {221}+15\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx}{\sqrt {221}}+\frac {14800}{73} \left (73-5 \sqrt {219}\right ) \int \frac {1}{\left (6 x-2 \sqrt {219}+30\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx+\frac {14800}{73} \left (73+5 \sqrt {219}\right ) \int \frac {1}{\left (6 x+2 \sqrt {219}+30\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-1000 \sqrt {\frac {3}{73}} \int \frac {1}{\left (6 x+2 \sqrt {219}+30\right ) \log ^2\left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx-100 \int \frac {x}{\log \left (\frac {3 x^2+30 x+2}{x^2+15 x+1}\right )}dx\) |
Input:
Int[(-100*x^5 - 750*x^6 + (300*x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300 *x^7)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)] + (-200*x - 6000*x^2 - 4550 0*x^3 - 7500*x^4 - 300*x^5)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^2)]^2)/(( 2 + 60*x + 455*x^2 + 75*x^3 + 3*x^4)*Log[(2 + 30*x + 3*x^2)/(1 + 15*x + x^ 2)]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).
Time = 2.87 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42
| method | result | size |
| risch | \(\frac {25 \left (x^{2}-2 \ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )\right ) x^{2}}{\ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )^{2}}\) | \(58\) |
| parallelrisch | \(-\frac {-225 x^{4}+450 x^{2} \ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )}{9 \ln \left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )^{2}}\) | \(60\) |
Input:
int(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*ln((3*x^2+30*x+2)/(x^2+1 5*x+1))^2+(300*x^7+7500*x^6+45500*x^5+6750*x^4+300*x^3)*ln((3*x^2+30*x+2)/ (x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/ln((3*x^2+30* x+2)/(x^2+15*x+1))^3,x,method=_RETURNVERBOSE)
Output:
25*(x^2-2*ln((3*x^2+30*x+2)/(x^2+15*x+1)))*x^2/ln((3*x^2+30*x+2)/(x^2+15*x +1))^2
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )\right )}}{\log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )^{2}} \] Input:
integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2) /(x^2+15*x+1))^2+(300*x^7+7500*x^6+45500*x^5+6750*x^4+300*x^3)*log((3*x^2+ 30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log(( 3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="fricas")
Output:
25*(x^4 - 2*x^2*log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1)))/log((3*x^2 + 30* x + 2)/(x^2 + 15*x + 1))^2
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {25 x^{4} - 50 x^{2} \log {\left (\frac {3 x^{2} + 30 x + 2}{x^{2} + 15 x + 1} \right )}}{\log {\left (\frac {3 x^{2} + 30 x + 2}{x^{2} + 15 x + 1} \right )}^{2}} \] Input:
integrate(((-300*x**5-7500*x**4-45500*x**3-6000*x**2-200*x)*ln((3*x**2+30* x+2)/(x**2+15*x+1))**2+(300*x**7+7500*x**6+45500*x**5+6750*x**4+300*x**3)* ln((3*x**2+30*x+2)/(x**2+15*x+1))-750*x**6-100*x**5)/(3*x**4+75*x**3+455*x **2+60*x+2)/ln((3*x**2+30*x+2)/(x**2+15*x+1))**3,x)
Output:
(25*x**4 - 50*x**2*log((3*x**2 + 30*x + 2)/(x**2 + 15*x + 1)))/log((3*x**2 + 30*x + 2)/(x**2 + 15*x + 1))**2
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.54 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (3 \, x^{2} + 30 \, x + 2\right ) + 2 \, x^{2} \log \left (x^{2} + 15 \, x + 1\right )\right )}}{\log \left (3 \, x^{2} + 30 \, x + 2\right )^{2} - 2 \, \log \left (3 \, x^{2} + 30 \, x + 2\right ) \log \left (x^{2} + 15 \, x + 1\right ) + \log \left (x^{2} + 15 \, x + 1\right )^{2}} \] Input:
integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2) /(x^2+15*x+1))^2+(300*x^7+7500*x^6+45500*x^5+6750*x^4+300*x^3)*log((3*x^2+ 30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log(( 3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="maxima")
Output:
25*(x^4 - 2*x^2*log(3*x^2 + 30*x + 2) + 2*x^2*log(x^2 + 15*x + 1))/(log(3* x^2 + 30*x + 2)^2 - 2*log(3*x^2 + 30*x + 2)*log(x^2 + 15*x + 1) + log(x^2 + 15*x + 1)^2)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.43 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {25 \, {\left (x^{4} - 2 \, x^{2} \log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )\right )}}{\log \left (\frac {3 \, x^{2} + 30 \, x + 2}{x^{2} + 15 \, x + 1}\right )^{2}} \] Input:
integrate(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2) /(x^2+15*x+1))^2+(300*x^7+7500*x^6+45500*x^5+6750*x^4+300*x^3)*log((3*x^2+ 30*x+2)/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log(( 3*x^2+30*x+2)/(x^2+15*x+1))^3,x, algorithm="giac")
Output:
25*(x^4 - 2*x^2*log((3*x^2 + 30*x + 2)/(x^2 + 15*x + 1)))/log((3*x^2 + 30* x + 2)/(x^2 + 15*x + 1))^2
Time = 4.95 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {2531672218\,\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}{50625}+\frac {25\,x^4-50\,x^2\,\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}{{\ln \left (\frac {3\,x^2+30\,x+2}{x^2+15\,x+1}\right )}^2}+\frac {\mathrm {atan}\left (\frac {x^2\,371{}\mathrm {i}+x\,2250{}\mathrm {i}+150{}\mathrm {i}}{955\,x^2+11010\,x+734}\right )\,5063344436{}\mathrm {i}}{50625} \] Input:
int(-(log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))^2*(200*x + 6000*x^2 + 45500 *x^3 + 7500*x^4 + 300*x^5) - log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))*(300 *x^3 + 6750*x^4 + 45500*x^5 + 7500*x^6 + 300*x^7) + 100*x^5 + 750*x^6)/(lo g((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))^3*(60*x + 455*x^2 + 75*x^3 + 3*x^4 + 2)),x)
Output:
(2531672218*log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1)))/50625 + (atan((x*225 0i + x^2*371i + 150i)/(11010*x + 955*x^2 + 734))*5063344436i)/50625 + (25* x^4 - 50*x^2*log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1)))/log((30*x + 3*x^2 + 2)/(15*x + x^2 + 1))^2
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {-100 x^5-750 x^6+\left (300 x^3+6750 x^4+45500 x^5+7500 x^6+300 x^7\right ) \log \left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )+\left (-200 x-6000 x^2-45500 x^3-7500 x^4-300 x^5\right ) \log ^2\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )}{\left (2+60 x+455 x^2+75 x^3+3 x^4\right ) \log ^3\left (\frac {2+30 x+3 x^2}{1+15 x+x^2}\right )} \, dx=\frac {25 x^{2} \left (-2 \,\mathrm {log}\left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )+x^{2}\right )}{\mathrm {log}\left (\frac {3 x^{2}+30 x +2}{x^{2}+15 x +1}\right )^{2}} \] Input:
int(((-300*x^5-7500*x^4-45500*x^3-6000*x^2-200*x)*log((3*x^2+30*x+2)/(x^2+ 15*x+1))^2+(300*x^7+7500*x^6+45500*x^5+6750*x^4+300*x^3)*log((3*x^2+30*x+2 )/(x^2+15*x+1))-750*x^6-100*x^5)/(3*x^4+75*x^3+455*x^2+60*x+2)/log((3*x^2+ 30*x+2)/(x^2+15*x+1))^3,x)
Output:
(25*x**2*( - 2*log((3*x**2 + 30*x + 2)/(x**2 + 15*x + 1)) + x**2))/log((3* x**2 + 30*x + 2)/(x**2 + 15*x + 1))**2